What is geometry?
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Andrius Kulikauskas
Jun 27, 2016, 6:45:27 AM6/27/16
to mathf...@googlegroups.com
Joe, Ted,
Thank you for your thoughts. I'm glad to be encouraged.
I posted a question, What is geometry? at Math Overflow and it was put
on "hold":
http://mathoverflow.net/questions/243109/what-is-geometry
I share it here. I've looked around and I'm concluding that Math
Overflow has become so popular among mathematicians that math listservs,
discussion groups, etc. no longer exist. So it's all the more precious
to be with people who care to learn advanced math and seek the light
within it.
I'm really interested to figure out the "six transformations" that I
mention below and in my essay. I think that Clifford algebras will
provide useful insights. I look forward to studying that together.
Joe, that seems like a well-written paper.
http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf
I will have to catch up!
Andrius,
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
--------------------------------------
What is geometry?
--------------------------------------
I wish to know what geometers and other mathematicians consider geometry.
Wikipedia defines geometry as "concerned with questions of shape,
size, relative position of figures, and the properties of space".
MathWorld defines geometry as "the study of figures in a space of a
given number of dimensions and of a given type", and formally, as "a
complete locally homogeneous Riemannian manifold".
nLab seems to define it as part of an Isbell duality between
geometry (presheaves) and algebra (copresheaves) where presheaves
(contravariant functors C->Set) and copresheaves (functors on C) are
identified with each other and thus glued together (for some category C).
https://ncatlab.org/nlab/show/geometry
I don't understand the latter but I would try if somebody might explain.
My own Ph.D. is in algebraic combinatorics. I will explain how I am
coming to think of geometry. I appreciate thoughts on how I might
develop my understanding further.
I study conceptual frameworks by which we think and live. I am applying
such frameworks to think about the "implicit math" in our minds by which
we figure things out in mathematics or interpret mathematics as, for
example, algebraic combinatorialists analyze an equation to figure out
what it is counting. I am also interested in how math can express and
model such conceptual frameworks.
As part of that, I made a diagram of the areas in math listed in the
Mathematics Subject Classification.
http://www.ms.lt/derlius/MatematikosSakosDidelis.png
https://en.wikipedia.org/wiki/Mathematics_Subject_Classification
I tried to identify which areas depended on which areas. I noticed that
differential geometry and algebraic geometry depend on geometry. But
what is geometry and what basic concepts does it contribute?
I am somewhat aware of Felix Klein's Erlangen program whereby we
consider transformation groups which leave geometric properties
invariant, and also groupoidification and geometric representation,
http://www.math.ucr.edu/home/baez/groupoidification/
moving frames, Cartan connection, principal connection and Ehresmann
connection. But I'm wondering if there is a more fundamental way to
think about geometry. I like the idea that we can get a geometry for
each of the Dynkin diagrams.
http://math.ucr.edu/home/baez/week181.html
In my essay, Discovery in Mathematics: A System of Deep Structure,
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
I notice that four infinite families of polytopes can be distinguished
by how they are extended in each new dimension. They seem to relate to
four different geometries and four different classical Lie algebras:
An - Simplexes are extended when the Center (the -1 simplex)
creates a new vertex and thereby defines direction, which is preserved
by affine geometry. Simplexes have both a Center and a Totality.
Cn - Cross-polytopes (such as the octahedron) are extended when the
Center creates two new vertices ("opposites") and thereby defines a line
in two directions, which is preserved by projective geometry.
Cross-polytopes have a Center but no Totality.
Bn - Cubes are extended when the Totality introduces a new mirror
and thereby defines right angles with previous mirrors, and the angles
are preserved by conformal geometry. Cubes have a Totality but no Center.
Dn - Demicubes have neither a Center nor a Totality. Instead of a
Center they have a collection of Origins and coordinate systems which
define simplexes that fit together to bound a space. We can think of the
demicubes as arising by introducing with each dimension a duality
mirror, that is, a mirror in which Origins become vertices and vertices
become Origins, and the new and old diagrams are joined. I don't yet
know but I suppose that the ambiguity of these demicubes could somehow
define areas, perhaps as oriented bounded spaces, in which case they
would be preserved by symplectic geometry.
Each of these four geometries would serve to define what we mean by
perspective, but especially, how a view from outside of a system (from a
higher dimension) and a view inside of a system (a lower dimension) can
be considered one and the same. In general, I am thinking that geometry
can be thought of as the ways of embedding one space into another space,
that is, a lower dimensional space into a higher dimensional space. I
imagine that tensors are important as the trivial, "plain vanilla"
version of this.
As I mention in my essay, I expect that there are six transformations by
which one geometry reinterprets a perspective from another geometry. And
I imagine that intuitively they may be precisely the various ways that
we interpret multiplication in arithmetic. I suppose that they may
include translation, rotation, scaling, homothety, similarity,
reflection and shear.
I thus ask how geometers think of geometry and what it contributes to
the big picture in math. I wish for my own philosophical investigations
to be more fruitful. I wonder how to pursue them further mathematically.
What should I study?
-------------------------
2016.06.27 04:20, Joseph Austin rašė:
> Andrius,
> I would love to continue the discussion of your work and Clifford Algebra, but we need a new thread.
> I found a CA paper I'm trying to work thru,
> http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf
> but got stuck at a certain point, p 14 eqn (11).
> Also have some question on your paper--more tomorrow--hopefully start a new thread.
>
> Meanwhile still working with Ted on MathPiper.
> Joe
>
Thank you for your thoughts. I'm glad to be encouraged.
I posted a question, What is geometry? at Math Overflow and it was put
on "hold":
http://mathoverflow.net/questions/243109/what-is-geometry
I share it here. I've looked around and I'm concluding that Math
Overflow has become so popular among mathematicians that math listservs,
discussion groups, etc. no longer exist. So it's all the more precious
to be with people who care to learn advanced math and seek the light
within it.
I'm really interested to figure out the "six transformations" that I
mention below and in my essay. I think that Clifford algebras will
provide useful insights. I look forward to studying that together.
Joe, that seems like a well-written paper.
http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf
I will have to catch up!
Andrius,
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
--------------------------------------
What is geometry?
--------------------------------------
I wish to know what geometers and other mathematicians consider geometry.
Wikipedia defines geometry as "concerned with questions of shape,
size, relative position of figures, and the properties of space".
MathWorld defines geometry as "the study of figures in a space of a
given number of dimensions and of a given type", and formally, as "a
complete locally homogeneous Riemannian manifold".
nLab seems to define it as part of an Isbell duality between
geometry (presheaves) and algebra (copresheaves) where presheaves
(contravariant functors C->Set) and copresheaves (functors on C) are
identified with each other and thus glued together (for some category C).
https://ncatlab.org/nlab/show/geometry
I don't understand the latter but I would try if somebody might explain.
