popup math book (from the past)

[kirby urner]

My friend Glenn rescued this beautiful popup Math Kit book:

https://flic.kr/p/GvUKnY
https://flic.kr/p/GvULNd
https://flic.kr/p/HknMvr

You get an Egyptian Pyramid right when you open it:

https://flic.kr/p/GvUNC5

Amazing.

I wonder what a future Math Kit might be like. 

Thinking of my "three chords" (a pun of sorts given

chords have geometric meaning, not just musical

meaning):

(i)  e ** (1j * τ) = 1 or e ** (1j ** π) = -1

(ii) data structures (such as polytopes / graphs / tables )

(iii) time dimension (e.g. rotations per time interval)

Parse trees are graphs too.

http://bit.ly/1XVMcoI  (interesting website, many exhibits!)

Kirby

[kirby urner]

Thinking of my "three chords" (a pun of sorts given

chords have geometric meaning, not just musical

meaning):

(i)  e ** (1j * τ) = 1 or e ** (1j ** π) = -1

(ii) data structures (such as polytopes / graphs / tables )

(iii) time dimension (e.g. rotations per time interval)

Parse trees are graphs too.

As we get into parse trees for mathematical expressions, we'll

want to get a good handle on the "tree" structure itself, as relayed

here:

https://youtu.be/qH6yxkw0u78  (builds vocabulary around "tree" data structure)

(I enjoy has accent, would assign this for homework, something

to get through before tackling CAS).

A "tree" is a subtype of "graph" which we'll also want to get to,

and relate to polytopes of various dimension.

Kirby

[Bradford Hansen-Smith]

Kirby asked; "I wonder what a future Math Kit might be like."  My experience tells me the ultimate future Math Kit is a stack of paper circles. This  makes no sense to any one out there because you have not been educated to think there is any value in folding circles. The experiential and comprehensive nature of the circle has yet to be explored given we have all been sold on the idea of the circle being a drawing, the concept itself plus the definition of a circle suggest proof that it is. Some time in the future we will find the need to fold circles as well as draw pictures of them if we are to move beyond the construction of units-to-unity and consider unity-to-units as part of the equation. Simply put, endless parts do not make a Whole, the Whole generates endless parts. One without the other is incomplete.

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[Joseph Austin]

Bradford,
I got to this discussion late.
Where can I find out more about "circle folding"?
Joe Austin

[kirby urner]

He's the top hit on Google if you input "folding circles":  http://wholemovement.com/

If wanting to start out with spheres before compressing them:   http://www.dividedspheres.com/

Kirby

[Joseph Austin]

Kriby,

I've always thought Euler's equation was a good summary of math:

If you can explain e, i, and π, and the operations of -, * and ^,

and what the i*π power of e means, you surely know a lot of math.

I'd like to explore the relationship between data structures and groups,

and I have in my reading stack an extensive post of yours on that subject.

I'm not as clear on your issue with the "time" dimension.

I believe "time" in physics relates to "order" in mathematics.

Temporal "before" is analogous to logically "prior" or "antecedent".

A differential equation in physics is analogous to a recurrence relation in mathematics.  By itself, neither defines a "state" until you specify a "boundary condition".  And in most physical equations I'm aware of, time is reversible,

so the distinction between "time" and "space" is merely one of interpretation,

not one of mathematics.

If there is a distinction between "delta" math and "lambda" math,

I'd say it's the distinction is between continuous vs. discrete.

But there are parallels as well, e.g. differential equations vs. difference equations; Lie groups vs. permutation groups.

Joe Austin

[Joseph Austin]

On May 27, 2016, at 1:01 AM, kirby urner <kirby...@gmail.com> wrote:

My friend Glenn rescued this beautiful popup Math Kit book:

https://flic.kr/p/GvUKnY
https://flic.kr/p/GvULNd
https://flic.kr/p/HknMvr

You get an Egyptian Pyramid right when you open it:

https://flic.kr/p/GvUNC5

Amazing.

I wonder what a future Math Kit might be like.  

Kriby,

There's an article in the latest Make Magazine re: pop-ups

http://makezine.com/2016/05/04/lovepop-reinvigorates-the-humble-greeting-card-with-engineered-pop-up-scenes/

[Bradford Hansen-Smith]

"If wanting to start out with spheres before compressing them:   http://www.dividedspheres.com/

Not a bad place to start a lot of good information. It falls short in that it attends mostly to the geodesic aspect or outside of the sphere and when it goes inside it takes the traditional approach of dissecting or cutting it apart into pieces thereby destroying unity. When reassembling the pieces it remains a collections of spherical pieces having lost unity. Think Humpty Dumpty.

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[kirby urner]

On Fri, May 27, 2016 at 8:31 AM, Joseph Austin <drtec...@gmail.com> wrote:

Kriby,

I've always thought Euler's equation was a good summary of math:

If you can explain e, i, and π, and the operations of -, * and ^,

and what the i*π power of e means, you surely know a lot of math.

Yeah, his V + F == E + 2 is core as well, although there's some evidence

Descartes also discovered this formula, but encrypted it, because having

seen what they did to Galileo, Mercator even, no intellectual felt comfortable

calling too much attention to himself (if one were a her, even more so keep
it quiet). 

Yes, I'm talking about the Inquisition which put a real damper on anyone's
inquisitiveness except theirs.  Glad those days are behind us.

http://www.amazon.com/Descartess-Secret-Notebook-Mathematics-Understand/dp/0767920341

 

I'd like to explore the relationship between data structures and groups,

and I have in my reading stack an extensive post of yours on that subject.

I think of a data structure as somewhat passive, like a chalk board, a holding

pattern, in contrast to what we call "operations" which in lambda calc may be

considered always unary i.e. of the form lambda(x).  Through currying, we can

always get there (a single input, a single output, i.e. result of evaluation).

Group Theory seeks to not include a unary operator but in talking about inverse

elements so much, the negative sign tends to creep in, calling attention to the

question of whether it's an operator or not.  To this day we can look at -3 in two

ways:  as 3 with an operator in front, or as the name of an element on the so-called

number line.

I do like that Group Theory introduces subtraction and division as syntactic sugar,

not as "additional operations", as A-B just means A+(-B) and A/B just means

A*(B^-1) where ^ sneaks in as another operator.

The point being: where we're overloading 'add' and 'multiply' operators in a
math language L (Mx and Hx i.e. both machine executable and human-readable),
we already have a design pattern handy for how to implement subtraction and

division, so long as we have what's called "taking the inverse" for each element,

with respect to add and multiply.  I've used this a lot in my code over the years,

in doing Vector classes or whatever.

Yay Abstract Algebra for steering us forward!

 

I'm not as clear on your issue with the "time" dimension.

I believe "time" in physics relates to "order" in mathematics.

Temporal "before" is analogous to logically "prior" or "antecedent".

It's not just my hangup. 

Did you see where I'm quoting Coxeter, page 119 of Regular Polytopes,
Dover Edition?  Euclidean Geometry is simply not about time or motion,
velocity or acceleration.  Newton added all that stuff.  The Euclidean stuff
is all still life photography, no movies. 

OK, constructions (so-called "proofs") are *like* movies, as we go step by
step, applying the axioms, getting the theorems by pure reason.  I admit
we develop our theorems through time.  But the theorems themselves will
not be about time.  They're "always true" (timelessly true).  There's no
entropy.  No energy.  Euclidean geometry is not physics.  No clocks are

winding down irreversibly or anything like that.

N-D Euclidean geometry is simply not about time. 

The stuff it proves is true "in Eternity" i.e. the Platonic Realm.

Don't blame me for this way of thinking, it was already entrenched

when I got here in 1958.  I'm just putting together the puzzle, as

constructivists tell me I must (we "construct our own reality" right?).

That kind of snobbishness, of mathematicians towards physicists and

engineers, has mostly gone away with the collapse of the trivium-quadrivium,

praise Allah.  Mostly it's just lingering reflex-conditioning with people

sort of forgetting where it came from.

We used to have theologians at the top of the totem pole, and anything

smelling of time was too vested in mortality (tempo-reality) to really matter

to "pure math" (cleanliness is next to Godliness and time was unclean).

Still, it's important to point out that in the 4D (four dimensions) of

Coxeter, none are time-like.  This is *not* Einstein's relativity theory
(which also uses 4D) in some other guise.  Not realizing that, one becomes
hopelessly confused, as are many Youtubers (yes, people are wrong on
the Internet :-D).  We need to teach about namespaces!

https://xkcd.com/386/

In the humanities a namespace is a context, something shared, like

what a "tribe" has in anthropology.  When in the tribe of the 4D Euclideans,

don't make the mistake of thinking you're in a room full of physicists

necessarily.  Different 4D.

 

A differential equation in physics is analogous to a recurrence relation in mathematics.  By itself, neither defines a "state" until you specify a "boundary condition".  And in most physical equations I'm aware of, time is reversible,

so the distinction between "time" and "space" is merely one of interpretation,

not one of mathematics.

If there is a distinction between "delta" math and "lambda" math,

I'd say it's the distinction is between continuous vs. discrete.

But there are parallels as well, e.g. differential equations vs. difference equations; Lie groups vs. permutation groups.

Joe Austin

You're lucky you never had a case of time-phobia.  Some people run

towards math precisely to get away from physics and all its mundane

constraints (so-called laws, never applicable in cartoons it seems).

