anonymous said:

more 4D moments please!! i absolutely adore them ^ ^anonymous said:

more 4D moments please!! i absolutely adore them ^ ^…and one more, by

**very**popular request:- prior to prep time for hair, a female individual absolutely
*insisted*in having her hair combed**exactly**25 times before beginning to styling process. - one particular individual i worked with regularly,
**never failed**to smear, smudge, or rub off their fully stage prepped make up before a performance or recording. - that same individual
**always**carried a small compact mirror in their pocket/bag*everywhere*they went. - an individual who was fully fluent in english favored popular rap and hiphop songs of american artists. they would
**purposely**teach members of their group wrong pronunciation of english lyrics for their own entertainment. - upon visiting different cities (including overseas), one individual habitually collected phone cases for their phone, from
*every*location. - with a particular group
**always**being over energetic and rowdy in the prep room and/or practice room, the leader of the group would get their attention by flickering the lights several times. - during commute to local events and venues, one individual
**always insisted**on sitting in a specific seat in the van because they believed it would bring good luck to the event and performance. - one particular individual would use
*no less than*three tissues to blow/wipe their nose after each time they sneezed.

- prior to prep time for hair, a female individual absolutely
- Hacks for thinking about high-dimensional space
High-dimensional Euclidean space is ℝ×ℝ×ℝ×ℝ×ℝ×…. Cartesian product of many continuous quantities.

You are already familiar with the concept via “an arbitrary number of sliders” or “an arbitrary number of columns in a spreadsheet”. Pivot tables are the normal way business people navigate a low-dimensional subset of a high-dimensional space.

External imageExternal imageExternal imageExternal imageExternal imageOne database or slider-config would be one point in a high-dimensional space. But what do these spaces look like overall?

# Lots of Corners, Lots of Directions

Think about the graph-skeleton of cubes and squares. Zoom in on a corner.

Look at how the number of neighbours increases as you increase dimension. The corner of a box in 2-D has two sticks coming out of it; in 3-D three sticks come out of it. In 30 dimensions there would be 30 sticks coming out of each ball. (Toward the interior. You’d need 2•30 sticks if we’re talking about a grid rather than a box.)

The sticks are dual to the corners. The number of corners

**doubles**each time you go up a dimension. Imagine`. → | → □ → cube`

as a video.External imageExternal imageExternal imageThe corners are chosen from the power set of

`{} → {0,1} → {0,1}² → {0,1}³`

and so on.- You can also think about numbering each corner of a cube: let’s say
`(1,0,1) ≝ (top,left,back)`

so that`(0,1,1)`

would be`(bottom,right,back)`

. - There are
`2•2•2`

corners to a cube, - just like there are
`2•2•2`

ways to for three switches to be flipped, `2•2•2`

choices of sign in ℝ³,- and
`2•2•2`

ways for three sliders`|||`

to be maxed-out.

But once you added a fourth dimension you would need to add another pair of direction words—how about “ana” and “kata”?—doubling the number of options by going up a dimension. (That’s why there are 2⁴=16 personality groups in the MBTI typology.)

# More on the outside, less on the inside.

Think about a Matryushka of nested spheres ⬤⊂⬤⊂⬤⊂… or, ok, even dolls.

External imageThe 10th one requires more plastic than the 1st one.

External image

Picture by Borb. Licensed under CC BY-SA 3.0 via Wikimedia Commons.In 3-D a two-dimensional surface gets larger as

`radius²`

. In 30 dimensions surface areas will go up like`radius²⁹`

.# Fatter at the equator.

Circles of latitude are wider in the middle of the ball than at the top.

External imageBut this is a consequence of the coördinatisation

`(lat,long)`

, not of the object itself. A hectare in Finland is as big as a hectare in Kenya. And I’m not suddenly faster running in Finland either.(Manifolds solve the problem by patching together different charts, each of which shows one area more faithfully.)