My own Ph.D. is in algebraic combinatorics. I will explain how I am
coming to think of geometry. I appreciate thoughts on how I might
develop my understanding further.
I study conceptual frameworks by which we think and live. I am applying
such frameworks to think about the "implicit math" in our minds by which
we figure things out in mathematics or interpret mathematics as, for
example, algebraic combinatorialists analyze an equation to figure out
what it is counting. I am also interested in how math can express and
model such conceptual frameworks.
As part of that, I made a diagram of the areas in math listed in the
Mathematics Subject Classification.
http://www.ms.lt/derlius/MatematikosSakosDidelis.png
https://en.wikipedia.org/wiki/Mathematics_Subject_Classification
I tried to identify which areas depended on which areas. I noticed that
differential geometry and algebraic geometry depend on geometry. But
what is geometry and what basic concepts does it contribute?
I am somewhat aware of Felix Klein's Erlangen program whereby we
consider transformation groups which leave geometric properties
invariant, and also groupoidification and geometric representation,
http://www.math.ucr.edu/home/baez/groupoidification/
moving frames, Cartan connection, principal connection and Ehresmann
connection. But I'm wondering if there is a more fundamental way to
think about geometry. I like the idea that we can get a geometry for
each of the Dynkin diagrams.
http://math.ucr.edu/home/baez/week181.html
In my essay, Discovery in Mathematics: A System of Deep Structure,
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
I notice that four infinite families of polytopes can be distinguished
by how they are extended in each new dimension. They seem to relate to
four different geometries and four different classical Lie algebras:
An - Simplexes are extended when the Center (the -1 simplex)
creates a new vertex and thereby defines direction, which is preserved
by affine geometry. Simplexes have both a Center and a Totality.
Cn - Cross-polytopes (such as the octahedron) are extended when the
Center creates two new vertices ("opposites") and thereby defines a line
in two directions, which is preserved by projective geometry.
Cross-polytopes have a Center but no Totality.
Bn - Cubes are extended when the Totality introduces a new mirror
and thereby defines right angles with previous mirrors, and the angles
are preserved by conformal geometry. Cubes have a Totality but no Center.
Dn - Demicubes have neither a Center nor a Totality. Instead of a
Center they have a collection of Origins and coordinate systems which
define simplexes that fit together to bound a space. We can think of the
demicubes as arising by introducing with each dimension a duality
mirror, that is, a mirror in which Origins become vertices and vertices
become Origins, and the new and old diagrams are joined. I don't yet
know but I suppose that the ambiguity of these demicubes could somehow
define areas, perhaps as oriented bounded spaces, in which case they
would be preserved by symplectic geometry.
Each of these four geometries would serve to define what we mean by
perspective, but especially, how a view from outside of a system (from a
higher dimension) and a view inside of a system (a lower dimension) can
be considered one and the same. In general, I am thinking that geometry
can be thought of as the ways of embedding one space into another space,
that is, a lower dimensional space into a higher dimensional space. I
imagine that tensors are important as the trivial, "plain vanilla"
version of this.
As I mention in my essay, I expect that there are six transformations by
which one geometry reinterprets a perspective from another geometry. And
I imagine that intuitively they may be precisely the various ways that
we interpret multiplication in arithmetic. I suppose that they may
include translation, rotation, scaling, homothety, similarity,
reflection and shear.
I thus ask how geometers think of geometry and what it contributes to
the big picture in math. I wish for my own philosophical investigations
to be more fruitful. I wonder how to pursue them further mathematically.
What should I study?
-------------------------
2016.06.27 04:20, Joseph Austin rašė:
> Andrius,
> I would love to continue the discussion of your work and Clifford Algebra, but we need a new thread.
> I found a CA paper I'm trying to work thru,
> http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf
> but got stuck at a certain point, p 14 eqn (11).
> Also have some question on your paper--more tomorrow--hopefully start a new thread.
>
> Meanwhile still working with Ted on MathPiper.
> Joe
>
Maria Droujkova
Jun 27, 2016, 6:57:00 AM6/27/16
to mathf...@googlegroups.com
Andrius,
MathOverflow is a very specialized forum for work on research math within current academic standards. It doesn't welcome any other endeavors, because the leaders aim to keep very tight adherence to that goal and that goal only.
You may find better success with exploratory questions on Math Forum, The Art of Problem Solving forum, XKCD forum, or http://matheducators.stackexchange.com/, or Twitter hashtag MTBoS.
In general, it may help to define a more narrow target audience and a more particular WHY (intent, purpose) before asking people to define geometry. That's because geometry, like laughter or song or dream, is a lot of things for different people in different circumstances.
Cheers,
Dr. Maria Droujkova
Dr. Maria Droujkova
NaturalMath.com
Make math your own, to make your own math!
Make math your own, to make your own math!
-- .- - ....
--
kirby urner
Jun 27, 2016, 12:53:22 PM6/27/16
to MathFuture
On Monday, June 27, 2016 at 3:45:27 AM UTC-7, Andrius Kulikauskas wrote:
<< SNIP >>
<< SNIP >>
--------------------------------------
What is geometry?
--------------------------------------
I wish to know what geometers and other mathematicians consider geometry.
You might get more interesting answers if you rephrased it to "I wish to know,
first of all, who consider themselves geometers in this world and why." That
might get put on hold too though.
Yes I know: "this world" is ambiguous as to its denotation, however I'm betting
enough readers of MathOverFlow would think of the listserv itself as the global
context, and by extension the people who might read it. They wouldn't call you
out on it.
A few might think by "this world" you mean "the universe" or "all that is the case",
but that'd be a minority I'm thinking, and even then they'd maybe have an
answer to your question.
For example do architects in general consider themselves geometers?
We'd need to poll to find out. It's an empirical question.
However to run this survey, we'd first we'd have to see whom consider
themselves architects and that takes us a next step back. "In what world?"
Andrius World? Do eskimos building igloos think of themselves as
architects? Beavers don't think they're dam builders, as they don't
think in English (I wouldn't say they don't think).
Given you're a cross-dresser in the sense of mentioning "God" now
and then (you won't get published in Nature or Scientific American that
way), my bias is to pigeon-hole you as into a blend of Sacred Geometry
and Extended Euclidean.
The "extended Euclideans" are the ones working with extended
Euclideanism, meaning juggling many more mutual perpendiculars than
the Greek fathers ever dreamed were either imaginable or manageable.
n-D timeless. Hypercross stuff. Not like what Einstein was into, and
not quantum mechanics either, although Clifford Algebra may have
applications in both areas.
You're also really into graphing all the different maths and how they
talk to each other (interesting stuff -- I always liked that book Mind Maps),
which makes you more of a meta-mathematician (one who maps maths),
meaning you'll likely to "get kicked upstairs" (colloquialism) to become
a philosopher of some kind (we oversee / supervise the mathematicians
though they'd bristle at this suggestion -- sounds too much like the bad old
days).