Kirby

[kirby urner]

On Fri, May 27, 2016 at 9:08 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

"If wanting to start out with spheres before compressing them:   http://www.dividedspheres.com/

Not a bad place to start a lot of good information. It falls short in that it attends mostly to the geodesic aspect or outside of the sphere and when it goes inside it takes the traditional approach of dissecting or cutting it apart into pieces thereby destroying unity. When reassembling the pieces it remains a collections of spherical pieces having lost unity. Think Humpty Dumpty.

Yeah good point Bradford, I hadn't really considered the planet interior, one might say. 

Popko's primer is very much surface-oriented, not mapping sphere internals as much, I share your impression.  He was one of those working on the radomes after all.  The book goes into more of the technical details.  Cite my earlier stories about Coxeter and how he came to learn of Fuller through the geodesic sphere patents the Pentagon was honoring.

When it comes to modular dissections, I'm sure you're familiar with the synergetics approach, which I think is about what anyone would do asked to subdivide a regular tetrahedron: 

(A) take those four quadrants those quadrays give us, and
(B) divide those each into tripods by dropping a perpendicular to each face center;
(C) then cut each in half for six modules per quadrant, three left and three right handed, for a total of 24.

http://www.rwgrayprojects.com/synergetics/s09/figs/f1301.html

Given the canonical octahedron has the same equi-triangular face as the tetrahedron, same edge-dimensions, it makes sense to brick those in with these same left and right handed "A modules" so far as possible. 

A gap remains, filled by another more slanty tetrahedron paired with each A: the so-called B-modules of equal volume. 

So (6 A modules + 6 B modules) x 8 faces = 12 x 8 =   96. 

http://www.rwgrayprojects.com/synergetics/s09/figs/f1601.html

Volume of each module: 1/24. 

Volume of Tetrahedron: 1 (24 A modules). 

Volume of Octahedron: 4 (48A + 48B modules). 

What any Martian schoolkid knows, well before puberty.

Now wouldn't it be great for Earthlings if their beloved Cube could dissect into the same As and Bs.  Well it can.  The so-called Minimum Tetrahedron, the so-called Mite, one of Sommerville's space-fillers [1], is comprise of a left and right A, plus a left or right B, for three modules total = 1/8. 

Exactly 24 of these make a Cube in a very logical way, giving the expected canonical volume of 3 in our Art School sculpture (the basis of IVM CAD).  What every art student should know.

So that leaves two Platonics to go and the As + Bs will not work. 

The solution Fuller adopts is to merge the Pentagonal Dodecahedron with its dual, the Icosahedron, also in the Platonic set, for the Rhombic Triacontahedron of 30 faces, diamonds with phi:1 long:short diagonals. 

Setting this shape to exactly wrap a unit radius sphere is one option, then exploding it apart into 120 so-called E modules.

The rhombic triacontahedron also nests with the rhombic dodecahedron (volume 6) in a pleasing way (as volume 7.5) and scaling that by a simple ratio shrinks it to volume 5 precisely, and the so-called T-modules. 5/120 = 1/24.

So voila, we've got the same volume as A and B, but with five-fold symmetry going.  What's more, the T is close to the E in volume by an interestingly tiny amount.  Mind the gap!

As it turns out, given the self-similarity of the phi-scaled E-module, shaving a rhombic triacontahedron to express Platonic volumes in terms of sums of phi-scaled Es is very doable. 

The S modules come into play here too, with S:E volume == to Cubocta:Icosa volume. 

David Koski is a lead explorer into this territory.  He and I both got Synergetics Explorer awards from BFI.  I don't know if anyone else has ever gotten one.  They changed how they do awards, calling it the BFI Challenge (an annual contest).

So that's a quick review of how we do it in Synergetics.  See the chart in Wikipedia for a summary, any time.

The sphere then comes about more from spinning.  The canonical shapes of the 4D IVM CAD sculpture are more sharp-featured than curved.  It's through spinning around axes that we get these blurred and buzzing balls.  We may then perform surface pattern analyses more like Popko's.

Kirby

[1] the 1/4 Rite (Fuller's terminology) is also a space-filler.
http://demonstrations.wolfram.com/SpaceFillingTetrahedra/

[kirby urner]

On Fri, May 27, 2016 at 9:21 AM, kirby urner <kirby...@gmail.com> wrote:

 

Group Theory seeks to not include a unary operator but in talking about inverse

elements so much, the negative sign tends to creep in, calling attention to the

question of whether it's an operator or not.  To this day we can look at -3 in two

ways:  as 3 with an operator in front, or as the name of an element on the so-called

number line.

The way Group Theory gets around having only a binary operator such that

(op A B) where A, B are in the group, is to have the group elements be

themselves unary operators on something. 

Then the binary op becomes "composition" or even "then" e.g. (op A B)
means (A then B) or (then A B) or perhaps B(A(x)) where x is some object
on which A acts, then B on the result.

This application of group elements is called "group action" and is typically
applied as rotations about an axis e.g. twisting a cube at 90 degrees around
some face axis is a "group element". 

I've also given the example of permutations or scrambles, where the element
is some switching around of letters per a substitution plan. 

Perm(plaintext) -> scrambled text. 

Such "perms" may be composed.  I have a Python type defined up for this,
a great entry point to group theory. [1]

What's important to have group-hood is that to every "do" (unary op) there

corresponds an "undo" (inverse op) such that (undo (do(x))) == x i.e. no change.

"No change" is like a "no op" such that no-op(x) -> x.  (then do undo)(x) -> x,

where "then" is the binary operator and do, undo are unary group actions.

What might be confusing to students though, when "then" is considered the

binary op, is how to talk about "associativity".  How does one distinguish

((A then B) then C) from (A then (B then C)).  Our concept of time makes
these indistinguishable since we have only the one chronology. 

If we do (B then C) first, we're testing ((B then C) then A) which is not what
we want to test.

If "to the left of" means "prior in time" then we'll need a way to test associativity

that's able to distinguish ((A then B) then C) from ((B then C) then A).  Good luck

with that.

Typically time is not involved and we simply use composition of functions i.e.

f(g(h(x))) may be written (f * g * h)(x) which may then be tested for associativity

such that (f * (g * h))(x) == ((f * g) * h)(x) where x is some object of group action.

Commutativity is not necessary for group-hood, and indeed spatial rotation ops

are famously not commutative (order matters, i.e. A then B is not the same as

B then A).

Kirby

[kirby urner]

On Sat, May 28, 2016 at 7:50 AM, kirby urner <kirby...@gmail.com> wrote:

Such "perms" may be composed.  I have a Python type defined up for this,
a great entry point to group theory. [1]

[1]  https://github.com/4dsolutions/Python5/blob/master/px_class.py

[Joseph Austin]

On May 27, 2016, at 12:21 PM, kirby urner <kirby...@gmail.com> wrote:

On Fri, May 27, 2016 at 8:31 AM, Joseph Austin <drtec...@gmail.com> wrote:

Kriby,

I've always thought Euler's equation was a good summary of math:

If you can explain e, i, and π, and the operations of -, * and ^,

and what the i*π power of e means, you surely know a lot of math.

Yeah, his V + F == E + 2 is core as well, although there's some evidence

Descartes also discovered this formula, but encrypted it, because having

seen what they did to Galileo, Mercator even, no intellectual felt comfortable

calling too much attention to himself (if one were a her, even more so keep
it quiet). 

I hadn't heard about "him" being "her".

Although I'm suspicious that some of Einstein was really his wife.

Yes, I'm talking about the Inquisition which put a real damper on anyone's
inquisitiveness except theirs.  Glad those days are behind us.

http://www.amazon.com/Descartess-Secret-Notebook-Mathematics-Understand/dp/0767920341

 

I'd like to explore the relationship between data structures and groups,

and I have in my reading stack an extensive post of yours on that subject.

I think of a data structure as somewhat passive, like a chalk board, a holding

pattern, in contrast to what we call "operations" which in lambda calc may be

considered always unary i.e. of the form lambda(x).  Through currying, we can

always get there (a single input, a single output, i.e. result of evaluation).

Whatever you call it, aren't there are two items of information used to produce a third? 

call it state.operator -> state"

or call in state x input -> state'

or call it f(x) -> y [where "f" conveys information in that is isn't "g"]

Group Theory seeks to not include a unary operator but in talking about inverse

elements so much, the negative sign tends to creep in, calling attention to the

question of whether it's an operator or not.  To this day we can look at -3 in two

ways:  as 3 with an operator in front, or as the name of an element on the so-called

number line.

Doesn't it become a question of chicken and egg?

I do like that Group Theory introduces subtraction and division as syntactic sugar,

not as "additional operations", as A-B just means A+(-B) and A/B just means

A*(B^-1) where ^ sneaks in as another operator.

The point being: where we're overloading 'add' and 'multiply' operators in a
math language L (Mx and Hx i.e. both machine executable and human-readable),
we already have a design pattern handy for how to implement subtraction and

division, so long as we have what's called "taking the inverse" for each element,

with respect to add and multiply.  I've used this a lot in my code over the years,

in doing Vector classes or whatever.

Yay Abstract Algebra for steering us forward!

By one line of thinking, negative numbers arise from the closure of "subtract," or additive inverse, on the naturals:  

What is the solution to  3 + x = 0?

