Posts tagged with 4D

Art, unlike life, is measured by its maxima.





[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …
A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …
We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory
(I rearranged their words liberally but the substance is theirs.)
in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.

[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …

A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …

We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory

(I rearranged their words liberally but the substance is theirs.)

in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.




Tony Robbin

hi-res




J is hott. Some highlights from the Wikipedia article and J's homepage:

  • you can do a lot with just a few characters in J. Define a moving average in 8 characters, including spaces, for example.
  • Have you ever felt like whether it’s Java or C, Python or Ruby, all these languages are just the Same Old Thing?

J makes thinking in high-dimensional arrays easy.

  1. The sentence .i 7 8 means “Show me a 7×8 two-array” (ok, “matrix” but … matrices are verbs and arrays are nouns)
  2. The sentence .i 7 8 3 means “Show me a 7×8×3 three-array”.
  3. The sentence .i 7 8 3 4 13 2 66 means "Show me a 7×8×3×4×13×2×66 dimensional seven-array”.

I won’t reprint the long outputs but here’s a shorter one.

   i.4 5 3
 0  1  2
 3  4  5
 6  7  8
 9 10 11
12 13 14

15 16 17
18 19 20
21 22 23
24 25 26
27 28 29

30 31 32
33 34 35
36 37 38
39 40 41
42 43 44

45 46 47
48 49 50
51 52 53
54 55 56
57 58 59

And another for clarity:

   i.3 5 4
 0  1  2  3
 4  5  6  7
 8  9 10 11
12 13 14 15
16 17 18 19

20 21 22 23
24 25 26 27
28 29 30 31
32 33 34 35
36 37 38 39

40 41 42 43
44 45 46 47
48 49 50 51
52 53 54 55
56 57 58 59

This is reminiscent of using R's combn function to visualise higher-dimensional stuff, right?

I guess this is how computers think all the time! I wonder what they say about us when we’re not around.




Apparently the “extra” dimensions of string theory are only supposed to be a few millimetres thick.

If that’s the case, could you dodge a bullet by moving a millimetre in the 10th dimension?

I guess it would depend on how wide the bullet and your liver are in the 10th dimension. Could lead to an interesting superpower: move in hidden dimensions.

  • The hero wouldn’t be invulnerable but would be less vulnerable. Still get an exit wound but maybe she’d only be grazed through the interior rather than completely ripped to shreds.
  • Still worth dodging/blocking a fist in the normal-sized 3 dimensions, but even a “direct” uppercut or body blow could become more of a “glancing blow” if she dodged in the thin directions. (NB: If ∃ 7 extra thin dimensions, each 1mm wide, and she dodged at once to “the other side” of all 7 at once—assuming, as well, that we’re “all the way to one side” of each of the extra dimensions—then she’d have made a total distance of √7mm between her and us.)
  • Joint locks—could she put someone in a joint lock they couldn’t get out of? Couldn’t she also get out of joint locks that no-one else could?
  • Couldn’t become invisible but become less visible.
  • Couldn’t pass through walls but could reach into crevices easier.
  • Could swim faster (twist her torso in the 10th dimension so the hands & feet still pull water, but less resistance on the mass of the body).

Am I thinking about this right?







John Baez:

To get [the D4 lattice], first take a bunch of equal-sized spheres in 4 dimensions. Stack them in a hypercubical pattern, so their centers lie at the points with integer coordinates. A bit surprisingly, there’s a lot of room left over - enough to fit in another copy of this whole pattern: a bunch of spheres whose centers lie at the points with half-integer coordinates!
If you stick in these extra spheres, you get the densest known packing of spheres in 4 dimensions. Their centers form the “D4 lattice”. It’s an easy exercise to check that each sphere touches 24 others. The centers of these 24 are the vertices of a marvelous shape called the “24-cell” - one of the six 4-dimensional Platonic solids. It looks like this:







…
Colour images by eusebia

John Baez:

To get [the D4 lattice], first take a bunch of equal-sized spheres in 4 dimensions. Stack them in a hypercubical pattern, so their centers lie at the points with integer coordinates. A bit surprisingly, there’s a lot of room left over - enough to fit in another copy of this whole pattern: a bunch of spheres whose centers lie at the points with half-integer coordinates!

If you stick in these extra spheres, you get the densest known packing of spheres in 4 dimensions. Their centers form the “D4 lattice”. It’s an easy exercise to check that each sphere touches 24 others. The centers of these 24 are the vertices of a marvelous shape called the “24-cell” - one of the six 4-dimensional Platonic solids. It looks like this:

Colour images by eusebia




Buckminster Fuller, 4D Tower: Time Interval 1 Meter, 1928
Gouache and graphite over positive Photostat on paper, 14 x 10 7/8”, Avery Architectural and Fine Arts Library, Columbia University in the City of New York, Image courtesy Avery Architectural and Fine Arts Library, Columbia University in the City of New York.
via newspace

Buckminster Fuller, 4D Tower: Time Interval 1 Meter, 1928

Gouache and graphite over positive Photostat on paper, 14 x 10 7/8”, Avery Architectural and Fine Arts Library, Columbia University in the City of New York, Image courtesy Avery Architectural and Fine Arts Library, Columbia University in the City of New York.

via newspace


hi-res




If you don’t read Sketches of Low-Dimensional Topology already, your personal astrologer advises that you start doing so before the next ecliptic.