On that score, you're definitely not a Logical Positivist (more how I'd think
of most OWL people), more of a Neo-Platonist (ergo more Continental as
the UKers think of it, with USAer philo profs mostly happy to ape the Brits,
Rorty, one of my mentors, somewhat an exception in reading more of the
Continental stuff)?
I'd be interested to see how you thinking might be shaped by an encounter
with the new Germanic blend, coming from the Globes, Bubbles and Foam
volumes. We have at least one serious reader of that material in 97214
(an Asylum District zip code **).
https://mitpress.mit.edu/books/bubbles
About the AuthorPeter Sloterdijk (b. 1947) is one of the best known and widely read German
intellectuals writing today. His 1983 publication of Critique of Cynical Reason
(published in English in 1988) became the best-selling German book of
philosophy since World War II. He became president of the State Academy
of Design at the Center for Art and Media in Karlsruhe in 2001. He has
been cohost of a discussion program, Das Philosophische Quartett
(Philosophical Quartet) on German television since 2002.
I'd be interested to learn more about the influence the Ray Kurzweil
transhumanism in Continental circles or is that memeplex more
parochial to the Anglophones. Also, have you studied Ken Wilbur?
Kirby
PS: interesting to learn how MathOverflow has become so central among
those of a mathematical bent (dint?) of mind. StackOverflow has achieved
a lot of providence in CS circles, which claims to be math on pseudo-random
days of the year based on some opaque #DeepLearning curve.
** Hawthorne District is named for Dr. Hawthorne, who helped
established one of Oregon's first, if not the first, mental hospitals,
on contract from Salem, however "Asylum" is also a pun on
"sanctuary for those with weird ideas".
John Bibby
Jun 27, 2016, 1:02:44 PM6/27/16
to mathf...@googlegroups.com
More interesting, I feel, is what ordinary people mean by this and other terms. (And how does it apply in other languages? I still recall my surprise in seeing "M. Hulot, Geometre" outside somebody's residence in France or was it Quebec?) JOHN BIBBY
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kirby urner
Jun 27, 2016, 1:32:47 PM6/27/16
to MathFuture
I'd be interested to learn more about the influence the Ray Kurzweil
transhumanism in Continental circles or is that memeplex more
parochial to the Anglophones. Also, have you studied Ken Wilbur?
In writing about (RI + AI) I'm also a transhumanist. Basically, those who think
AI can "take our jobs" are in cahoots with those of us "giving away our jobs"
to AI wherever possible i.e. it's more a case of "here, you can have it", versus
feel threatened in any way. AI is cute, like a baby dolphin. I'm not about to
run screaming from a favorite pet.
Kirby
Kirby
PS: interesting to learn how MathOverflow has become so central among
those of a mathematical bent (dint?) of mind. StackOverflow has achieved
a lot of providence in CS circles, which claims to be math on pseudo-random
days of the year based on some opaque #DeepLearning curve."
"has achieved provenance" is what I should have said. My "achieved a lot
of providence" is not grammatical English really.
articles.philly.com/1996.../25649938_1_curator-items-valley-forge-historical-society
Nov 29, 1996 - Swigart said that the museum had achieved provenance; that is,
authority to make claims based on exhaustive research that the items on ...
Just got my Pi3 through Amazon this morning (an Amazon van, not a delivery
drone quadcopter) so I can now fire up Pi3D and start working through examples
in Peter Farrell's Pythonic Math book. Pix in Photostream:
https://flic.kr/p/Jz86tT
Bradford Hansen-Smith
Jun 28, 2016, 1:21:55 AM6/28/16
to mathf...@googlegroups.com
Andrius and others, just saw this post "What is Geometry?" Couldn't resist. Short answer, an "overflow."
There are too many kinds of geometries ranging from sacred to academic to even try to make sense of it. Early on I realized geometry is an earth centered concept that should have been dropped years ago, but math is slow to change and we like to argue about what it was. We have been off the earth for a number of years and measuring the unmeasurable towards both ends of the into /out from spectrum. To update the word I figured it must means wholemovement; the sphere is whole as nothing else demonstrates and measuring is about movement. In my mind wholemovement give us more than we can imagine, to explore, go deeper into what works and what does not with a more comprehensive, even cosmic perspective, what we can not see and really have no way to talk about, which seems to be in favor now days. Imagination is a great escape from the reality of experience, before it becomes our reality.
I look at your map and don’t understand most of the words there but suspect it is all covered by the geometry of wholemovement. There is no where indication of origin or principles that are inherent in this net of fragmented concepts. I have often asked about the principles of math? As yet have gotten no answers that make sense. I guess that relates to your need to make a map and trace out what is fundamental. Nobody else seems to know.
Four different geometries? The concept of center comes with the compass. What about the circle without a center, or the one that is its own center by virtue of concentric movement in both directions. There are too many kinds of geometries ranging from sacred to academic to even try to make sense of it. Early on I realized geometry is an earth centered concept that should have been dropped years ago, but math is slow to change and we like to argue about what it was. We have been off the earth for a number of years and measuring the unmeasurable towards both ends of the into /out from spectrum. To update the word I figured it must means wholemovement; the sphere is whole as nothing else demonstrates and measuring is about movement. In my mind wholemovement give us more than we can imagine, to explore, go deeper into what works and what does not with a more comprehensive, even cosmic perspective, what we can not see and really have no way to talk about, which seems to be in favor now days. Imagination is a great escape from the reality of experience, before it becomes our reality.
I look at your map and don’t understand most of the words there but suspect it is all covered by the geometry of wholemovement. There is no where indication of origin or principles that are inherent in this net of fragmented concepts. I have often asked about the principles of math? As yet have gotten no answers that make sense. I guess that relates to your need to make a map and trace out what is fundamental. Nobody else seems to know.
Or the totality of nothing that has the same properties of the circle that has been removed from the paper.
If mirrors are reflection then 1 (totality) reflecting itself is 3, origin + reflection + mirror. 1+1=3 . Without the mirror there is no totality. We always see what we want in a mirror and it is never true, but close enough to virtual reality that it will pass as the real thing.
Does this mean that origin is principle to all right angle differentiation through multiple reflection and is context for all possibilities of geometries and maths, or is it where you start when drawing a line or vector? Probably both, at the same time. Geometry originally seemed to imply generalizations about temporal experience, maybe spiritual as well. Did it happen when the first of our species pick up a piece of charcoal and drew a circle of the sun in the dark on the cave wall? Did that circle have a center, or did it start with a center and rounding itself back to start again?