Or, what is 0 minus 3?

I say "negative three" is the number defined by "zero minus three", 

the result of the unevaluated inverse operation: -3 === "0-3".

Similarly, 1/3 is the result of "one divided by three."

So we suppress the "0" in "0-3" and write "-3" for the additive inverse of 3.

So let's suppress the 1 in "1/3" and write "/3" for the multiplicative inverse of 3.

The advantage of this line of reasoning is that all non-naturals suddenly become pairs of naturals.

And "integers" have the general form (n-m) where n and m are any natural.

In pair form, the integers are thus equivalence classes of differences,

with the normal form having either n or m = 0.

If this seems unusual, consider that the rationals are defined as equivalences classes of ratios n/m.

So suppose we make "difference" and "ratio" the primary operation and "sum" and "product" are their degenerate inverses?

What kind of "algebra" do we have then?

And suppose we extend the approach to powers, roots, and logs. &ct.

The distinction between differences and sums is that differences have a direction!  They are nascent vectors!

I started down this track with the Ackermann (i.e. the "arithmetic" function).

http://drtechdaddy.com/2013/06/07/replicount-a-recursive-function-for-natural-number-arithmetic/

Of course, we would lose commutativity, but there are a lot in interesting non-commutative or anti-commutative systems.

What we might end up with, though, is a "dual" system,

such as points and lines (in space) or instants and durations (in time).

Let me wax philosophical for a moment.

Suppose that "difference" and "ratio" really are more "naturally" fundamental than "sum" and "product".

Does that mean we are short-changing our students by starting them on add and multiply and then confusing them with signed numbers and fractions?

Ask any four year old:

If your brother had 4 candy bars and you have 2...?

He's not thinking: "that's 6 altogether."

He's thinking "he has more than I do."

Obviously, I've hit a hot button. But I still don't get what you're railing against.

Are you saying Quadrays is just a namespace, don't try attaching a metric to it?  Don't understand "dimension" or "volume" in the Euclidean sense?

OK, so how many "dimensions" does a tree have?

Just two--width and height? Or do we get a new "dimension" at every fork?

Maybe there's no "time", but not all math is reversible.

2+3 =5, but so is 1+4 and 0+5 and -2+7.

Once you have the 5, you can't know whether it was 2+3  or one of the other infinite possibilities.

So what if I label one axis "time" and another "distance"?

Or if I add two more and call one "momentum" and the other "energy"?

The traces of my bollard balls in the coordinate space still show conservation of mass and energy and momentum, and if you tell we where they are and which way they are going, I can tell you where they've been. 

And if you include the pool cue in the system, I can even tell you when was the break.  Or if not, we will get a very interesting "glimpse into the beginning of the universe" , the supposed "chaos" before the balls were even racked up,

all hypothetical without any supposition of a "god" with a rack and cue stick.

A differential equation in physics is analogous to a recurrence relation in mathematics.  By itself, neither defines a "state" until you specify a "boundary condition".  And in most physical equations I'm aware of, time is reversible,

so the distinction between "time" and "space" is merely one of interpretation,

not one of mathematics.

If there is a distinction between "delta" math and "lambda" math,

I'd say it's the distinction is between continuous vs. discrete.

But there are parallels as well, e.g. differential equations vs. difference equations; Lie groups vs. permutation groups.

Joe Austin

You're lucky you never had a case of time-phobia.  Some people run

towards math precisely to get away from physics and all its mundane

constraints (so-called laws, never applicable in cartoons it seems).

Kirby

I abandoned physics because I was seduced by computers.

I never considered myself a "mathematician", though in the course of my other pursuits I suspect I was involved with more math than most math majors.

I had BOTH delta and lambda.

Joe

[kirby urner]

On Sat, May 28, 2016 at 10:43 AM, Joseph Austin <drtec...@gmail.com> wrote:

On May 27, 2016, at 12:21 PM, kirby urner <kirby...@gmail.com> wrote:

On Fri, May 27, 2016 at 8:31 AM, Joseph Austin <drtec...@gmail.com> wrote:

Kriby,

I've always thought Euler's equation was a good summary of math:

If you can explain e, i, and π, and the operations of -, * and ^,

and what the i*π power of e means, you surely know a lot of math.

Yeah, his V + F == E + 2 is core as well, although there's some evidence

Descartes also discovered this formula, but encrypted it, because having

seen what they did to Galileo, Mercator even, no intellectual felt comfortable

calling too much attention to himself (if one were a her, even more so keep
it quiet). 

I hadn't heard about "him" being "her".

Although I'm suspicious that some of Einstein was really his wife.

That's not what I meant.  I meant if you were an intellectual AND a woman,

watch out!  The Inquisition was very interested in uppity women.  You might

have heard.

They threw Mercator in jail because his maps were better than anything

coming out of the Vatican.  Anything eroding the supreme authority of Rome

was an inconvenient truth at that point.

For this reason, Descartes, always hungering for less visibility, apparently

encrypting his discovery that V + F == E + 2, leaving it to Euler to cash in

on that one.
 

Yes, I'm talking about the Inquisition which put a real damper on anyone's
inquisitiveness except theirs.  Glad those days are behind us.

http://www.amazon.com/Descartess-Secret-Notebook-Mathematics-Understand/dp/0767920341

 

I'd like to explore the relationship between data structures and groups,

and I have in my reading stack an extensive post of yours on that subject.

I think of a data structure as somewhat passive, like a chalk board, a holding

pattern, in contrast to what we call "operations" which in lambda calc may be

considered always unary i.e. of the form lambda(x).  Through currying, we can

always get there (a single input, a single output, i.e. result of evaluation).

Whatever you call it, aren't there are two items of information used to produce a third? 

call it state.operator -> state"

or call in state x input -> state'

or call it f(x) -> y [where "f" conveys information in that is isn't "g"]

We can think of state as a noun (a state of affairs) and actions as verbs (they make

changes to state).

The logical question that arises when you change state, is are you "throwing away"

the previous object and carrying the changes forward in "the new object".  Except

we wouldn't normally call it a "new object" in that case, but "the same one".

In other words, one might go:

tetrahedron = Polyhedron(V=4, F=4, E=6)

tetrahedron.rotate(axis = Q0, degrees = 60)

and see that as a tetrahedron changing state.

OR

one may go:

 

tetrahedron = Polyhedron(V=4, F=4, E=6)

new_tetra = tetrahedron.rotate(axis = Q0, degrees = 60)

in which case you now have two tetrahedrons:  the one before the operation

and the one after.

Makes a difference.

Functional programmers tend to encourage the latter.  They'd rather

focus on immutability such that any changes result in something brand

new, not something old now different.  Bookkeepers tend to agree.

Group Theory seeks to not include a unary operator but in talking about inverse

elements so much, the negative sign tends to creep in, calling attention to the

question of whether it's an operator or not.  To this day we can look at -3 in two

ways:  as 3 with an operator in front, or as the name of an element on the so-called

number line.

Doesn't it become a question of chicken and egg?

The difference between binary and unary operators is important at some level.

We need to keep revisiting that difference.

If every element has an inverse element such that the binary operation with those

two results in Identity, per any group, then I don't think we can really avoid the

idea of a "unary operator" i.e. "that which takes an element to its own inverse".

Not that finding an inverse is always trivial, even if one exists.
 

The point being: where we're overloading 'add' and 'multiply' operators in a
math language L (Mx and Hx i.e. both machine executable and human-readable),
we already have a design pattern handy for how to implement subtraction and

division, so long as we have what's called "taking the inverse" for each element,

with respect to add and multiply.  I've used this a lot in my code over the years,

in doing Vector classes or whatever.

Yay Abstract Algebra for steering us forward!

By one line of thinking, negative numbers arise from the closure of "subtract," or additive inverse, on the naturals:  

What is the solution to  3 + x = 0?

Or, what is 0 minus 3?

Indeed.  In order to obtain groups and fields, we'll introduce new types of element if necessary.

The complex numbers came along to satisfy closure in various ways.

We like group symmetry don't we.
 

I say "negative three" is the number defined by "zero minus three", 

the result of the unevaluated inverse operation: -3 === "0-3".

Similarly, 1/3 is the result of "one divided by three."

Or 1/3 is 1 times the multiplicative inverse of 3.  That makes it look like an operation.

We like to mix algorithmic and naming notations.  What if we never wrote pi (the Greek letter)

but always pi( ), implying it was a process, not a noun.  Different way of thinking.

 

So we suppress the "0" in "0-3" and write "-3" for the additive inverse of 3.

So let's suppress the 1 in "1/3" and write "/3" for the multiplicative inverse of 3.

We could do that, sure.  1/3 = 1 * (/3).
 

The advantage of this line of reasoning is that all non-naturals suddenly become pairs of naturals.

And "integers" have the general form (n-m) where n and m are any natural.

In pair form, the integers are thus equivalence classes of differences,

with the normal form having either n or m = 0.

Vectors really show the value of A + B = A + (-B) as we know what -B is:  a Vector pointing the other way,

such that B + (-B) = zero-vector.
 

If this seems unusual, consider that the rationals are defined as equivalences classes of ratios n/m.

So suppose we make "difference" and "ratio" the primary operation and "sum" and "product" are their degenerate inverses?

What kind of "algebra" do we have then?

