Tadashi Tokieda: Thinking with Pictures
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Andrius Kulikauskas
Aug 18, 2016, 8:04:47 AM8/18/16
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I've been trying to figure out the essence of geometry. I've been
finding helpful videos.
Tadashi Tokieda is a creative lecturer who gave a series of talks on
"Geometry and Topology" in South Africa:
https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ
His first talk is a wonderful exploration of the Moebius strip which
made that truly interesting to me for the first time.
His second talk is about his main idea, that he wants us to think
pictorially, to draw and especially, to draw ideas which are more than
just photos.
Personally, a reward for me was his idea that geometry and topology
"expand our vision". That's a very profound way to define geometry. I
have been thinking of geometry as the ways of embedding a lower
dimensional space into a higher dimensional space. But "expand our
vision" means to expand a "smaller" space with a "larger" one, in
whatever sense. It could mean, for example, adding a point or line to
represent the infinite horizon.
I will add a few more ideas that came up where I think of us.
I have been watching Norman Wildberger's video lectures about Universal
Hyperbolic Geometry.
https://www.youtube.com/playlist?list=PLC37ED4C488778E7E
He thinks of it as projective geometry "with a distinguished circle"
added. So that distinguished circle made me think of Bradford and
circle folding. What that God-given circle does is to relate points
inside the circle with points outside the circle, which reminds me of
"conformal" geometry. In other words, folds within a circle would have
meaning beyond the circle, too.
Joe wrote about his interest in triples such as addition, multiplication
and exponentiation. I watched three video lectures on algebraic
geometry by T.E. Venkata Balaji in India. They considerably illuminated
this very opaque subject.
https://www.youtube.com/watch?v=-MASKnQriQo
I gathered that perhaps the central idea that organizes mathematical
areas is the Fundamental Theorem of Algebra, which says that any
polynomial (with complex coefficients) can be factored completely
(yielding complex roots). The root (the "zeros") are thought of as what
can be geometrically constructed. The polynomial with its coefficients
are what can be algebraically described. In studying this, the Zariski
topology is very important and natural. It has us consider on the
roots, that is, the inputs for which polynomials output zero. And it
says that if we have sets of roots, then multiplying polynomials
corresponds to uniting sets of roots, and adding polynomials corresponds
to intersecting sets of roots. So addition and multiplication
correspond to intersection and union. The "algebraic numbers" are the
ones constructed by equations given by addition and multiplication, that
is, polynomials. If we allow equations with exponentials, then we can
get at least some of the "transcendental numbers". So it's interesting
to consider what would correspond to exponentiation. In combinatorics,
the power series of the exponential function is related to labeling or
unlabeling things (I forget which).
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
finding helpful videos.
Tadashi Tokieda is a creative lecturer who gave a series of talks on
"Geometry and Topology" in South Africa:
https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ
His first talk is a wonderful exploration of the Moebius strip which
made that truly interesting to me for the first time.
His second talk is about his main idea, that he wants us to think
pictorially, to draw and especially, to draw ideas which are more than
just photos.
Personally, a reward for me was his idea that geometry and topology
"expand our vision". That's a very profound way to define geometry. I
have been thinking of geometry as the ways of embedding a lower
dimensional space into a higher dimensional space. But "expand our
vision" means to expand a "smaller" space with a "larger" one, in
whatever sense. It could mean, for example, adding a point or line to
represent the infinite horizon.
I will add a few more ideas that came up where I think of us.
I have been watching Norman Wildberger's video lectures about Universal
Hyperbolic Geometry.
https://www.youtube.com/playlist?list=PLC37ED4C488778E7E
He thinks of it as projective geometry "with a distinguished circle"
added. So that distinguished circle made me think of Bradford and
circle folding. What that God-given circle does is to relate points
inside the circle with points outside the circle, which reminds me of
"conformal" geometry. In other words, folds within a circle would have
meaning beyond the circle, too.