The six transformation you refer to; “by which one geometry reinterprets a perspective from another geometry.” I do not understand. But I do know there are six transformations a circle makes with three folded diameters that are fundamental to all subsequent folding and symmetries of the sphere, they also demonstrate the other functions you mentioned, and more.
"What should I study? you ask." Study the circle as compression from
spherical unity. There is nothing more comprehensive and inclusive. It is the biggest playing field you can step into and you get to pick any position you want to play. To be more
precise, as Maria said, geometry is “like laughter or song or dream.” Left
field is great place to do that. There are not too many of us out here, the pay is lousy and you don't get much action.
As to
Kirby’s question, since I do not consider myself a geometer, only a left fielder, I am unable to give a reasonable answer.
Yes, John, ordinary people, I’m with you on that generalization.
Brad
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Paul Libbrecht
Jun 28, 2016, 7:27:15 AM6/28/16
to mathf...@googlegroups.com
John, can you say more about this movie?
Was it about the lines in the car showroom?
I do not know of such a movie and googling did not bring me any such movie.
I believe Tatie's movies are all googlable.
Paul
Was it about the lines in the car showroom?
I do not know of such a movie and googling did not bring me any such movie.
I believe Tatie's movies are all googlable.
Paul
John Bibby
Jun 28, 2016, 7:33:12 AM6/28/16
to mathf...@googlegroups.com
No - despite the name "Hulot" it was not a movie! Just a genuine sign on a street. (To understand, you need to know what "geometre" means in French: see links below.) JOHN
Links:
---------------------------------------------------------------------------
- York Left Discussion Forum: inaugural meeting (on Brexit): Wednesday 29th June (7.30 at Shoulder of Mutton, Heworth) - please tell your friends! See www.YLLyork1.eventbrite.co.uk
- York Bus Forum: you can join or donate here. We meet at 6pm on the 2nd Tuesday of every month (West Offices, Station Rise, York YO1 6GA). Support our campaign for NO BUS CUTS !!
kirby urner
Jul 2, 2016, 8:25:10 PM7/2/16
to mathf...@googlegroups.com
I'd be interested to see how you thinking might be shaped by an encounter
with the new Germanic blend, coming from the Globes, Bubbles and Foam
volumes. We have at least one serious reader of that material in 97214
(an Asylum District zip code **).
https://mitpress.mit.edu/books/bubblesAbout the AuthorPeter Sloterdijk (b. 1947) is one of the best known and widely read German
intellectuals writing today. His 1983 publication of Critique of Cynical Reason
(published in English in 1988) became the best-selling German book of
philosophy since World War II. He became president of the State Academy
of Design at the Center for Art and Media in Karlsruhe in 2001. He has
been cohost of a discussion program, Das Philosophische Quartett
(Philosophical Quartet) on German television since 2002.
More weaving...
Kirby
Message has been deleted
Maria Droujkova
Jul 2, 2016, 8:45:35 PM7/2/16
to mathf...@googlegroups.com
Kirby, Thank you for your analysis. It's interesting to know that Peter Slojterdik's "Spheres" (Bubbles!) is finally in English. I have a volume in Spanish but have not read much from it. Maybe I should look again but I'm not sure if there is any meaningful idea there. Is there? I read Ken Wilbur's book "The Theory of Everything"
This wasn't supposed to go to the group - it's being edited. Sorry about that.
kirby urner
Jul 2, 2016, 9:37:36 PM7/2/16
to mathf...@googlegroups.com
I've not studied either thinker concertedly.
Again, thank you.
Joseph Austin
Jul 5, 2016, 7:31:53 PM7/5/16
to mathf...@googlegroups.com
On Jun 27, 2016, at 2:40 AM, Andrius Kulikauskas <m...@ms.lt> wrote:I'm really interested to figure out the "six transformations" that I mention below and in my essay. I think that Clifford algebras will provide useful insights. I look forward to studying that together.
Andrius,
I've been meaning to respond to your paper, but haven't got my thoughts together yet.
Just a couple of speculations:
1. I'm not a geometer, but as I remember HS geometry, it involved constructions (of geometric figures) and deductions (proofs).
When I learned computing, I was struck by a parallel--computing is a constructive kind of math, in many ways it is like constructing a proof. Unfortunately, we have not been so careful to build programs on axioms and proofs.
2. The six transformations and their possible relation to products suggest to me perhaps a relation to my idea that the "product" is all that exists a priori,
and the actual operations are projections and transformations of a multidimensional something. In other words, though we tend to think of multiplication as putting things together, in fact it's philosophical dubious that you can create a surface from two lines; it's make more "physical" sense to think of creating a line from two surfaces.
In robotics, we have "line following" programs. We put a piece of black tape on a light floor to define the path. But the tape isn't the line--it's much to wide.
The "line" that the robot actually follows is the boundary between the black tape and the white floor.
What if "geometry" is ultimately not so much the study of points on planes or spheres or hyperboloids, but the study of "space" and the ways one space can be transformed into another, and the kinds of figures that arise from identifying "geodesics" and "equipotentials" in such spaces?
But before I can make further comments, I need to dig into your paper some more.
Joe
Andrius Kulikauskas
Jul 6, 2016, 8:09:30 PM7/6/16
to mathf...@googlegroups.com
Hi Joe,
Thank you for keeping in touch. I appreciate your thoughts.
I'm in the Lithuanian port of Klaipeda this week. Last Friday, at a
theology conference, I gave a talk on "God's dance":
http://www.ms.lt/sodas/Book/GodsDance
I need to add diagrams...
It's how I imagine God's point of view, how God investigates, "Is God
necessary? Would God be even if he wasn't?" I think this question
drives the unfolding of all kinds of conceptual structure that is at the
heart of my mathematics paper.
A main conclusion is that the basis for those four geometries and six
transformations are the ten ways of "loving all" as given by the "ten
commandments". There are four ways of "loving God" and six ways of
"loving your neighbor as yourself". And the upshot is that when we love
our neighbor, we oblige God to be good. So these can be understood as
constraints for God, commandments onto God, which flow out of our
positive behavior. Our negative behavior is of secondary importance and
the prohibitions likewise.
I have a lot to write about and a lot I want to work further on. But I
have to focus on my work-for-pay, helping write articles in substance
use/abuse/control. It's an opportunity to think about behavior and
ethics, which will be the second part of my book (after God's dance),
and related to those ten commandments/geometries & transformations.
I'm also reading about Clifford algebras... I'm intrigued by their
connection with Pascal's triangle.
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
For example, Steven Lehar says the complex numbers can be identified
with the row 1-2-1. I'm wondering if the quaternions can be related
with the row 1-3-3-1. And so on?
Andrius
Thank you for keeping in touch. I appreciate your thoughts.
I'm in the Lithuanian port of Klaipeda this week. Last Friday, at a
theology conference, I gave a talk on "God's dance":
http://www.ms.lt/sodas/Book/GodsDance
I need to add diagrams...