And suppose we extend the approach to powers, roots, and logs. &ct.

The distinction between differences and sums is that differences have a direction!  They are nascent vectors!

If the difference is thirty dollars, what direction is that?  I think we need to know more.
 

I started down this track with the Ackermann (i.e. the "arithmetic" function).

http://drtechdaddy.com/2013/06/07/replicount-a-recursive-function-for-natural-number-arithmetic/

Of course, we would lose commutativity, but there are a lot in interesting non-commutative or anti-commutative systems.

What we might end up with, though, is a "dual" system,

such as points and lines (in space) or instants and durations (in time).

Let me wax philosophical for a moment.

Suppose that "difference" and "ratio" really are more "naturally" fundamental than "sum" and "product".

Does that mean we are short-changing our students by starting them on add and multiply and then confusing them with signed numbers and fractions?

We do tell them that a ratio is another name for a rational number i.e. any p:q is p/q is a member of Q.

Ratios are a number type.  Differences?  Well, we have "absolute value" precisely to speak only of the "positive" distance.
 

Ask any four year old:

If your brother had 4 candy bars and you have 2...?

He's not thinking: "that's 6 altogether."

He's thinking "he has more than I do."

Depends on the story problem.  They may be about to pool them and divide them up among the whole cub scout den.
 

 

I'm not as clear on your issue with the "time" dimension.

I believe "time" in physics relates to "order" in mathematics.

Temporal "before" is analogous to logically "prior" or "antecedent".

It's not just my hangup. 

Did you see where I'm quoting Coxeter, page 119 of Regular Polytopes,
Dover Edition?  Euclidean Geometry is simply not about time or motion,
velocity or acceleration.  Newton added all that stuff.  The Euclidean stuff
is all still life photography, no movies. 

OK, constructions (so-called "proofs") are *like* movies, as we go step by
step, applying the axioms, getting the theorems by pure reason.  I admit
we develop our theorems through time.  But the theorems themselves will
not be about time.  They're "always true" (timelessly true).  There's no
entropy.  No energy.  Euclidean geometry is not physics.  No clocks are

winding down irreversibly or anything like that.

N-D Euclidean geometry is simply not about time. 

The stuff it proves is true "in Eternity" i.e. the Platonic Realm.

Don't blame me for this way of thinking, it was already entrenched

when I got here in 1958.  I'm just putting together the puzzle, as

constructivists tell me I must (we "construct our own reality" right?).

<< SNIP >>

Obviously, I've hit a hot button. But I still don't get what you're railing against.

Sigh.  It's not a hot button.  It's the history of western civilization.  I learned

all this in school and am just repeated what I think "everybody knows" but

I guess not.

N-D Euclidean geometry is not about any time dimensions.  Coxeter is like the

father of that discipline so he oughta know.

Einstein.4D != Coxeter.4D  (see page 119, Regular Polytopes, Dover edition).

 

Are you saying Quadrays is just a namespace, don't try attaching a metric to it?  Don't understand "dimension" or "volume" in the Euclidean sense?

No.

Quadrays have a metric.  It's very easy to compute the distance given

(2,1,1,0) and (3,0,1,0).  Same as in XYZ.

Quadrays address the same space XYZ does and have many of the very same properties.

What I'm talking about is the difference between "pure math" and physics.

Pure math does not involve energetic dimensions, time chief among them.  No mass either.   Not if its pure.

I'm not saying I'm a "pure math" snob, just there's this electric fence in academia left over from like since Roman times.

 

OK, so how many "dimensions" does a tree have?

Just two--width and height? Or do we get a new "dimension" at every fork?

I'd say we live irreducibly in volume.  Can you put a tree in a room?  Yes.  Can a tree contain a room?  Yes.

Therefore a tree is of the same dimension as the room that contains it and the room it contains.

How many dimensions is that?  Do we count time?

Different tribes have different answers.  The standard answer is "all rooms are 3D" (XYZ thinking) and time is an add-on.

The Fuller.4D answer is all rooms are topologically tetrahedral at minimum, because the tetrahedron is the first enclosure (a triangle is what we, the observer, might see, so the observer in adding another point, the viewpoint, creates a tetrahedron).

Fuller's way of thinking is never taught except in art schools but quadrays help us wrap our heads around what he means by 4D. 

The amputated XYZ system, minus its negative flows, is not going to "flow" in every direction. 

The four flows from (0,0,0,0) will actually take you anywhere, if added correctly.  Just extend each quadray as much as needed and add tip to tail.

XYZ will take you anywhere *provided* you're permitted to reverse a basis vector and not call that a new basis vector, as that would screw up the count (there can be only three, given the "three dee" dogma).
 

 

Maybe there's no "time", but not all math is reversible.

2+3 =5, but so is 1+4 and 0+5 and -2+7.

Once you have the 5, you can't know whether it was 2+3  or one of the other infinite possibilities.

Not "maybe" there's no time.  There isn't.  Not in N-D Euclidean geometry.

Of course in *applications* one may introduce time.

In Fuller.4D, we add time *after* we have pure volume, i.e. the tetrahedron.  4D is just another name for 3D, given we're counting how many walls the room has, not half the spokes of a "jack" shape (3 of 6, the positive arms).

 

So what if I label one axis "time" and another "distance"?

Or if I add two more and call one "momentum" and the other "energy"?

The traces of my bollard balls in the coordinate space still show conservation of mass and energy and momentum, and if you tell we where they are and which way they are going, I can tell you where they've been. 

This is physics, includes a time dimension.

Math applies
 

And if you include the pool cue in the system, I can even tell you when was the break.  Or if not, we will get a very interesting "glimpse into the beginning of the universe" , the supposed "chaos" before the balls were even racked up,

all hypothetical without any supposition of a "god" with a rack and cue stick.

A differential equation in physics is analogous to a recurrence relation in mathematics.  By itself, neither defines a "state" until you specify a "boundary condition".  And in most physical equations I'm aware of, time is reversible,

so the distinction between "time" and "space" is merely one of interpretation,

not one of mathematics.

If there is a distinction between "delta" math and "lambda" math,

I'd say it's the distinction is between continuous vs. discrete.

But there are parallels as well, e.g. differential equations vs. difference equations; Lie groups vs. permutation groups.

Joe Austin

You're lucky you never had a case of time-phobia.  Some people run

towards math precisely to get away from physics and all its mundane

constraints (so-called laws, never applicable in cartoons it seems).

Kirby

I abandoned physics because I was seduced by computers.

I never considered myself a "mathematician", though in the course of my other pursuits I suspect I was involved with more math than most math majors.

I had BOTH delta and lambda.

Joe

I'm just an anthropologist trying to clarify the meaning of 4D as used by three

distinct tribes:  Einstein.4D; Coxeter.4D; Fuller.4D.

The first two are routinely confused.  The latter is never taught.  So it's a long slog.

Kirby

[Joseph Austin]

Kirby,

Thanks for this explanation of the difference between the group "operation" and the group "action".  Even I can understand it!

As for associativity, isn't this a property of the group "operations"

rather than of the group "actions"?

Suppose we have group elements A B C, and group operation *, where

A*B = X, B*C = Y; 

Then as I understand it, associativity implies:

X*C = A*Y = A*B*C .

So isn't the "action"  f(A*B)  the action of some other element f(X), 

a single action, rather than a sequence of two, and f(B*C) likewise?

So "associativity" of the group operation implies two different sets of group actions which both end up at the same point. e.g. suppose f(A)=90º, f(B)=-180º, f(C)=+270º.  f(A*B) = -90º, f(B*C) = +90º, f((AB)C)= -90+270 = 180;  f(A(BC))= +90+90=180.

Joe

[Joseph Austin]

On May 28, 2016, at 2:47 PM, kirby urner <kirby...@gmail.com> wrote:

I'm just an anthropologist trying to clarify the meaning of 4D as used by three

distinct tribes:  Einstein.4D; Coxeter.4D; Fuller.4D.

The first two are routinely confused.  The latter is never taught.  So it's a long slog.

I"m just trying to understand what it is about "time" that makes Einstein 4D different from Coxeter.  

But my real issue is "product".

What is a "square second"?  

By my understanding, the product of two physical measurements is a measurement in a different "dimension,"  e.g. length x length = area;

mass x velocity = momentum.

But the field axioms say the product of two numbers is a number, 

not a "square" number.

So what does it mean that 1 gram x 10 cm/sec is the "same" momentum as 10 grams x 1 cm/sec, and the "momentum" can transfer from 1 object to another resulting in velocities in inverse proportion to their masses?

Or take 10cm x 1cm vs. 5cm x 2cm.  Both are the same amount of "surface"

but different amounts of "length". (22cm vs. 14cm).

I don't really think it has to do with one of the dimensions being "time",

so much as the idea that we have a conserved quantity that distributes over space time and mass.

So what math calls "factors", physics calls "projections" on a "multi-dimensional" coordinate system.

So what kind of "math" is that?  What kind of "product" is that?

Joe Austin

[kirby urner]

On Sat, May 28, 2016 at 4:59 PM, Joseph Austin <drtec...@gmail.com> wrote:

On May 28, 2016, at 2:47 PM, kirby urner <kirby...@gmail.com> wrote:

I'm just an anthropologist trying to clarify the meaning of 4D as used by three

distinct tribes:  Einstein.4D; Coxeter.4D; Fuller.4D.

