Image of a three-dimensional net of a tesseract , created by Dmn with Paint Shop Pro . The net of a tesseract is the unfolding of a tesseract into 3-D space. Let the dimension from left to right be labeled x , the dimension from bottom to top be labeled z , and the dimension from front to back be labeled y . Let coordinates by ( x , y , z ). Let the top cube have coordinates (0,0,1), the cube below it have coordinates (0,0,0), the cube in front of it have coordinates (0,−1,0), the cube behind it have coordinates (0,1,0), the cube to the left (−1,0,0), the one to the right (1,0,0). Let the cube below the central one have coordinates (0,0,−1) and the one at the bottom (0,0,−2).

The central cube (0,0,0) is seen to be connected to six other cubes, but when folded in 4-D every cube connects to six other cubes. The frontal cube (0,−1,0) connects in the −Y direction to (0,0,−2), in the +Y direction to (0,0,0), in the +X direction to (1,0,0), in the −X direction to (−1,0,0), in the +Z direction to (0,0,1), in the −Z direction to (0,0,−1).

There are twelve different ways in which the tesseract can be rotated (in 4-D) by 90 degrees in such a way that four of the cubes exchange positions cyclically while the remaining four cubes stay in place but rotate (in 3-D). For example, one 4-D rotation causes the following four-cube exchange: (0,0,1)→(0,0,0)→(0,0,−1)→(0,0,−2)→(0,0,1). Meanwhile, the same rotation causes cube (0,1,0) to rotate 90 degrees around the +X axis, the (0,−1,0) cube to rotate 90 degrees around the −X axis, the (1,0,0) cube to rotate 90 degrees in the −Y direction and the (−1,0,0) cube to rotate 90 degrees in the +Y direction.

The twelve 4-D rotations are:
1: (0,0,1)→(0,0,0)→(0,0,−1)→(0,0,−2)→(0,0,1),
9: (0,0,1)→(1,0,0)→(0,0,−1)→(−1,0,0)→(0,0,1),
10: (0,0,1)←(1,0,0)←(0,0,−1)←(−1,0,0)←(0,0,1),
11: (0,0,1)→(0,1,0)→(0,0,−1)→(0,−1,0)→(0,0,1),
12: (0,0,1)←(0,1,0)←(0,0,−1)←(0,−1,0)←(0,0,1).

Each 4-D rotation has a "dual" which is perpendicular to the 3-D rotation of the stationary cubes. There are six pairs of dual (4-D) rotations:

• 1 ↔ 4,
  
• 2 ↔ 3,
  
• 5 ↔ 12,
  
• 6 ↔ 11,
  
• 7 ↔ 9,
  
• 8 ↔ 10.
  

The dual of a 4-D rotation implies, by means of the right-hand rule, how the stationary cubes are supposed to rotate in 3-D.

Since there are eight cubes and each cube connects to six other cubes, then each cube has a pair of cubes to which it does not connect: (1) itself, and (2) its opposite. Thus there are four pairs of opposite cubes:
1: (0,0,1) ↔ (0,0,−1),
2: (0,0,0) ↔ (0,0,−2),
3: (−1,0,0) ↔ (1,0,0),
4: (0,−1,0) ↔ (0,1,0).

Each pair of opposite cubes aligns itself along opposite sides of one of four orthogonal axis of 4-D space. Therefore it is possible to establish a one-to-one onto mapping f between the unfolded positions of the cubes in 3-D and the canonical coordinates of their folded positions in 4-D, viz.

${\displaystyle f:(-1,0,0)\mapsto (-1,0,0,0)=-I,}$

${\displaystyle f:(1,0,0)\mapsto (1,0,0,0)=I,}$

${\displaystyle f:(0,-1,0)\mapsto (0,-1,0,0)=-J,}$

${\displaystyle f:(0,1,0)\mapsto (0,1,0,0)=J,}$

${\displaystyle f:(0,0,1)\mapsto (0,0,1,0)=K,}$

${\displaystyle f:(0,0,-1)\mapsto (0,0,-1,0)=-K,}$

${\displaystyle f:(0,0,0)\mapsto (0,0,0,1)=1,}$

${\displaystyle f:(0,0,-2)\mapsto (0,0,0,-1)=-1.}$

The canonical 4-D coordinates have been given labels corresponding to basis quaternions (and their negatives). Using these labels, the 4-D rotations can be expressed more simply as
1: K → 1 → −K → −1 → K,
2: K → −1 → −K → L → K,
3: I → J → −I → −J → I,
4: I → −J → −I → J → I,
5: −I → 1 → I → −1 → −I,
6: −I → −1 → I → 1 → −I,
7: −J → 1 → J → −1 → −J,
8: −J → −1 → J → 1 → −J,
9: K → I → −K → −I → K,
10: K → −I → −K → I → K,
11: K → J → −K → −J → K,
12: K → −J → −K → J → K.

All of these rotations follow a pattern A → B →− A →− B → A , so that each one can be abbreviated as an ordered pair ( A , B ). Then, each rotation can be abbreviated furthest as the product of the ordered pair of quaternions, which yields an imaginary quaternion:
1: (K,1) = K
2: (K,−1) = −K
3: (I,J) = K
4: (I,−J) = −K
5: (−I,1) = −I
6: (−I,−1) = I
7: (−J,1) = −J
8: (−J,−1) = J
9: (K,I) = J
10: (K,−I) = −J
11: (K,J) = −I
12: (K,−J) = I

The pairs of dual quaternions are then seen to have the following properties: the products of their single-quaternion abbreviations are always unity:

• 1 ↔ 4 : K (− K) = 1,
  
• 2 ↔ 3 : (−K) K = 1,
  
• 5 ↔ 12 : (− I) I = 1,
  
• 6 ↔ 11 : I (−I) = 1,
  
• 7 ↔ 9 : (−J) J = 1,
  
• 8 ↔ 10 : J (−J) = 1.
  

Each of the twelve rotations has a pair of candidate duals, but one of them is the reversal of the rotation, i.e. given rotation ( A , B ), its reverse is ( A , − B ), so it is disqualified as the dual of ( A , B ), leaving only one possible dual.

Summary

Description Tesseract2.svg	(See above, text taken from cited source)
Date	1 April 2007
Source	Own work based on: Tesseract2.png
Author	Traced by Stannered
Other versions	Tesseract2.png , Tesseract net Crooked House.svg
SVG development InfoField	The SVG code is valid . This map was created with Inkscape .

Licensing

This work has been released into the public domain by its author, Dmn , at the English Wikipedia project . This applies worldwide. In case this is not legally possible: Dmn grants anyone the right to use this work for any purpose , without any conditions, unless such conditions are required by law.