Joe wrote about his interest in triples such as addition, multiplication
and exponentiation. I watched three video lectures on algebraic
geometry by T.E. Venkata Balaji in India. They considerably illuminated
this very opaque subject.
https://www.youtube.com/watch?v=-MASKnQriQo
I gathered that perhaps the central idea that organizes mathematical
areas is the Fundamental Theorem of Algebra, which says that any
polynomial (with complex coefficients) can be factored completely
(yielding complex roots). The root (the "zeros") are thought of as what
can be geometrically constructed. The polynomial with its coefficients
are what can be algebraically described. In studying this, the Zariski
topology is very important and natural. It has us consider on the
roots, that is, the inputs for which polynomials output zero. And it
says that if we have sets of roots, then multiplying polynomials
corresponds to uniting sets of roots, and adding polynomials corresponds
to intersecting sets of roots. So addition and multiplication
correspond to intersection and union. The "algebraic numbers" are the
ones constructed by equations given by addition and multiplication, that
is, polynomials. If we allow equations with exponentials, then we can
get at least some of the "transcendental numbers". So it's interesting
to consider what would correspond to exponentiation. In combinatorics,
the power series of the exponential function is related to labeling or
unlabeling things (I forget which).
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Bradford Hansen-Smith
Aug 18, 2016, 12:02:37 PM8/18/16
to mathf...@googlegroups.com
Andrius, Norman Wildberger's Universal Hyperbolic Geometry is in the right direction.
"What that God-given circle does is to relate points inside the circle
with points outside the circle, which reminds me of "conformal"
geometry. In other words, folds within a circle would have meaning
beyond the circle, too."
An easy way to understand this is everything that is beyond the circle is not. The concentric nature of the circle is infinite, therefore everything "outside' of the circle is inside a larger circle aligned to the measure of the given circle unit. Projective geometry works because of the circle, it just got left out because it was already a "part" of Euclidean geometry. Every tangent to a given circle is a chord to a larger circle. By eliminating the importance of the circle we confined ourselves to focus on fragments, loosing context and origin, except as we designate by definitions.
To get a better idea of this inside and outside of the circle take a look at:
http://wholemovement.com/blog/item/735-the-other-circle
Understand that everything on the rectangle paper, or of any polygon, is simply the truncation of the circle where all sense of origin has been eliminated and all the inter-relationships between all parts and pieces is lost. We are left having to figure out how they all go together, and of course the more pieces we identify the more confusing it becomes. By not recognizing common origin within sphere-to-circle compression that reveals the concentric nature of the circle, we are always trying to create unity that is already there to be observed if only we could detach ourselves for a moment from the beloved fragments and constructing methods we have developed and so desperately hang onto.
I am curious about your intuitive interest in the circle, I don't find this with most math people, maybe it is and they repress it.
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Andrius Kulikauskas
Aug 25, 2016, 5:41:27 PM8/25/16
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Bradford,
Thank you for your response! I glanced through your blog post. There is
much to absorb and that will require time at some point. This Fall I am
supposed to be teaching a course at Vilnius Gediminas Technical
University for students on how to investigate a question. And I will
devote one class to "artistic" and "tactile" approaches. I will talk a
bit about artist Natalie d'Arbeloff's book on artistic investigation,
"Designing with Natural Forms" where she has chapters on investigating
(or simply playing around with) water, a pineapple, one's hand and
eggs. So then in class we'll spend some time folding circles. I will
indeed have to prepare for that! Maybe you will have ideas. It's just
not clear if my class has been moved to the Spring or cancelled entirely.
Personally, I respect the circle. But often it does seem unclear what
to do with it. So it's noteworthy that you've successfully pursued that
question. Really, my intuitive interest is in people, that is, in you
and your own interest in circle folding. I want to believe that there is
indeed some deep point to that. And I'm finding hints of that as I
survey abstract math.
I have my own preferences for a comprehensive abstract thinking that is
engaged and not detached. I have personally found this successful and
want to see it through to complete my fundamental results and make that
understandable to others. That includes trying to understand the big
picture for abstract math and its roots and unfolding. So, for example,
my philosophy guides me to think that there should be four geometries
that relate to four different levels of knowledge (whether, what, how,
why) and they seem to be: affine, projective, conformal, symplectic.
However, there seems to be, from my point of view, various uses of these
terms which obscures the distinct natures of each of these geometries.
So, for example, I am thinking that adding a circle to projective
geometry actually makes it a conformal geometry or even a symplectic
geometry, but it is still called "projective geometry with a circle".
Recently, I sensed a connection with the basic number systems:
* Affine geometry - preserves vectors - constructs natural numbers or
integers
* Projective geometry - preserves lines - constructs rational numbers
* Conformal geometry - preserves angles - constructs real numbers
* Symplectic geometry - preserves areas, or actually, angular momentum =
position x momentum - constructs complex numbers (I am thinking that the
Clifford algebras with their "geometric product" are this sort of geometry).