Sample pics and text:

Klein Bottle000

Picture 9

Picture 5

Picture 26

18foldliftP-237rotate

The 3-sphere as two balls

Picture 14

Trefoil with monodromy sweep ghostedTrefoil with monodromy annulusTrefoild with mondoromy sweep 2

Trefoil with monodromy annulus 2

Lineartoruswithctctstr

Picture 47

Medial surface of the Whitehead link.

StandardContactStructure4 (by epsilon_is_afraid_of_zeta)

annulusinsolidtorus

annulusinsolidtorus-chopped

6jsymbols-equivalence

image

image

Photo 8

image

image
ctctstr4, originally uploaded by epsilon_is_afraid_of_zeta.

ctctstr2

ctctstr3

conformal supports

T3singlefullside2

hopf band monodromy

Picture 7

Picture 6

Ko's Octahedral relation-2

7_7-Sutured Manifold Decomp

P-237onfiber-trefoil

View this 3-manifold as an interval of concentric spheres where you have to imagine gluing the inner sphere to the outer sphere.

aaaaqqaq

Near each point on a singular fiber, a regular fiber passes by some fixed number of times, the order of the singular fiber. In the picture above this number is 5 for both singular fibers.

Here they have order 2.
2 with 5 copies origview

Here they have order 3.
3 with 4 copies origview

Here they have order 1 and so they aren’t that special. A homeomorphism would make all the fibers appear as radial arcs, the S^1‘s of the S^1 x S^2.
1 with 5 copies origview

For the conference honoring the 60th birthday ofCaroline Series(only a German wiki?!?), I was one of a handful asked to contribute pictures inspired by her work. First up is my contribution followed by a description. After that are a few more.

Triangulation I, II, III, IV

WHtangle-top

WHBig2

WHdeepinside7

Whiteheadtangle-squarespherebig

Whiteheadtangle-squareF2

Whitehead flythrough

 

I don’t think I will ever be this awesome. Not only are the pictures way easier to understand than some scary symbols, but the text explains their meaning really clearly and in not-too-many words.

Pass the acid—I mean, the advanced mathematics—please. People thought Grigory Perelman was crazy for turning down a million-dollar prize and living like an ascetic. "Why should I jump for a million dollars, when I can control the vacuum space in between the quarks of the universe?” is my paraphrase of his reply. I don’t have a million dollars, nor do I understand all of this ring fiber link knot book page contact braid surgery stuff. But right now I’m honestly not sure which I would prefer: the imagination, or the dinero.

Thanks to Maxime (@2_43112609_1 on twitter) for the pointer.




Three and a half centuries before Christ, Plato outlined his conception of a great society in The Republic.

imageimage

The important decisions are made by “philosopher-kings”—an overclass who arguably represent Plato’s idea of human perfection.

image

The philosopher-kings train both their bodies and their minds—like Leonardo da Vinci, or a young American hoping to attend Harvard. The pinnacle of their education is geometry.

image

Although circles don’t actually exist, one can conceive of a circle and Plato thinks there’s something important about that.

image

Taking the reins back from Plato: what I’ve found in my life is that concepts I learn, feelings I feel, stories I hear, experiences I remember, newspaper crap or history books I read — all inputs expand my own private languageAlmost like adding elements to a basis?  Doug Hofstadter told a story that illustrates “personal metaphors” well.

As I learn more mathematics, I find my internal vocabulary expanding quite a lot — faster than through any other learning activity, and definitely faster than just-experiencing life passively. (“Personal” and “private”really are the right words for it; sharing these thoughts is really hard! Hence the writing.)

 

That’s background. Now, on to the topic of originality.

Question: Are there any truly new ideas? I sometimes sense that a few Big Ideas occur to many, many people, and that the history of philosophy is just an exercise in rehashing them through different filters. I got the same feeling reading A Canticle for Leibowitz, a work of speculative fiction that spans several millennia and displays what I felt was a 16-year-old history nerd’s overly simplistic view of the broad trends of history.

image

Is there any music that isn’t totally derivative of something else? Questions like that turn artists to LSD in search of originality.

image

I have a different suggestion: there are truly new ideas today, ideas that demonstrably could never have occurred to Plato, and they come from 20th-century mathematics. (I’m not saying there aren’t new ideas elsewhere—just that with the mathematical ideas it’s quite clear that nobody could have thought of them before.) So we of the 21st century essentially have a reserve of raw idea-ore which we can mine and smelt for use in some creative pursuit—even if it’s just thinking about life differently.

 

I’m sort of relying on the Edward Gorey theory of creativity here: that people don’t necessarily generate original ideas, but they can filter their sensory input, and refilter / resample their internal dynamics, and still make output that’s visibly different than everything else that’s come before.

image

Does a unique piece of music count as original, even if it’s just a convex combination of musics that previously existed? In the biopic about him, Philip Glass described his music as a fusion of East and West (like ragas and haute couture strict piano).

image

In that case, throwing some indisputably new ideas into the hopper (adding more orthogonal elements to the basis) has to increase the volume of the space of potential outputs.