External imageExternal imageExternal image# The sphere is tiny or the cube’s mass is in its spikes.

This is justified by comparing spheres with cubes.

External imageIn the 3-D the corners of the cube stick out beyond the sphere. I could measure the size of these corners by putting another sphere outside the corners, and calculating

`volume(circumsphere) − volume(insphere)`

. The Earth has more volume in its mantle than in its inner core.External imageCombine this with what I said above in the Beatles Matryushka section. I’m seeing the mantle and core as successions of nested shells of

`volume ≈ ∫radius²`

, each bigger than the last. Once again this pattern will only get more extreme in high dimensions.# Simplices

These are the spaces on which probability happens. In 3 variables I can think about

`(⅓,⅓,⅓)`

,`(⅔,0,⅓)`

,`(0,1,0)`

, and`(0,0,1)`

.External imageDo they have to get more complicated when we pad them with zeroes?

`(0,0,⅓,0,0,⅓,0,0,0,⅓)`

and`(0,0,⅔,0,0,0,0,0,0,⅓)`

tell me that a 10-simplex has 10 choose 3 = 120 3-simplices embedded in it. But`(.1,.1,.1,.1,.1,.1,.1,.1,.1,.1)`

tells me it contains more than that.I can still think about the 10-simplex as connections between

`(1,0,0,0,0,0,0,0,0,0)`

,`(0,1,0,0,0,0,0,0,0,0)`

, …`(0,0,0,0,0,0,0,0,0,1)`

, but now there are more ways to connect—since each of those points has up to 9 buddies it can be heading toward at a time.External imageWhen 2 entries are nonzero, that’s a 2-D surface. When 7 entries are nonzero, that’s a 7-D surface. (Just think of 7 sliders.)

External image# 3-Spheres on Up

**Added, 2015:**Don’t listen to me; listen to Jack Milnor.All of this mention of spheres, it’d be nice to have an alternative description to

`w²+x²+y²+z²=1`

and`sin θ₁ sin θ₂ sin θ₃ … cos θₙ`

, which I’ve always found hard to digest.External imageI know that

`S¹×S¹`

isn’t`S²`

, because that would be a torus instead. So what*do*I combine, and how, to get S²?External imageI like the Drawstring Bag model:

That takes me from the open disk to S² (= the

*surface*of the Earth—sans magma). This is a topological construction so small perturbations are ok. Calling the geoid a “sphere” is ok. The disk doesn’t have to be exactly circular.External imageTo get to the 3-sphere I then imagine a

*solid*sphere (= Earth + magma), attach*another*point ∞, and**identify all of the Earth’s crust with ∞.**Going back to the open disk, here’s another imagination-exercise I do to clear this up for myself. In some of the early Mario games, a glitch you could do was to put half of Mario’s body on one side of the screen and half on the right. This is what I think of when I imagine two sides of a square being identified.

External image(You can play inside an octagon-with-identifications too. Thicken this octagon and imagine yourself flying around inside it. Imagine waving your arm through eg

`a̅₁`

. Where does it come out? Turn your head and watch what it’s doing. BTW, just like Mario and Pac–Man, this identification manifold was considered by entertainment artists around the same time period—remember Scooby-Doo and the Gang running in and out of hotel doors? But the animators will change the rules either over time or for different people or introduce noncommutativity—part of the humour, which proves that people*do*intuitively understand identification polyhedra (as well as, obviously, object permanence), or else those changes wouldn’t be funny.)

“DunceHatSpace”. Licensed under Public Domain via Wikimedia Commons.To do this with the drawstring-bag, I imagine a circle of friends. Even though they

*look*like they’re standing all the way across the disk from each other, they’re actually immediately next to each other at ★, and they could make a ring and touch hands and dance around it.Or, a solitary explorer at ★ could take small steps and

*look*, in the disk model, like he’s jumped all the way across the Earth. He stretches his hand out and, like Mario, it shows up on the other side. But this is just an aberrance of the disk model, not of S². (This also tells you how the gridding should go on the disk.)So in symbols, T² = ◯×◯ ≠ S² = ⬤/◯ = ⬤/∂⬤ (The pattern

`Im ƒ / ker ƒ`

is much broader, including the`+c`

of integration.)# Specific dimension sometimes matters.