Thus are just my hunches. I have to look over and systematize anew
various expositions, such as Wildberger's videos on Universal hyperbolic
geometry. I am hoping to apply symmetric functions to restate his many
classic theorems and concepts in a way that I might unify.
I learned about Grothendieck's "six operations"
https://en.wikipedia.org/wiki/Six_operations
It's mysterious to me, but it may perhaps be the six transformations
that I'm looking for between the four kinds of geometry.
I also learned about the Geometrization conjecture (related to the
Poincare conjecture) and the eight Thurston geometries:
https://en.wikipedia.org/wiki/Geometrization_conjecture
So that is something to understand.
I am currently doing a study of variables. See the diagram here:
http://www.ms.lt/sodas/Book/Variables
I thought there should be six types of variables. Or rather, pairs, for
example, "dependent" and "independent". Basically, there should be six
ways of thinking about variables. So I wanted to investigate this,
survey the ways that variables are used, be insightful about the
distinctions, and systematize the ways.
But I got about twice as many pairs as I expected. Now, it seems that
in my mind, the variables reference different mental structures which I
refer to in my article:
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
Namely, sometimes a variable seems to refer to an item in a list; or an
element of a set; or one copy among several (as with a multiset
consisting of the same element); or of an "original" "origin" that may
be all alone. Well, in my mind, for example, a variable may be
"arbitrary" (an arbitrary value) or "particular" (a particular value).
If I think it is "particular", then it refers to a unique value that is
distinct from all the others, which is to say, it is an element of a
set. But if it is "arbitrary", then it is as if there is no distinction
to be made, it is just a "thing that exists" among a multiset of "things
that exist". So one way to consider what is going on here is to suppose
that our mind is taking an "arbitrary" variable (among the "copies") and
then refining it to be a "particular" variable (understood to be an
element in a set). So there are six ways to make such refinements.
But there also seem to be six ways to go in the reverse direction! For
example, a variable may be "typed" or "untyped". The type might be
"integer" or "mammal" etc. Well, we could imagine that what's happening
here is that an unrestricted (untyped) variable is given a type. My
mind may be thinking of the untyped variable as refering to some
idiosyncratic thing that is understood to be different from everything
else, which is to say, it is an element of a set. But then if I give it
a type, I am thinking of it as having that "type" in common with
everything else of that type. They are all just copies of the same type
(inasmuch as I neglect their other content). So the notion of "type"
may allow my mind to "unrefine" the variable, that is, to distinguish
the "copy" aspect from the "element" aspect. Certainly, that kind of
mental move could be very useful. And having these six refinements and
six unrefinements could make for a system of basic transformations that
could yield very complex and meaningful constructions.
The "proof" will be when the system as a whole turns out to be
insightful and fruitful. But these first steps are encouraging for me.
I will need to think and write more about the distinctions I am making
and considering. But I appreciate if anybody has ideas on ways of
thinking about different kinds of variables.
Eiciunai, Lithuania
> m...@ms.lt <mailto:m...@ms.lt>
> +370 607 27 665 <tel:%2B370%20607%2027%20665>
> <mailto:mathfuture%2Bunsu...@googlegroups.com>.
>
>
>
>
> --
> Bradford Hansen-Smith
> www.wholemovement.com <http://www.wholemovement.com>
> <mailto:mathfuture+...@googlegroups.com>.
Thank you for your response! I glanced through your blog post. There is
much to absorb and that will require time at some point. This Fall I am
supposed to be teaching a course at Vilnius Gediminas Technical
University for students on how to investigate a question. And I will
devote one class to "artistic" and "tactile" approaches. I will talk a
bit about artist Natalie d'Arbeloff's book on artistic investigation,
"Designing with Natural Forms" where she has chapters on investigating
(or simply playing around with) water, a pineapple, one's hand and
eggs. So then in class we'll spend some time folding circles. I will
indeed have to prepare for that! Maybe you will have ideas. It's just
not clear if my class has been moved to the Spring or cancelled entirely.
Personally, I respect the circle. But often it does seem unclear what
to do with it. So it's noteworthy that you've successfully pursued that
question. Really, my intuitive interest is in people, that is, in you
and your own interest in circle folding. I want to believe that there is
indeed some deep point to that. And I'm finding hints of that as I
survey abstract math.