It's how I imagine God's point of view, how God investigates, "Is God
necessary? Would God be even if he wasn't?" I think this question
drives the unfolding of all kinds of conceptual structure that is at the
heart of my mathematics paper.
A main conclusion is that the basis for those four geometries and six
transformations are the ten ways of "loving all" as given by the "ten
commandments". There are four ways of "loving God" and six ways of
"loving your neighbor as yourself". And the upshot is that when we love
our neighbor, we oblige God to be good. So these can be understood as
constraints for God, commandments onto God, which flow out of our
positive behavior. Our negative behavior is of secondary importance and
the prohibitions likewise.
I have a lot to write about and a lot I want to work further on. But I
have to focus on my work-for-pay, helping write articles in substance
use/abuse/control. It's an opportunity to think about behavior and
ethics, which will be the second part of my book (after God's dance),
and related to those ten commandments/geometries & transformations.
I'm also reading about Clifford algebras... I'm intrigued by their
connection with Pascal's triangle.
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
For example, Steven Lehar says the complex numbers can be identified
with the row 1-2-1. I'm wondering if the quaternions can be related
with the row 1-3-3-1. And so on?
Andrius
2016.07.06 02:31, Joseph Austin rašė:
>
>> On Jun 27, 2016, at 2:40 AM, Andrius Kulikauskas <m...@ms.lt
>
>> On Jun 27, 2016, at 2:40 AM, Andrius Kulikauskas <m...@ms.lt
>> <mailto:m...@ms.lt>> wrote:
>>
>> I'm really interested to figure out the "six transformations" that I
>> mention below and in my essay. I think that Clifford algebras will
>> provide useful insights. I look forward to studying that together.
>
> Andrius,
> I've been meaning to respond to your paper, but haven't got my
> thoughts together yet.
> Just a couple of speculations:
>
> 1. I'm not a geometer, but as I remember HS geometry, it involved
> *constructions (*of geometric figures) and *deductions* (proofs).
>>
>> I'm really interested to figure out the "six transformations" that I
>> mention below and in my essay. I think that Clifford algebras will
>> provide useful insights. I look forward to studying that together.
>
> Andrius,
> I've been meaning to respond to your paper, but haven't got my
> thoughts together yet.
> Just a couple of speculations:
>
> 1. I'm not a geometer, but as I remember HS geometry, it involved
> --
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Joseph Austin
Jul 7, 2016, 8:29:48 AM7/7/16
to mathf...@googlegroups.com
On Jul 6, 2016, at 4:47 PM, Andrius Kulikauskas <m...@ms.lt> wrote:I'm also reading about Clifford algebras... I'm intrigued by their connection with Pascal's triangle.
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
For example, Steven Lehar says the complex numbers can be identified with the row 1-2-1. I'm wondering if the quaternions can be related with the row 1-3-3-1. And so on?
My understanding is that the correspondence comes from the dimensionality of space.
In 3D we have volume (1 orientation), area (3 orientations), length (3 orientations), and scalar (1 orientation).
Physical quantities can be associated with any of these.
In physics, in priciple, we observe how observable quantities interact, then go looking for a mathematical system that behaves that way.
So if we combine the motion of a boat (say going across still water such as a lake) and a flowing river,
we discover they combine "geometrically" with the resulting motion on a diagonal, and the math becomes what we call vectors.
In the case of magnets in proximity to electric currents, we observe "mangetic forces" perpendicular (or circling around) the flow of current,
and end up with Maxwell's equations, with a mysterious "cross-product" that has "ugly" mathematical properties. So Clifford come along and offers a more elegant way of understanding the mathematics. Quantum Mechanics introduces a lot more curious mathematical devices which, according to Hestenes,
can all be unified within the Clifford or "Geometric" product.
I favor the view that the N (whatever N is) dimensional universe, and everything in it, are always fully N dimensions, and the lower-dimensional quantities we measure are prjections or boundaries within those N dimensions. Which is to say that they represent a kind of "factoring" of the N-D object into a "product" of smaller-dimensional perceptions.
The full Geometric Product therefore hints at manifestations of physical reality in all the dimensionalities of the product--manifestations we may have at first missed or confused with a differet order due to our inappropriate mathematical models,
e.g. confusing a quantity related to a directed area (2D) with a normal vector (1D). Of course it made no intuitive sense to regard cicular motion around a line (angular momentum) as "pointing" in the direction of the axis, but with a representational repetoire limited to scalars and vectors, that's what we came up with! Clifford Algebra offers a more intuitively satisfying alternative, the oriented chiral area.
In quantum mechanics, we choose mathematical systems that have the behavior (group properties) of the physical observables,
whether or not we can attach intutive significance to the mathematical structures themselves. This lead to the principle "don't ask what it means,
just do the math." But with Geometric Algebra, we may at last have a mathematical framework that does the same math but also provides a structure that suggests an intuitive "meaning," specifically, a relationship to the geometry of "space".
This is why I am so intrigued by your classification of groups and transformations and geometry--perhaps it can all be unified within Geometric Algebra,
or if not, suggest a way of extending it.
Joe Austin
Andrius Kulikauskas
Jul 25, 2016, 10:09:48 AM7/25/16
to mathf...@googlegroups.com
Joe,
Thank you very much for your helpful letter. I will still have to
absorb it along with what I've been learning about Clifford algebras.
I found Alan Macdonald's videos 0-5 on Geometric Algebra very helpful.
https://www.youtube.com/watch?v=srwoPQfWWS8
I'm also learning a lot watching Norman Wildberger's videos. Here's a
couple on Lie groups and symmetry:
https://www.youtube.com/watch?v=MApjCNK_Nrk
It's outstanding how many subjects he covers with his videos.
I'm trying to think of what I'm looking for in math. I suppose I wish
for a unified understanding of math, what it is and what it is not, how
it unfolds. What is natural math (all of implicit math in the mind that
we use to figure things out) and what is contrived math (much of what
gets manipulated on paper) and how are they related?
Pascal's triangle seems very basic. At the heart is a recursion
relation. One way it appears is as in Euler's characteristic, the
formula that Vertices - Edges + Faces = 2. If we add the unique Center
and the unique Totality (the Volume), and we change the signs around,
then we get:
Center - Vertices + Edges - Faces + Totality = 0
which seems a lot more natural. And that's just one row (the fifth row)
in Pascal's triangle where we are expanding (1-1)^N.
There's a lot to study what Euler's characteristic means for different
shapes with holes, etc.
Just some thoughts for now.
Thank you very much for your helpful letter. I will still have to
absorb it along with what I've been learning about Clifford algebras.