A lot of what I’m walking you through is

`dimension → ∞`

kind of stuff. But sometimes odd or even matters. And sometimes there are structures that show up in like dimension 7 or even seemingly random high numbers. Just a warning.# Computing on the surface of a high-dimensional sphere.

Persi Diaconis explains how to do this. The trick is to think of “matrices with determinant one” as the group

`SO(n)`

, which represents either a point on a sphere or the action of twisting a sphere so ★ moves to that point. (The “noun” and “verb” versions are so easily interchangeable that mathematicians will usually elide the two.) If you allow determinant ±1, that’s`O(n)`

—so allowing reflection—and`O(n)`

is what the Gram-Schmidt algorithm reduces matrices to in order to solve`[A]⋅x⃗ = b⃗`

.To do what Dr Diaconis says in

`R`

you would run this 2-liner:`n % matrix(n,n) %>% qr %>% qr.Q`

which you can verify has determinant ±1 and is orthogonal with things like:

`rnorm(n**2) %>% matrix(n,n) %>% qr %>% qr.Q -> diaconis diaconis %>% eigen {diaconis %>% eigen}$values %>% Mod #eigenvalues of an O(n) are length 1 diaconis %>% det diaconis %>% crossprod %>% zapsmall #diaconis %*% t(diaconis)`

@zentree @isomorphisms in stat computing I get students to find nearest neighbor distances for points from U[0,1]^k, k=1,2,10,100

— Thomas Lumley (@tslumley)

June 2, 2015@isomorphisms For stats, the fact that all points approx equidistant is most important @zentree

— Thomas Lumley (@tslumley)

June 2, 2015# Use rotations.

A heap of linear algebra—and therefore the logic of high-dimensional Euclidean space—is simplified if you can imagine quotienting by

`SO(n)`

.Having the drawstring-bag model and the stick+ball model

*and*the ability to compute examples with matrices*and*the rectangular-grid model helps me imagine high-dimensional space. Hope that was useful for you.## Next steps

- Tits Buildings
- Coxeter-Dynkin diagrams
- Polytopes
- cell complexes

- You can also think about numbering each corner of a cube: let’s say
## In my philosophy class today:

The four dimensions and the difficulty for us, inhabitants of a 3D world, it is almost impossible for us to imagine how things look in 4D.

In the same way, if someone who lives in a flat dimension can not understand our world and its beauty.

Now, the Tesseract is an object that can only be appreciated and understood in a fourth dimension.

Loki saw this, saw things that mortals could not imagine.

He saw and learned beyond the known, he was able to do it.

I refuse to believe that such an evolved being was killed by a damn Titan using only his hands, even when they say that this Titan was powerful, I refuse to accept this.

This scene must hide something or is an absolute lack of respect and common sense.

I will cling to this because my brain can not process the opposite.

- An extra dimension
A group of researchers at ETH Zurich, Switzerland, have added an additional dimension to 3D printing. They have created moveable and shape variable 4D objects that can be folded into 3D shapes or can change shape as a reaction to external influences such as temperature. The team also developed a construction principle that allows them to control the objects deformation. Each structure is designed to change configuration in a certain way, and are able to support weight, making them the first load-bearing 4D objects.

*Credit: ETH Zurich / Tian Chen*The objects are made on the structural principle that depends on two states, retracted or extended. Combining these states allows for the creation of more complex structures, and gives them the ability to form several stable forms. A multi-material 3D printer, using a rigid polymer and an elastic polymer, printed the structures.

The space saving potential of these structures makes them a candidate for aerospace applications. The researchers are also considering their use in ventilation systems and medicine.

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