I have my own preferences for a comprehensive abstract thinking that is
engaged and not detached. I have personally found this successful and
want to see it through to complete my fundamental results and make that
understandable to others. That includes trying to understand the big
picture for abstract math and its roots and unfolding. So, for example,
my philosophy guides me to think that there should be four geometries
that relate to four different levels of knowledge (whether, what, how,
why) and they seem to be: affine, projective, conformal, symplectic.
However, there seems to be, from my point of view, various uses of these
terms which obscures the distinct natures of each of these geometries.
So, for example, I am thinking that adding a circle to projective
geometry actually makes it a conformal geometry or even a symplectic
geometry, but it is still called "projective geometry with a circle".
Recently, I sensed a connection with the basic number systems:
* Affine geometry - preserves vectors - constructs natural numbers or
integers
* Projective geometry - preserves lines - constructs rational numbers
* Conformal geometry - preserves angles - constructs real numbers
* Symplectic geometry - preserves areas, or actually, angular momentum =
position x momentum - constructs complex numbers (I am thinking that the
Clifford algebras with their "geometric product" are this sort of geometry).
Thus are just my hunches. I have to look over and systematize anew
various expositions, such as Wildberger's videos on Universal hyperbolic
geometry. I am hoping to apply symmetric functions to restate his many
classic theorems and concepts in a way that I might unify.
I learned about Grothendieck's "six operations"
https://en.wikipedia.org/wiki/Six_operations
It's mysterious to me, but it may perhaps be the six transformations
that I'm looking for between the four kinds of geometry.
I also learned about the Geometrization conjecture (related to the
Poincare conjecture) and the eight Thurston geometries:
https://en.wikipedia.org/wiki/Geometrization_conjecture
So that is something to understand.
I am currently doing a study of variables. See the diagram here:
http://www.ms.lt/sodas/Book/Variables
I thought there should be six types of variables. Or rather, pairs, for
example, "dependent" and "independent". Basically, there should be six
ways of thinking about variables. So I wanted to investigate this,
survey the ways that variables are used, be insightful about the
distinctions, and systematize the ways.
But I got about twice as many pairs as I expected. Now, it seems that
in my mind, the variables reference different mental structures which I
refer to in my article:
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
Namely, sometimes a variable seems to refer to an item in a list; or an
element of a set; or one copy among several (as with a multiset
consisting of the same element); or of an "original" "origin" that may
be all alone. Well, in my mind, for example, a variable may be
"arbitrary" (an arbitrary value) or "particular" (a particular value).
If I think it is "particular", then it refers to a unique value that is
distinct from all the others, which is to say, it is an element of a
set. But if it is "arbitrary", then it is as if there is no distinction
to be made, it is just a "thing that exists" among a multiset of "things
that exist". So one way to consider what is going on here is to suppose
that our mind is taking an "arbitrary" variable (among the "copies") and
then refining it to be a "particular" variable (understood to be an
element in a set). So there are six ways to make such refinements.
But there also seem to be six ways to go in the reverse direction! For
example, a variable may be "typed" or "untyped". The type might be
"integer" or "mammal" etc. Well, we could imagine that what's happening
here is that an unrestricted (untyped) variable is given a type. My
mind may be thinking of the untyped variable as refering to some
idiosyncratic thing that is understood to be different from everything
else, which is to say, it is an element of a set. But then if I give it
a type, I am thinking of it as having that "type" in common with
everything else of that type. They are all just copies of the same type
(inasmuch as I neglect their other content). So the notion of "type"
may allow my mind to "unrefine" the variable, that is, to distinguish
the "copy" aspect from the "element" aspect. Certainly, that kind of
mental move could be very useful. And having these six refinements and
six unrefinements could make for a system of basic transformations that
could yield very complex and meaningful constructions.
The "proof" will be when the system as a whole turns out to be
insightful and fruitful. But these first steps are encouraging for me.
I will need to think and write more about the distinctions I am making
and considering. But I appreciate if anybody has ideas on ways of
thinking about different kinds of variables.
Eiciunai, Lithuania
> +370 607 27 665 <tel:%2B370%20607%2027%20665>
>
> --
> You received this message because you are subscribed to the Google
> Groups "MathFuture" group.
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> --
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> <https://groups.google.com/d/optout>.
>
>
>
>
> --
> Bradford Hansen-Smith
> www.wholemovement.com <http://www.wholemovement.com>
> --
> You received this message because you are subscribed to the Google
> Groups "MathFuture" group.
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