I found Alan Macdonald's videos 0-5 on Geometric Algebra very helpful.
https://www.youtube.com/watch?v=srwoPQfWWS8
I'm also learning a lot watching Norman Wildberger's videos. Here's a
couple on Lie groups and symmetry:
https://www.youtube.com/watch?v=MApjCNK_Nrk
It's outstanding how many subjects he covers with his videos.
I'm trying to think of what I'm looking for in math. I suppose I wish
for a unified understanding of math, what it is and what it is not, how
it unfolds. What is natural math (all of implicit math in the mind that
we use to figure things out) and what is contrived math (much of what
gets manipulated on paper) and how are they related?
Pascal's triangle seems very basic. At the heart is a recursion
relation. One way it appears is as in Euler's characteristic, the
formula that Vertices - Edges + Faces = 2. If we add the unique Center
and the unique Totality (the Volume), and we change the signs around,
then we get:
Center - Vertices + Edges - Faces + Totality = 0
which seems a lot more natural. And that's just one row (the fifth row)
in Pascal's triangle where we are expanding (1-1)^N.
There's a lot to study what Euler's characteristic means for different
shapes with holes, etc.
Just some thoughts for now.
2016.07.07 15:29, Joseph Austin rašė:
>
>> On Jul 6, 2016, at 4:47 PM, Andrius Kulikauskas <m...@ms.lt
>
>> On Jul 6, 2016, at 4:47 PM, Andrius Kulikauskas <m...@ms.lt
kirby urner
Jul 25, 2016, 12:03:26 PM7/25/16
to mathf...@googlegroups.com
Pascal's triangle seems very basic. At the heart is a recursion relation. One way it appears is as in Euler's characteristic, the formula that Vertices - Edges + Faces = 2. If we add the unique Center and the unique Totality (the Volume), and we change the signs around, then we get:
Center - Vertices + Edges - Faces + Totality = 0
So you're saying Center + Totality == 2 ?
In Synergetics, we have the formula (2*P) * (F^2) + 2 where P is a composite made of low-order primes.[1]
This is used to compute the number of balls in a next layer, where the shape is governed by P.
When P = 5 the formula is of course 10 * F^2 + 2 giving the series 1, 12, 42, 92, 162... https://oeis.org/A005901
When P = 5 the formula is of course 10 * F^2 + 2 giving the series 1, 12, 42, 92, 162... https://oeis.org/A005901
The +2 is associated with axial opposite vertexes. When you want to spin a shape, pick two opposite vertexes.
The x2 is associated with 2nd power exponential magnification and shrinkage.
The general pattern: (multiplicative 2) * (shape controller) * (frequency)^2 + (additive 2)
multiplicative 2: convex / concave (relates to inside / outside)
additive 2: axial spin (rotation)
The additive 2 (polar) is then connected with V + F = E + 2 (same additive 2) e.g. we may consider N to mean "nonpolar" such that:
N + F = E
where N = V - 2
Frequency (F) is how much we subdivide, like when a hexapent cage becomes more and more like a sphere with an increase in F, with Descartes Deficit staying 720 degrees (one tetrahedron) throughout.
A "high frequency icosasphere" is the typical geodesic sphere (known as Class I in the literature).
The use of the term "frequency" is deliberate, for its connection to the electromagnetic spectrum etc.
Speaking of high frequency icosapheres (omni-triangulated almost-spheres): N:F:E == 1:2:3
i.e. the number of non-polar vertexes : number of facets : number of edges == 1:2:3 (ratio)
We're thinking of any conceptual system (belief system) as a network of nodes on a "planet" (Little Prince) meaning a graph that "connects around in all circumferential directions".
A typical scenario is to start with an spherically symmetrical polyhedron and jab sticks through opposite edges, opposite face centers, opposite vertexes.
In spinning around each of these axes, we generate great circle networks.
The cuboctahedron generates 25 such great circles; the icosahedron generates 31.
We may superimpose these two networks. We may spin again, around intersection points, to create secondary networks.
We find LCD (lowest common denominator) triangles, left and right, which, repeated, recreate these networks.
Synergetics contains rather extensive tabulations of these LCD triangles.
Kirby
[1] http://www.4dsolutions.net/synergetica/synergetica2.html#top
[1] http://www.4dsolutions.net/synergetica/synergetica2.html#top
By setting P=5 in the expression 2PF^2+2, the number of spheres in each successive layer of the cuboctahedron may be derived. F stands for "frequency" and is the number of intervals between spheres along any edge of the cuboctahedron or, alternatively, is the number of intervals along any radius between a corner sphere and the central one.
In a virus, the RNA-protecting shell or capsid is made from sub-units called capsomeres. By taking F as the number of between-capsomere intervals, and using 10F^2+2 on capsid "shell frequencies" of 1,2,3,4,5 and 6, we obtain corresponding counts of 12, 42, 92, 162, 252 and 812 capsomeres. "All of these numbers are in fact found in actual viruses, 12 for certain bacteriophages, 42 for wart viruses, 92 for reovirus, 162 for herpesvirus, 252 for adenovirus and 812 for a virus attacking crane-flies (Tipula or daddy-long-legs)" - The Natural History of Viruses by C.H. Andrews (W.W. Norton R Co., 1967).
Andrius Kulikauskas
Jul 25, 2016, 1:47:15 PM7/25/16
to mathf...@googlegroups.com
Kirby,
It will be very interesting to go through your detailed letter. I was
struck by your operations +2 and x2, so I want to share my thought on that.
First, I want to quickly explain what I mean by the "Center" and the
"Totality" which you asked about. It all started with my interest in
the "simplex" which came from my interest in trying to understand the
"field with one element". A basic situation where the latter comes up
is when we count the number of k-dimensional subspaces of an
n-dimensional vector space over a finite field with q elements. That
number is given by the Gaussian binomial coefficient which is a
polynomial in q. And when q=1 we get the usual binomial coefficients,
given by Pascal's triangle, which count the number of subsets of size k
of a set of size n. The upshot of which is that underlying vector
spaces is an intrinsically ordered basis e1, e2, ..., en. But when you
only have one choice of scalar (the field has size 1), then you really
don't have a choice, and so the order of the basis doesn't matter, and
so we get subsets of an (unordered) set instead of sublists of an
(ordered) list.
Now the "field with one element" also comes up in all kinds of algebraic
geometry questions, which is how I learned about the simplex. And when
I read about the simplex, I saw that its parts (its sub-simplexes), are
counted by Pascal's triangle. See the table here:
https://en.wikipedia.org/wiki/Simplex
Also, I was interested because the simplex is a generalization of the
tetrahedron, which you've been championing. Indeed, note that we can
read off from Pascal's triangle that a tetrahedron has: 1 (what?) 4
vertices 6 edges 4 faces 1 volume. So what is the "what"? Note that:
* each vertex is a zero-dimensional 0-simplex
* each edge is a one-dimensional 1-simplex with 2 vertices connected to
each other
* each face is a two-dimensional 2-simplex, a triangle with 3 vertices
connected to each other in all possible ways
* the volume is a three-dimensional 3-simplex, a tetrahedron with 4
vertices connected to each other in all possible ways.