The first two are routinely confused.  The latter is never taught.  So it's a long slog.

I"m just trying to understand what it is about "time" that makes Einstein 4D different from Coxeter.  

Probably I'm making it too hard by bringing in all this history.  We should just stick to units,

such as of distance, of time, of mass etc. 

I like the questions you raise regarding this units below. 

What does it mean to multiply time by time?

Such questions aside, you'll likely agree that the Pythagorean Theorem is only about spatial
quantities of lengths and areas.

We typically say "A squared plus B squared equals C squared" where A, B are the legs of

a right triangle and C is the hypotenuse.

What's not so often stated is building an equilateral triangle on A, B and C also expresses the
same identity, i.e. the shape that we use for "second powering" does not *have* to be a square.

http://www.grunch.net/synergetics/quadray/pythag2.gif

But that's beside the point.  The point is that time is not involved as a variable. 

There's no room for it to insert itself anywhere in the theorem.

That's what N-D Euclidean geometry is like.  A time variable is not part of what's proved.

So when a Euclidean introduces a fourth dimension, another after three, it's just as "space-like" as the others.  It gets no special treatment as "time".

In Relativity on the other hand, computing the interval shared across reference frames is *not* simply the distance formula bumped up by one dimension.

https://en.wikipedia.org/wiki/Euclidean_distance

Versus:

https://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals

 

But my real issue is "product".

What is a "square second"?  

By my understanding, the product of two physical measurements is a measurement in a different "dimension,"  e.g. length x length = area;

mass x velocity = momentum.

But the field axioms say the product of two numbers is a number, 

not a "square" number.

These are good questions.
 

So what does it mean that 1 gram x 10 cm/sec is the "same" momentum as 10 grams x 1 cm/sec, and the "momentum" can transfer from 1 object to another resulting in velocities in inverse proportion to their masses?

Or take 10cm x 1cm vs. 5cm x 2cm.  Both are the same amount of "surface"

but different amounts of "length". (22cm vs. 14cm).

I don't really think it has to do with one of the dimensions being "time",

If we forbid time in any form, that's a quick way to razor off physics from pure math.

Velocity is not a feature in any of Euclid's theorems.

I'm not saying there's anything wrong with having time (do we have a choice?).

I'm just saying here's a way to distinguish the different meanings of 4D, based on the tribe using it.

XYZ 3D + Time is Einstein.4D  (physics tribe)

XYZ 3D + another spatial dimension (not time) is Coxeter.4D ... continuing on to N-D (pure math tribe)

Tetrahedron as Minimum Conceptual Room is Fuller.4D, using caltrop (4 directions), instead of jack (6 directions, 3 basis).  More the art school / code school crowd?

Fuller.4D is more like Karl Mengers "geometry of lumps" in which points, lines, planes and boxes are all of the same dimensionality, just differently shaped.  Menger says that's Non-Euclidean.  I'd say he's right.

 

so much as the idea that we have a conserved quantity that distributes over space time and mass.

So what math calls "factors", physics calls "projections" on a "multi-dimensional" coordinate system.

So what kind of "math" is that?  What kind of "product" is that?

Joe Austin

A worthy investigation.

Kirby

[kirby urner]

On Sat, May 28, 2016 at 4:35 PM, Joseph Austin <drtec...@gmail.com> wrote:

 

Kirby,

Thanks for this explanation of the difference between the group "operation" and the group "action".  Even I can understand it!

As for associativity, isn't this a property of the group "operations"

rather than of the group "actions"?

Right.  When I chain permutations together I get another permutation within the group.

To "chain" is to "compose" and/or to "multiply".
 

Suppose we have group elements A B C, and group operation *, where

A*B = X, B*C = Y; 

Then as I understand it, associativity implies:

X*C = A*Y = A*B*C .

Yes.

It's just I found a Youtube awhile back where the mentor was saying we could
substitute the word "then" for composition.

B then A = X.  C then B = Y.

That confused me, as I couldn't then understanding how ((B then A) then C)

could be distinguished from (B then (A then C)).  Even thought the parens say

to do (A then C) "first", the fact that (A then C) comes after B was messing me up.

Conclusion:  I should avoid the suggestion to use "then" for composition.

 

So isn't the "action"  f(A*B)  the action of some other element f(X), 

a single action, rather than a sequence of two, and f(B*C) likewise?

Right.  As I put it, chain together all the permutations you like, when you apply

the end result to X, a string (plaintext), you apply it as a unary operator.  sigma(X)

a lot of the books say.  g ^ x -> new x.
 

So "associativity" of the group operation implies two different sets of group actions which both end up at the same point. e.g. suppose f(A)=90º, f(B)=-180º, f(C)=+270º.  f(A*B) = -90º, f(B*C) = +90º, f((AB)C)= -90+270 = 180;  f(A(BC))= +90+90=180.

Joe

Right.

I just couldn't get my mind around "then" when trying to do it in two ways

per the associative rule.  ((A then B) then C) versus (A then (B then C)).

My bad.

Kirby

[kirby urner]

On Sat, May 28, 2016 at 7:25 PM, kirby urner <kirby...@gmail.com> wrote:

 

 

So isn't the "action"  f(A*B)  the action of some other element f(X), 

a single action, rather than a sequence of two, and f(B*C) likewise?

Right.  As I put it, chain together all the permutations you like, when you apply

the end result to X, a string (plaintext), you apply it as a unary operator.  sigma(X)

a lot of the books say.  g ^ x -> new x.
 

Back to the binary versus unary operator topic, I suppose there's ambiguity

in saying g ^ x -> new x is "unary" as we could see it as (^ g x) with ^ the

operator and g, x the two inputs.  That's like saying -3 is really (- 0 3) perhaps.

But when we go f(x) i.e. apply a function to x, we don't usually think of that

as a binary operation, even though we could write something such as

(apply f x) or (do f x) where we make f "do something" with x.

Typically we say f(x) is a "function call" i.e. we a are "calling f with x".

That sounds like f is a unary operator on x, but again, every F(x) might

be written (call F x).

Below I show Group Action by members of the group G that is all

permutations of lowercase letters plus space to a rearrangement of

same.  Out[12] shows the result of calling p1 or applying p1, a

group element, to a phrase, a member of set X:  all strings of
lowercase letters and space.

Set X, the space of phrases we might want to scramble, and

the permutations such as p1 and p2, both in G, are *not* the same

set. 

Many of the texts on Group Theory are eager to show that G may
act on itself.  Fine, but lets not start there.  G and X are distinct

in this example:

In [6]: phrase = "the rain in spain stays mainly in the plain"

In [10]: p1 = P().shuffle()

In [11]: p2 = P().shuffle()

In [12]: p1(phrase)

Out[12]: 'yuilnfovlovlxwfovlxyfgxlmfovcglovlyuilwcfov'

In [13]: p2(phrase)

Out[13]: 'guyrkexirxir oexir gez rmexinzrxirguyronexi'

In [14]: p3 = p1 * p2  # permutations may be composed (use multiply symbol)

In [15]: p3(phrase)  # unary or binary operation?

Out[15]: 'zlxnipsqnsqnwhpsqnwzptwnmpsqdtnsqnzlxnhdpsq'

In [16]: p1

Out[16]: P class: (('e', 'i'), ('m', 'm'), ('t', 'y'))...

In [17]: p2

Out[17]: P class: (('e', 'y'), ('m', 'm'), ('t', 'g'))...

In [18]: p4 = ~(p1 * p2)  # get the inverse or "undo" of (p1 * p2)

In [19]: p4('zlxnipsqnsqnwhpsqnwzptwnmpsqdtnsqnzlxnhdpsq')  # decode the ciphertext

Out[19]: 'the rain in spain stays mainly in the plain'

Kirby

[Andrius Kulikauskas]

Joe, Kirby,

I'm thinking that this distinction between ratios and products comes up
as the distinction between "contravariants" and "covariants". And I'm
imagining that a (p,q) tensor tells you that p dimensions are to be
understood in terms of "division" (contravariants, as with vectors) and
q dimensions are to be understood in terms of "multiplication"
(covariants, as with covectors - reflections). I'm still trying to
figure it out. For example, if you have an answer A = 5 / 7 then on
the one hand you are dividing, and in fact, the denominator 1/7 is the
"unit", that is, the "denominated" whereas the 5 is the amount, the
"numerated". If you want the fraction to stay the same then you have to
multiply the top and bottom by the same. I mean to say that I don't
understand but I think that tensors are relevant to this question.

The link between difference/sum and ratio/product is given by the
exponential/logarithm function. In particular, the Lie group G and the
Lie algebra A are related by:

e**A = G

So this is a key equation for relating the discrete world (Lie algebra
A) and the continuous world (Lie group G). Addition/subtraction in the
discrete world is matched by multiplication/division in the continuous
world.

The equation above involves matrices. In general, there is the very
meaningful "polar decomposition" of matrices:

M = P U = P e**iH analogous to polar coordinates for a complex
number: z = R e**i t

P is a positive semi-definite Hermitian matrix, which means that all of
its eigenvalues are nonnegative real numbers, which means that the
effect of the matrix P is simply to distort the lengths of vectors in
various directions (which is analogous to R).

