6 vs 8

You cannot picture 8 dimensions. You can still count them, and counting is enough.

Part 1

What "n-dimensional" actually means

Forget pictures. Dimension is just this: how many numbers do I have to tell you so you know exactly where I am?

A bead on a rope

n = 1 number

That is the whole idea. 6-dimensional space is nothing more than the collection of every possible list of 6 numbers. 8-dimensional space is every possible list of 8 numbers. A robot arm with 6 joints lives in a 6-dimensional space, because 6 joint angles pin it down completely. Add two more joints and the same arm now lives in an 8-dimensional space. Nothing spooky, just a longer address.

Part 2

First, convince yourself 6 and 8 really are different

Build a cube in n dimensions and count its corners. In 1D a "cube" is a stick with 2 ends. In 2D it is a square with 4 corners. In 3D, 8 corners. Each new dimension doubles the corners, because every old corner gets a twin pushed into the new direction.

n = n = 3

Corners (2^n)

Edges

Directions from any corner

Careful, this is intuition and not the proof. A 6-cube has 64 corners and a 8-cube has 256, so the two spaces are clearly not built the same way. But counting corners is about the cube, not about the space. Topology lets you squish the cube into a ball and the corners vanish. So we need a test that survives squishing. That is Part 3.

Part 3

The proof machine: take away one point

In the last lesson you pulled one point out of a string and it fell into 2 pieces, while the pancake stayed in 1. That trick has a grown-up version that works in every dimension. Pull the middle point out of n-dimensional space, then let what is left shrink as far as it will go. It always shrinks down to a sphere with n minus 1 dimensions, written S^n-1.

n = n = 1 Shrink

Space	Remove one point, then shrink	What you get	Can you draw it?
1D line	falls apart	S⁰: two dots	yes
2D flat	collapses inward	S¹: a circle	yes
3D room	collapses inward	S²: a balloon skin	yes
6D	same move	S⁵	no, and that is fine
8D	same move	S⁷	no, and that is fine

Why this is allowed. If 6D space and 8D space really were the same shape in disguise, then punching a point out of each would have to leave you with the same shape too. So the question "is 6D the same as 8D?" turns into the easier question "is S⁵ the same as S⁷?"

Part 4

Every shape carries a hole ID card

Counting holes was easy for a donut because there was only one kind of hole. In higher dimensions there are several kinds, so a shape's ID card is a list: how many pieces, how many loop-holes, how many balloon-holes, and so on up the ladder. A sphere S^k has a very clean card. It lights up in exactly two places: slot 0, because it is one connected piece, and slot k, because it wraps around one k-shaped hole. Everywhere else it is blank.

S⁵ (from 6D space)

S⁷ (from 8D space)

first space 6D

second space 8D

And that is the answer to your quiz question. Squishing can never change a hole ID card, so two shapes with different cards can never be the same shape. 6D space punched leaves a card lit at slot 5. 8D space punched leaves a card lit at slot 7. Different card, different shape. It works for any two different n and m, which is why the answer was False. The grown-up name for this result is Brouwer's invariance of domain, from around 1911, and it is proved with exactly this hole-counting machinery.