The question then is what is a negative-one-dimensional -1 simplex? I
realized also that the parts of the simplex are generated by the
expansion (include + exclude)^N. For example, if we expand:
(include OR exclude) AND (include OR exclude) AND (include OR exclude)
AND (include OR exclude)
then each term is a subsimplex that is defined by a set of "included"
vertices. So there are:
1 way to include NO vertices
4 ways to include 1 vertex (they are the vertices)
6 ways to include 2 vertices (they are the edges)
4 ways to include 3 vertices (they are the faces)
1 way to include all 4 vertices (and the entire tetrahedron)
So what is the unique way to include NO vertices of this simplex? and
any simplex? The Wikipedia article states that nobody really knows.
But I reckon that it is the unique "center" of the simplex. Note that
each simplex has a unique center but that it generally isn't any of the
vertices. And then I can imagine that center as the initial
negative-one dimensional simplex that generates each new vertex and
shifts accordingly.
Note also that if we draw the center, for example, the center of the
tetrahedron, then we can think of it as the new, in this case, "fifth"
vertex. And if we link it to the other vertices then we can imagine the
new edges, the new faces, and the four new cells, the fifth cell being
the original tetrahedron which will be apparent when we move the new
vertex to the fourth dimension. In this sense, truly the "center" is a
negative-one-dimension! And it lets us think in the next dimension.
Then the dual of the center is the way to include all of the vertices,
which in the case of the tetrahedron is the volume or cell, but in
general, I call the "totality".
So I think that's a nice geometric way to think. And it carries over to
other infinite families of polytopes. The cross-polytopes (like the
octahedron) are gotten by adding not one vertex at a time but rather two
vertices at a time, as if in opposite directions. So the resulting
Pascal's triangle gets skewed by powers of 2, as I note in the
corresponding diagram of my paper:
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
http://www.ms.lt/derlius/crosspolytopes.png
And studying that diagram and "Pascal's triangle" leads to the
conclusion that the octahedron etc. has no volume! Instead, it has 8
faces. In other words, the cross-polytopes have no "totality", if we
are to consider them defined by their "Pascal's triangle". And the
cubes are the dual, and they have no "center", likewise.
I was struck by your operations +2 and x2. I realized that the
cross-polytopes are built up by the operation +2. We keep adding a pair
of vertices (like a particle and anti-particle, or like the two rays of
a line). For the cube, we keep multiplying the vertices by 2 as we
build it. I suppose we can think of this version of Pascal's triangle
as interpolating between +2 and x2. That is, the cross-polytopes and
cubes are defined by the same version of Pascal's triangle, just from
opposite sides.
Does this help make more sense of what I mean by "center" and "totality"?
It will be very interesting to go through your detailed letter. I was
struck by your operations +2 and x2, so I want to share my thought on that.
First, I want to quickly explain what I mean by the "Center" and the
"Totality" which you asked about. It all started with my interest in
the "simplex" which came from my interest in trying to understand the
"field with one element". A basic situation where the latter comes up
is when we count the number of k-dimensional subspaces of an
n-dimensional vector space over a finite field with q elements. That
number is given by the Gaussian binomial coefficient which is a
polynomial in q. And when q=1 we get the usual binomial coefficients,
given by Pascal's triangle, which count the number of subsets of size k
of a set of size n. The upshot of which is that underlying vector
spaces is an intrinsically ordered basis e1, e2, ..., en. But when you
only have one choice of scalar (the field has size 1), then you really
don't have a choice, and so the order of the basis doesn't matter, and
so we get subsets of an (unordered) set instead of sublists of an
(ordered) list.
Now the "field with one element" also comes up in all kinds of algebraic
geometry questions, which is how I learned about the simplex. And when
I read about the simplex, I saw that its parts (its sub-simplexes), are
counted by Pascal's triangle. See the table here:
https://en.wikipedia.org/wiki/Simplex
Also, I was interested because the simplex is a generalization of the
tetrahedron, which you've been championing. Indeed, note that we can
read off from Pascal's triangle that a tetrahedron has: 1 (what?) 4
vertices 6 edges 4 faces 1 volume. So what is the "what"? Note that:
* each vertex is a zero-dimensional 0-simplex
* each edge is a one-dimensional 1-simplex with 2 vertices connected to
each other
* each face is a two-dimensional 2-simplex, a triangle with 3 vertices
connected to each other in all possible ways
* the volume is a three-dimensional 3-simplex, a tetrahedron with 4
vertices connected to each other in all possible ways.
The question then is what is a negative-one-dimensional -1 simplex? I
realized also that the parts of the simplex are generated by the
expansion (include + exclude)^N. For example, if we expand:
(include OR exclude) AND (include OR exclude) AND (include OR exclude)
AND (include OR exclude)
then each term is a subsimplex that is defined by a set of "included"
vertices. So there are:
1 way to include NO vertices
4 ways to include 1 vertex (they are the vertices)
6 ways to include 2 vertices (they are the edges)
4 ways to include 3 vertices (they are the faces)
1 way to include all 4 vertices (and the entire tetrahedron)
So what is the unique way to include NO vertices of this simplex? and
any simplex? The Wikipedia article states that nobody really knows.
But I reckon that it is the unique "center" of the simplex. Note that
each simplex has a unique center but that it generally isn't any of the
vertices. And then I can imagine that center as the initial
negative-one dimensional simplex that generates each new vertex and
shifts accordingly.
Note also that if we draw the center, for example, the center of the
tetrahedron, then we can think of it as the new, in this case, "fifth"
vertex. And if we link it to the other vertices then we can imagine the
new edges, the new faces, and the four new cells, the fifth cell being
the original tetrahedron which will be apparent when we move the new
vertex to the fourth dimension. In this sense, truly the "center" is a
negative-one-dimension! And it lets us think in the next dimension.
Then the dual of the center is the way to include all of the vertices,
which in the case of the tetrahedron is the volume or cell, but in
general, I call the "totality".
So I think that's a nice geometric way to think. And it carries over to
other infinite families of polytopes. The cross-polytopes (like the
octahedron) are gotten by adding not one vertex at a time but rather two
vertices at a time, as if in opposite directions. So the resulting
Pascal's triangle gets skewed by powers of 2, as I note in the
corresponding diagram of my paper:
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
http://www.ms.lt/derlius/crosspolytopes.png
And studying that diagram and "Pascal's triangle" leads to the
conclusion that the octahedron etc. has no volume! Instead, it has 8
faces. In other words, the cross-polytopes have no "totality", if we
are to consider them defined by their "Pascal's triangle". And the
cubes are the dual, and they have no "center", likewise.