U is a unitary matrix, which means that it preserves lengths but may be
a rotation, for example, as given by the Hermitian matrix H, whose
eigenvalues are real.

Well, for Lie groups (continuous groups) to exist their actions (their
elements) need to have counteractions, that is, inverses. And when those
actions are described as matrices, it turns out that there can't be any
radial component P. That is, the matrix can't stretch vectors bigger or
smaller. Otherwise, apparently, the action would rip the group apart,
it would not be continuous. All that can exist is the angular
component. In other words, the volumes (bound by a set of vectors) have
to be preserved. These volumes are given by the determinant, which I
think detects what is "inside" the volume and what is "outside" of it.
The determinant has to be nonzero (so that the volume doesn't collapse,
and thus the matrix is reversible), but also it has to have absolute
value 1 (so that there is no stretching bigger or smaller). We can also
relate this to Cramer's rule for calculating the inverse of the matrix,
where the denominator is the determinant, and thus in our case there is
no denominator to speak of.

What this means is that for Lie groups there is always a "short cut" for
calculating the inverse of the action. In the case of the circle group,
for example, it means that an action doesn't have to be thought of as a
big matrix that needs to be inverted. Instead, in that case we can
think of the action as rotating by an angle, and the inverse is simply
rotating back. Thus these "short cuts" are given by the adjoint
matrix. For example, for unitary matrices the short cut for calculating
the inverses is to take the conjugate transpose.

So now I'm trying to understand what "short cuts" are allowed. That
apparently classifies the Lie groups. The way that classification is
made is instead to look at the Lie algebras. Instead of looking at
multiplication (in Lie groups) we look at addition (in Lie algebras).
The addition is I think described by crystallographic lattices, which is
where the tetrahedral vs. Euclidean geometries come up, for example.
And so it is possible to calculate the limited possibilities for the
geometry. So I will try to figure that out and report back.

A related way to understand this is to look at the "normal forms"
preserved by the Lie groups. I suppose this means that each Lie group
preserves not only the lengths (and volumes) but something more
precise. There aren't many possibiities, though.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

>
> Let me wax philosophical for a moment.
> Suppose that "difference" and "ratio" really are more "naturally"
> fundamental than "sum" and "product".
> Does that mean we are short-changing our students by starting them
> on add and multiply and then confusing them with signed numbers
> and fractions?
>

[Andrius Kulikauskas]

Joseph, Kirby,

Thank you for teasing out these issues.

Some thoughts that came up... As regards why 10 gram x 1 cm / sec is
the same as 1 gram x 10 cm / sec, the "conservation of momentum" is a
consequence of the symmetry in space, in that the outcomes don't depend
on any particular coordinate in space, as per:
https://en.wikipedia.org/wiki/Noether's_theorem
Emily Noether was an inspiring woman mathematician.

"Conservation of energy" is a consequence of the symmetry in time, in
that the outcomes don't depend on any particular time coordinates. So
the difference between space and time may perhaps be thought of as the
difference between momentum and energy.

Energy also gives an example of what "second squared" can mean. I just
wonder what energy means. The Wikipedia article on energy is mystifying:
https://en.wikipedia.org/wiki/Energy
Mathematically, kinetic energy K = 1/2 m v**2 is the integral (by
velocity) of momentum M = m v. Velocity is the expressed relationship
between space and time. The higher the velocity, the more weight is
placed on time, and the lower the velocity, the more weight is placed on
space. Kinetic energy is what it takes to go from velocity v = 0 to v =
V. Potential energy is the background energy that explains for us that
total energy is conserved.

Perhaps one difference in interpreting the fourth dimension is whether
time has a plus or a minus sign. In Minkowski space, used by Einstein,
time and space have opposite signs:
https://en.wikipedia.org/wiki/Minkowski_space
Whereas in Euclidean space/time I imagine they have the same sign. I'm
curious to learn more about the reason for that distinction.

Joseph, I like very much your emphasis on units. I found that helpful
in learning physics and as a tutor I developed some general principles
for my students:

"Every answer is an amount and a unit"
(3 is not an answer but rather 3 feet, 3 seconds, etc.)
But then (by sleight of hand) a number can become a unit: 3 millions, 3
sevenths, etc.

I'm wondering about the purpose of this breakdown. It's probably partly
to distinguish between what we attribute to our mental world (the
amounts) and to the physical world (the units). And perhaps it helps us
distinguish between answers and questions. Answers are fixed and so they
are "contravariant": if we divide up the units by 1000, and go from
kilograms to grams, then we have to multiply the amounts by 1000.
Whereas questions are not fixed and they are often phrased in terms of
(1/unit) as "per unit": How many miles per hour? And if we divide up the
hour into 60 minutes, and we get "per minute", then we have to divide up
our amount (How many) by 60. I'm just thinking out loud.

I taught that
"You combine like units", (to calculate), for example:
3 sec + 2 sec = 5 sec but 3 sec + 2 feet isn't anything (to combine)
3 million + 2 million = 5 million
3 sevenths + 2 sevenths = 5 sevenths
3 X + 2 X = 5 X but 3 X + 2 Y doesn't combine

"You list different units" (to make your answer easy to understand)
the marathon was won in: 2 hours + 12 minutes + 8 seconds

You convert different units to same units in order to combine and, in
general, to make it simpler to calculate.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

2016.05.29 02:59, Joseph Austin rašė:
>
>> On May 28, 2016, at 2:47 PM, kirby urner <kirby...@gmail.com

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[kirby urner]

On Sat, May 28, 2016 at 11:30 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

 

Perhaps one difference in interpreting the fourth dimension is whether time has a plus or a minus sign.  In Minkowski space, used by Einstein, time and space have opposite signs:
https://en.wikipedia.org/wiki/Minkowski_space
Whereas in Euclidean space/time I imagine they have the same sign.  I'm curious to learn more about the reason for that distinction.

Right, you're putting your finger on the two meanings of 4D here.

In 4-D Euclidean geometry, the hypercube or tesseract has only spatial dimensions, not time-like axis,  Just like a regular cube.  It exists in space only.  Time is not of interest.

Minkowski space-time uses a non-Euclidean geometry with a different algebra.  As you say, the sign is different.  Time is given special treatment.

There's no crime or issue in having two separate language games going.  They don't contradict each other.  It's not a requirement to "reconcile" these two ways of talking.

However, as a science fiction writer, you might want to deliberately confuse the hypercube with a time machine in some way, as that makes it all sound so believable.  Somehow a tesseract lets the hero go into the distant past or future.

Coxeter, a chief practitioner and developer in the space of N-D Euclidean geometry, was frustrated by this kind of deliberate confusion.  I don't blame him, as today it's rather hard to tease them apart again.   He wrote:

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H.G. Wells in The Time Machine, has led such authors as J. W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.
H.S.M. Coxeter. Regular Polytopes. Dover Publications, 1973. pg. 119

I'm not coming at this difference as if there's something that needs to be settled or fixed. 

We can play chess or checkers on the same board of 64 squares.

The important concept to introduce is that of "namespace" (similar to "context").  Are we playing chess or checkers?  We need to know that.

Likewise, Synergetics is a namespace, and in that context, "4D" as yet a different meaning, relating to our sense of volume connecting to our concept of container / containing.  The minimum "room" (container) has four corners and four faces so lets say space is 4D.

Quadrays are the equivalent of XYZ where instead of three basis vectors and their three negative counterparts, organized in a "jack" pattern (six spokes), we have four basis vectors, no need for negative counterparts, organized in a "caltrop" pattern.

Instead of eight octants around the origin like in XYZ we have just four quadrants.

Fuller believed our approach to STEM was too rectilinear, too cube-oriented.  It's a critique you won't find coming from many thinkers.  Given my focus on philosophy at Princeton, it stands to reason I was attracted by his unusual slant.

However, when I study how Fuller's philosophy was received, I find it getting caught up in the prevailing confusions already swirling in the awake of two other meanings of 4D, which I've labeled Coxeter.4D and Einstein.4D.

Where did all this talk of "higher dimensions" come from in the first place and how did we get into these confusions?  The book The Fourth Dimension and Non-Euclidean Geometry in Modern Art by Linda Darlrymple Henderson tackles precisely that question.  It's a book about art history.  Finally, a tribe or subculture that's able to help me trace the intellectual history of the 1900s.  Conclusion:  art history students are the ones most well positioned to grasp 4D versus 4D versus 4D.

Back to XYZ, consider it a space-filling "scaffolding" that grids volume with any number of cubes.  That's what XYZ looks like when we fly through it in some animation.

Now consider a different scaffolding that grids volume with tetrahedrons and octahedrons instead.  This is what in Synergetics we call the "isotropic vector matrix" (IVM).  Again, the quadrays language game helps us here, as we're able to assign all the hubs only-positive-integer coordinates (convenient).

In Synergetics, the 4D IVM is like XYZ in being a space-filling scaffolding that stretches indefinitely in every direction.  In XYZ, each hub connects to six neighbors in the surrounding space.  In the IVM, each hub connects to twelve neighbors.

The IVM is what Alexander Graham Bell was building and referred to as "kites".  Fuller was able to get a patent on this structure which he named the "octet truss" for patenting purposes.  In my own curriculum writing, the IVM is given "equal time" with XYZ one could say, and that marks me as "ethnically distinct" from those who ignore the IVM completely.

I use the word "ethnically" on purpose as the kind of math we choose and pass on to our progeny is indicative of our subculture or tribe.  Likewise along those lines I emphasize programming as integral with mathematics, the Python language in particular.  My Python tribe is here in Portland, Oregon this week, having a pow wow called Pycon.

To different ethnicities there correspond these various institutions, not just flavors of mathematics.

Adding anthropology to STEM is what I call STEAM, and in terms of understanding our world and its peoples, I find having this anthropological component is essential.  It's another way to keep all the language games (e.g. 4D vs 4D vs 4D) from getting all jumbled.

I knew this guy Bob Textor (Stanford based) who helped start the US Peace Corps.  He was also focused on anthropology and felt Peace Corps volunteers would strongly benefit from anthropological training.  I think he was correct.  Without a sense of ethnicities and how these differ, including in their approach to math, we come out thinking relatively incoherently about the world.

Kirby

[kirby urner]

On Sat, May 28, 2016 at 11:30 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

<< SNIP >>
 

Energy also gives an example of what "second squared" can mean.  I just wonder what energy means.  The Wikipedia article on energy is mystifying:
https://en.wikipedia.org/wiki/Energy

I'm on a physics teachers listserv, originally by invitation of Dr. Bob Fuller (distinct from RBF, the 4D guy), a famous physics teacher (master of pedagogy) from the midwest (so-called -- Illinois, Nebraska and like that). My blog has some more details about the guy.  He went to Burma with the Methodists at the start of his career as a science teacher, and had the Nobel Laureate Aung San Suu Kyi as his 6th grader student.  For the rest of his life, he followed her career.[1]

The physics tribe is very focused on Energy as a concept, so that physics might be cast in terms of energy conservation laws, for a basis.  But then Entropy is something else again, that may increase or decrease in some curious relationship with Energy.  Working with those two concepts is of core concern on that listserv.  Decreasing entropy or negentropy is also known as syntropy.  Synergetics as a word was meant to embrace both energy and entropy by taking the syn of syntropy and marrying it with energy for Synergetics (a study of energy & entropy/, amidst explorations in the geometry of thinking).

My own approach to physics, which I call First Person Physics sometimes, involves looking at "scenarios" or "films" which our just our day-to-day time tunnels, our own lives in action.  "Action" is the keyword here and has Newtonian units of mvd.  If we break our life into "frames" (x frames per second) then action gets quantized on a "per time frame" basis, and we write mvd/t which conveniently works out to units of Energy (mvv).  Energy per time (speed the film runs at) is Newton's power (E/t).  Finally, Planck's Constant h is in action units so E = hf where f = frequency = 1/t.

You might consider the above a "holding pattern" for helping to orient a student in terms of Newtonian units.  As you live your life, your scenario, there's an energy flow that breaks up into momenta, things going this way and that, arm moving, walking, talking etc.  Slice that thinly in time, like a movie camera does, to get "action frames" or "energy buckets".  Now you're thinking more like a physicist.  When the film runs too fast (let alone in reverse) you sense lots of physical laws being broken.  We know what "looks right" i.e. we have a sense of what's realistic in terms of energy expenditure rate E/t (power).

In Synergetics, we use time tunnel scenarios in an overlapping pattern (picture a spaghetti ball) to suggest Universe, eternally aconceptual i.e. not summarizable as a static picture with any reality.  Synergetics defines Universe in that way (eternally aconceptual, partially overlapping scenarios, each like a movie i.e. energy in action).

Kirby

from EduSummit / Pycon

Oregon Convention Center

Portland, Oregon


[1] https://en.wikipedia.org/wiki/Aung_San_Suu_Kyi
https://flic.kr/p/9vsoto

[Bradford Hansen-Smith]

Kirby: "In 4-D Euclidean geometry, the hypercube or tesseract has only spatial dimensions, not time-like axis,  Just like a regular cube.  It exists in space only.  Time is not of interest."
There is a problem here, the cube as all-space filling shown in the IVM, when conceptually isolated has no spacial context which otherwise it would totally occupy being all-space filling. For the development of higher dimensions, hypercube and the like, there must be space available for what the cube totally fills through both division and multiplication. Given the differences in language what are we really talking about?

"Action" is the keyword here and has Newtonian units of mvd.

To consider action as key is to suggest movement, (primal power) as the fourth dimension, not time or any other causality.

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[kirby urner]

On Sun, May 29, 2016 at 9:35 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

Kirby: "In 4-D Euclidean geometry, the hypercube or tesseract has only spatial dimensions, not time-like axis,  Just like a regular cube.  It exists in space only.  Time is not of interest."
There is a problem here, the cube as all-space filling shown in the IVM, when conceptually isolated has no spacial context which otherwise it would totally occupy being all-space filling. For the development of higher dimensions, hypercube and the like, there must be space available for what the cube totally fills through both division and multiplication. Given the differences in language what are we really talking about?

I'd say when the cube is completely isolated, space is the implied context. 

We might thank of that space as simply a larger cube?  The cube may have cubes inside as well.

There's no need for "higher dimensions" to imagine an expanding cube. 

For centuries, mathematics did fine without any hypercubes.

Of course the tetrahedron is more primitive than the cube so feel free to replace "cube" with "tetrahedron" above.

 

"Action" is the keyword here and has Newtonian units of mvd.

To consider action as key is to suggest movement, (primal power) as the fourth dimension, not time or any other causality.

So is this Bradford.4D?

As Linda's book makes clear, the 4D meme has been used many ways. 

My taxonomy of just three namespaces using 4D is meant to keep it simple, but the vista really isn't quite that simple.

Kirby

[Bradford Hansen-Smith]

Thanks, I wasn't sure, we are talking language.

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[kirby urner]

On Sun, May 29, 2016 at 11:36 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

Thanks, I wasn't sure, we are talking language.

We're each in our own namespace, but there's no doubt some overlap right?

Peter Farrell of this MathFuture group is here at Pycon.  He came to visit yesterday.

Unfortunately, he's not signed up for this EduSummit event, sold out (yet free) so wasn't allowed in.

I'm blogging about the summit as it unfolds (it's just today -- the conference continues).

http://worldgame.blogspot.com/2016/05/edusummit-pycon-2016.html

Kirby

[Joseph Austin]

I don't see what difference it makes how you philosophically view the "dimensions" as long as they are isotropic.

But I could point out that even physical 3-space is not isotropic--material objects behave much differently in the vertical/radial dimension than in the horizontal!

From a physical or metaphysical viewpoint, time is "different" than space in that we cannot "see" it all at once, though in truth we cannot see physical space all at once either--we only "see" a "frame" at a time, and are usually focused on only a small region of that.

Tell me, mathematically, what is important about the difference between dt and dx?

Joe

[Joseph Austin]

Kirby,

You may have inadvertently suggested an answer my question about the difference between "time" and "space".  "Time" is not commutative!

Perhaps the use of "then" confuses associativity with commutativity.

As you observe, "then" is non-commutative,

so we can't understand A * (B * C) to mean "B then C" happens "before" A in a temporal or logical sense, only that the equivalent action Y following A gives the same result as the action X equivalent to (A * B) followed by C.

There may be some non-trivial, even profound, implications of this understanding.

Joe

[kirby urner]

On Tue, May 31, 2016 at 9:05 AM, Joseph Austin <drtec...@gmail.com> wrote:

I don't see what difference it makes how you philosophically view the "dimensions" as long as they are isotropic.

But I could point out that even physical 3-space is not isotropic--material objects behave much differently in the vertical/radial dimension than in the horizontal!

Are you talking about gravitational field here?

When we write a game engine, say to play chess or checkers, we need to
define the location of pieces (state) and their legal moves (changes to state), but
we're not required to model what happens if a piece falls off the table and shatters
into smithereens (more likely it bounces).

There's this pristine concept of "space" inherited from Greek geometry which is the
3D equivalent of a 2D chalk board.

Why we call it "3D" in the first place is one of the questions to investigate. 

One way to proceed with that investigation (using "investigation" in the sense
Wittgenstein did i.e. as a philosophical endeavor) is to imagine a "tribe" which
thinks differently about space (Wittgenstein imagined tribes quite a bit, giving
his philosophy the flavor of anthropology). 

Rather than make up an example, I have such a tribe ready-made.  They say
space is "4D" because "space = room = tetrahedron" i.e. the paradigm polyhedron,

with the minimum features to qualify as such, has four faces, four corners and

six edges.  There's no three in this picture. 

The same tribe that says space is 4D in this sense also fills it with a different
space-filling scaffolding than XYZ (all cubes).  They fill space with octahedrons
and tetrahedrons instead (IVM).

However, the dominant orthodox culture in which all of us grow up and go to

school, insists that space is 3D.  The 3Dness of space was well established

by the 1800s and Euclidean geometry in its spatial form continued to evolve

in that context.