I was struck by your operations +2 and x2. I realized that the
cross-polytopes are built up by the operation +2. We keep adding a pair
of vertices (like a particle and anti-particle, or like the two rays of
a line). For the cube, we keep multiplying the vertices by 2 as we
build it. I suppose we can think of this version of Pascal's triangle
as interpolating between +2 and x2. That is, the cross-polytopes and
cubes are defined by the same version of Pascal's triangle, just from
opposite sides.
Does this help make more sense of what I mean by "center" and "totality"?
kirby urner
Jul 25, 2016, 2:31:06 PM7/25/16
to mathf...@googlegroups.com
On Mon, Jul 25, 2016 at 10:30 AM, Andrius Kulikauskas <m...@ms.lt> wrote:
Kirby,
It will be very interesting to go through your detailed letter. I was struck by your operations +2 and x2, so I want to share my thought on that.
First, I want to quickly explain what I mean by the "Center" and the "Totality" which you asked about. It all started with my interest in the "simplex" which came from my interest in trying to understand the "field with one element". A basic situation where the latter comes up is when we count the number of k-dimensional subspaces of an n-dimensional vector space over a finite field with q elements. That number is given by the Gaussian binomial coefficient which is a polynomial in q. And when q=1 we get the usual binomial coefficients, given by Pascal's triangle, which count the number of subsets of size k of a set of size n. The upshot of which is that underlying vector spaces is an intrinsically ordered basis e1, e2, ..., en. But when you only have one choice of scalar (the field has size 1), then you really don't have a choice, and so the order of the basis doesn't matter, and so we get subsets of an (unordered) set instead of sublists of an (ordered) list.
Now the "field with one element" also comes up in all kinds of algebraic geometry questions, which is how I learned about the simplex. And when I read about the simplex, I saw that its parts (its sub-simplexes), are counted by Pascal's triangle. See the table here:
https://en.wikipedia.org/wiki/Simplex
Also, I was interested because the simplex is a generalization of the tetrahedron, which you've been championing. Indeed, note that we can read off from Pascal's triangle that a tetrahedron has: 1 (what?) 4 vertices 6 edges 4 faces 1 volume. So what is the "what"? Note that:
* each vertex is a zero-dimensional 0-simplex
Alternative: each vertex is a 3-simplex of higher frequency (too small to "tune in")
* each edge is a one-dimensional 1-simplex with 2 vertices connected to each other
Each edge is a stretched out 3-simplex, like a pencil.
* each face is a two-dimensional 2-simplex, a triangle with 3 vertices connected to each other in all possible ways
Faces are windows, empty space. The simplex is a wireframe only.
* the volume is a three-dimensional 3-simplex, a tetrahedron with 4 vertices connected to each other in all possible ways.
The 3-simplex is a topological minimum for conceptuality i.e. all Russian Dolls are inside Russian Dolls and "the container" is the beginning of "concept" in the imagination.
Given everything is a lump (Menger), we have no 2-simplex or 1-simplex or 0-simplex. The "number of coordinates" is immaterial as in a finite space all locations may be serialized i.e. one may "tour" the points along some "itinerary". There's no distinguishing "dimensionality" based on coordinate addressing.
But that's just another namespace, one among many. Some IT shop / casino is using that, we might imagine, in ancient China.
We'll define the simplex as 4D because we want something as primitive as 3D which was based on cubic thinking. We talk about the 3D XYZ and 4D IVM as inhabiting the same conceptual Russian Doll space of contained containers. That's the only "theater" we've got, either purely conceptual (cartoon / claymation) or "made real" (energy added).
We'll define the simplex as 4D because we want something as primitive as 3D which was based on cubic thinking. We talk about the 3D XYZ and 4D IVM as inhabiting the same conceptual Russian Doll space of contained containers. That's the only "theater" we've got, either purely conceptual (cartoon / claymation) or "made real" (energy added).
Conceptual / Time-Size independent Tetrahedron <-- "pre-frequency": 4D
===============================================
Special-case mortal / experiential / informational Tetrahedron <- "frequency added": 4D+
Calling it 4D is more about putting distance, in semantic space, between a philosophy and mainstream mathematics, which is free to continue with exactly its own preferred conceptual apparatus.
We're free to jump into and out of that sandbox at will.
Starting fresh with a deliberately remote vocabulary is a way to disentangle a philosophy with pre-existing chalkboard scribbles.
Relevant psychology: http://www.grunch.net/synergetics/psych.html
I was struck by your operations +2 and x2. I realized that the cross-polytopes are built up by the operation +2. We keep adding a pair of vertices (like a particle and anti-particle, or like the two rays of a line). For the cube, we keep multiplying the vertices by 2 as we build it. I suppose we can think of this version of Pascal's triangle as interpolating between +2 and x2. That is, the cross-polytopes and cubes are defined by the same version of Pascal's triangle, just from opposite sides.
Yes, I associate +2 with translation, as all lateral movement may be cast as rotation, with a very large radius making for almost-straight paths ("purely straight" is not needed, lines remain "deliberately non-straight" in all cases).
I associate x2 with scaling, with 2nd power growth in area, with 3rd power growth in volume. The power rule.
The 1, 12, 42, 92, 162... series is 2nd power driven whereas the volume of the corresponding cuboctahedron is 20*F^3, where 12-around-1 is volume 20. Double the edges and the volume shoots up to 160 tetravolumes.
The 4D tetrahedron (what you call simplex-3) is the basis for our all-positive-number 4-tuples around origin / center (0,0,0,0).
Four basis vectors divide space into four quadrants, only three of which are needed to address points in any one quadrant, so every 4-tuple address has at least one zero in it (the vector entirely in the opposite quadrant). I have algorithms to jump back and forth from 4D IVM 4-tuples to 3D XYZ 3-tuples.
Does this help make more sense of what I mean by "center" and "totality"?
I think so.
In my own geometric philosophy (a theater for psychology, a framework for thinking), I don't use the addition of perpendiculars to define higher dimensions.
I see n-tuple data structures with an applied metric (Pythagorean) but don't look for "four mutually orthogonal perpendiculars" anywhere in this philosophy. That doesn't keep me from using polytope data structures. I just don't make "existence claims" for "higher dimensions" ("utility" and "existence" are orthogonal concepts).
I see n-tuple data structures with an applied metric (Pythagorean) but don't look for "four mutually orthogonal perpendiculars" anywhere in this philosophy. That doesn't keep me from using polytope data structures. I just don't make "existence claims" for "higher dimensions" ("utility" and "existence" are orthogonal concepts).
Again, the mainstream is free to continue with N-D extended Euclidean geometry, no one is damming that river, mine either.
Kirby
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