Then two things occurred:

(A)  physics needed to model space-time in which space-like XYZ distance
might appear as a time-like difference in a different reference frame.  What
you see as two events happening at the same time (simultaneously), I might
see as at a different distance apart, and happening at different times (not
simultaneously).  Neither one of us is "mistaken".  What we agree on is
an "interval" between these events, computed as a function of their (x,y,z,t)
coordinates in each reference frame.  This is geometry we attribute to
Minkowski and is what Einstein's relativity framework uses.  Minkowski's

apparatus is considered "non-Euclidean" because the interval between

events is not computed using the Euclidean distance formula.  A different

metric is applied.

https://en.wikipedia.org/wiki/Relativity_of_simultaneity

(B) mathematics needed to extend Euclidean geometry to higher dimensions,

embracing the so-called polytopes.  These higher dimensional data structures

also have real world applications, such as signal multiplexing (getting lots of

separate data streams sharing the same channel).  The Euclidean distance

formula, based on the Pythagorean Theorem, is used.  All the dimensions are

"space-like".

Keeping (A) and (B) separate in the public mind is what my quote from Coxeter

was aimed at doing.  (A) and (B) are different language games.

Both (A) and (B) overlap in agreeing that "space is 3D".  Where they differ

is in how they look at "4D" i.e. (x, y, z) with an added dimension.  In

Minkowski space, the added t is treated differently than the added spatial

dimension in Euclidean space.  A.4D is not equal to B.4D.

Do we so far agree?

In the late 1970s, another work was published, the philosophical writings

of R. Buckminster Fuller, widely recognized for his geodesic dome, leaving aside

whether it was really his invention.  His principal invention, in his own mind,
was a  way of thinking, which he ended up naming Synergetics.  This work

is not widely studied or appreciated, despite the convenient whole number

volumes it introduces, based on a tetrahedron of volume one.  As one of the

few people who has taken an interest in his work, I've found it necessary to

spell out A.4D and B.4D as distinct (as discussed above) in order to make

room for C.4D, which is neither of these but is influenced by both. In 1991,

I wrote a short booklet called 'Synergetics:  The Invention Behind the Inventions"

which is these days on-line:

http://www.4dsolutions.net/synergetica/synergetica1.html

In trying to make these distinctions clear, I introduce the notion of "namespaces"

from computer science, which is akin to "context" in the humanities.  I find it

necessary to make these distinctions in order to give Synergetics some

"space" in a more mental or metaphysical sense.  Fuller's usage of '4D' will

not come across as coherent as long as it's confused with either A.4D or B.4D.

Confusing A.4D with B.4D is ready rampant (what Coxeter was railing against

on page 119).  Teasing these meanings of '4D' apart would seem a useful

exercise even if it weren't motivated by my agenda to salvage Synergetics.

  

From a physical or metaphysical viewpoint, time is "different" than space in that we cannot "see" it all at once, though in truth we cannot see physical space all at once either--we only "see" a "frame" at a time, and are usually focused on only a small region of that.

Tell me, mathematically, what is important about the difference between dt and dx?

Joe

In Greek philosophy, the Pythagorean Theorem is proved, in many ways,

without recourse to any deltas representing the passage of time.  We consider

the Pythagorean Theorem to be timelessly true, given certain assumptions

i.e. axioms, which also do not involve time.  The difference between dt and dx

is that you don't need dt anywhere in a pure geometry book, whereas you do

need dx.  Distance in the spatial sense, as defined by the distance formula, is

devoid of any time variable.

However events do happen in some order, even if that order is not initially
agreed upon in all reference frames.  Achieving consensus about order, even
if that means fabricating the order somewhat arbitrarily (but in a way deemed
"fair"), is what so-called "blockchain computing" is all about.

Agreement on both the order of events and current state (some status quo)

is a common human need and we use computation to help determine that,

and to enforce it.  Steps must be taken in a certain order.  Even in proving

timeless truths, we go "step by step".

At no point am I trying to deny the importance or relevance of physics and

considerations about time.  Mainly I'm just pointing out language games such

as Euclidean geometry, taught to millions of school children over millions of

hours, along with an XYZ analytic geometry that goes along, is not billed as

physics but as math.  One of its hallmarks is it ignores the passage of time

as irrelevant to the timeless theorems it aims to prove.

Perhaps we don't see eye to eye on whether there's any really strong

metaphysical difference between time and space, but leaving that
discussion aside for now, maybe we can agree there's an anthropological
difference.  I think it's accurate to say that physics and the community of

physicists, is much more focused on language games that include time

in their computations, whereas in "pure math" one is likely to encounter

many games, played by tribes of mathematicians, that include the idea
of "logical order" and/or "step by step reasoning", but do not have any
sense of a clock.  Might we summarize the difference as between "timed
chess" and "chess with no timer"?

 

Earlier in this thread (and reproduced below) Bradford suggested "action"
is maybe more the conceptually primal than writing out the Newtonian
units mvd (i.e. mass * velocity * distance) would lead us to suggest. 

That's a good thought. 

With action (in Planck's sense), we get "momentum for a distance"

which is pretty primal in terms of "what else happens when state
changes".  In making action "per a time interval" (mvd/t), we

get Newtonian energy units, i.e. momentum for a distance in a time.

That's like a snapshot of an "energy delta" (an energy expense). 

The blurred bullet in the frame of film expresses an energy expenditure.

How many "dimensions" have we added with "action" and/or "energy".

Again, the Euclideans do not consider their points, lines and planes,

their polytopes, to have any "mass".  They're Platonic.  They're

templates.  In special case, in times and places, these things take

on specific physical characteristics perhaps, but the theorems ignore
them.  In the "mind's eye" we imagine them unhampered with such

attributes.

And yet such energetic dimensions are requirements for mere
existence are they not?  

We're back to Greek philosophy and discussions about the difference
between generalized ideas (the archetypal tetrahedron) and special
case energetic instantiations (instances) of same.  I think your average

Greek philosopher might agree to a diagram such as this:

Metaphysical / Timeless / General / Archetypal <--- Generic Tetrahedron
======================================================

Physical / Temporal / Special / Objective <--- any individual Tetrahedron

Does that make sense to you as well?  The ===== is meant to separate

what's "purely logical" from what's "existing" in space and time.

Kirby

[Joseph Austin]

On May 31, 2016, at 3:58 PM, kirby urner <kirby...@gmail.com> wrote:

 In 1991, 

I wrote a short booklet called 'Synergetics:  The Invention Behind the Inventions"

which is these days on-line:

http://www.4dsolutions.net/synergetica/synergetica1.html

Kirby,

While wending my way through your booklet on Synergetics, I'm reminded of an article from several years ago on the relation of Penrose's kites and darts to non-periodic patterns in Islamic art. It includes over a dozen full-color photographs and drawings. Sebastian R Prange, "The Tiles of Infinity", Saudi Aramco World 60,5 (Sep-Oct 2009) Aramco Services Co. Houston TX.  If perchance you are not familiar with it, I could send you a copy. 

Joe Austin

[kirby urner]

Hi Joe --

I'm familiar with the kites and darts certainly, but don't have the Aramco World thing you're talking about.  You have multiple copies?  I would gladly add one to my collection.

Speaking of Saudi Arabia, there's a geodesic sphere restaurant in Riyadh, known in diplomatic circles for its symbolic value.  The dots to connect here include Sir Norman Foster, one of Bucky's biggest fans.  Foster has created a mint condition Dymaxion Car [tm] based on the original designs.

https://en.wikiarquitectura.com/index.php/Al_Faisaliyah_Center
http://www.archdaily.com/121530/video-norman-foster-recreates-buckminster-fullers-dymaxion-car

Because RBF was so well known around the world, as a positive futurist, diplomats, e.g. cultural attaches, would frequently have Synergetics on a visible bookshelf, kind of that Expo / World's Fair mentality wherein every country puts its best foot forward.

The original design for the 1967 Montreal Expo pavilion was not a giant geodesic ball but a "geoscope" i.e. a geodesic globe showing global data (shades of ESRI).  The globe would every so often unfold into a flat map, a so-called Dymaxion Map aka Fuller Projection.

That's a lot of valuable heritage that's somewhat kept at bay (on the shelf) thanks to puzzling aspects of Synergetics that make it seem too off the deep end.  In part it's a genre issue.  It's a work in the humanities in the sense that it relies on prose to support a thought process, rather than a lot of special case math notations. 

Wittgenstein's Remarks on the Foundations of Mathematics, integral with Philosophical Investigations, is somewhat the same way, in making only sparse use of logic notations. 

As you may have seen from that booklet I wrote in 1991, I was confidant even then there'd be plenty to computerize going forward.  Over two decades later, looking back, we've seen quite a bit of that.  I'm still finding it necessary to talk about "4D versus 4D versus 4D" however, to help readers recognize how different namespaces make for different meanings.

My family has lived in Islamic-oriented parts of the world quite a bit:  many years in Cairo, a few in Bangladesh, plus my dad worked with Libya for six years before either of those.  I never made it to Libya but did kick around Cairo and Dhaka quite a bit.

I would think D. Koski's tiling of 3D space, using phi-scaled Synergetics E-mods and S-mods, might be interesting to Islamic artists.  I mention his studies in that booklet.  He's been productive since 1991.  He'll be at Bridges in Finland this summer. 

The math-art community is big on RBF memes at least superficially, although there seems to be a lot of resistance to adopting any of the Synergetics terminology beyond "frequency" for geodesic spheres.  Mentioning A, B, T, E or S modules would require a citation to the published work and if you check bibliographies that rarely happens. 

Again, I think confusions about basic meanings has a lot to do with it.  That '4D' would link to a tetrahedron is non-obvious to someone expecting either 3D + Time and/or a 4D Euclidean treatment.

Kirby