B*tchin’ six dimensional 6-cube. The rainbow colours and…

July 1, 2011
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(This article was first published on Mathematical Poetics, and kindly contributed to R-bloggers)



B*tchin’ six dimensional 6-cube. The rainbow colours and glass panes really help this visualisation.
 

Examples of 6-dimensional things

If it’s hard to envision 6 dimensions, consider this: the possible tunings of a guitar constitute a 6-dimensional space. You can tune to EADGBE (standard), DADGAB, drop-D, DADGAD, GCCGCC, BEBEBE, CGCFGE, and many others.

(If you consider notes an octave apart to be equivalent, then we’re talking about a quotient space, each distance being topologically on a loop. But that’s just one system of musical valuation — and like the winding number of a complex number, it’s totally apparent that high octaves do not sound exactly the same as low sounds. And doing a 720° is more impressive than a 360°. If the abstract “loop” is unwound, there is a highest note (“1”) and a lowest note (“0”) that can effectively be played on each string (dimension).)

You can also think about 6-D as being the six columns in a table or array. For example the { RBI, on-base percentage, fielding errors, stolen bases, sacrifice flies, and home-runs } for a number of baseball players.

Or you can think about six security prices moving in parallel, from bell to bell at the NYSE.

Again the lowest price is called “0” and the highest is called “1”. This renaming places the jumping Brownian motions inside a secteract. So instead of six 1-D paths it’s one 6-D path:

Enough examples of 6-dimensional things. Back to the 6-cube itself.

Let’s make one.

The bounds of the secteract (its “corners”? Or should I say its 6-corners.) come from filling in each of six slots with either 0 or 1.

There are 64 ways to do this. (two options for each of six slots = 2^6.) For example (0,0,0,0,0,1) is one, (0,0,0,0,1,0) is another, and (0,1,1,0,1,0) is a third out of the 64.

The R programming language was nice enough to write out all of the vertices for me without my having to type much. Here they are:

> booty=c(0,1)
> expand.grid(booty,booty,booty,booty,booty,booty) #rockin everywhere

   Var1 Var2 Var3 Var4 Var5 Var6
1     0    0    0    0    0    0
2     1    0    0    0    0    0
3     0    1    0    0    0    0
4     1    1    0    0    0    0
5     0    0    1    0    0    0
6     1    0    1    0    0    0
7     0    1    1    0    0    0
8     1    1    1    0    0    0
9     0    0    0    1    0    0
10    1    0    0    1    0    0
11    0    1    0    1    0    0
12    1    1    0    1    0    0
13    0    0    1    1    0    0
14    1    0    1    1    0    0
15    0    1    1    1    0    0
16    1    1    1    1    0    0
17    0    0    0    0    1    0
18    1    0    0    0    1    0
19    0    1    0    0    1    0
20    1    1    0    0    1    0
21    0    0    1    0    1    0
22    1    0    1    0    1    0
23    0    1    1    0    1    0
24    1    1    1    0    1    0
25    0    0    0    1    1    0
26    1    0    0    1    1    0
27    0    1    0    1    1    0
28    1    1    0    1    1    0
29    0    0    1    1    1    0
30    1    0    1    1    1    0
31    0    1    1    1    1    0
32    1    1    1    1    1    0
33    0    0    0    0    0    1
34    1    0    0    0    0    1
35    0    1    0    0    0    1
36    1    1    0    0    0    1
37    0    0    1    0    0    1
38    1    0    1    0    0    1
39    0    1    1    0    0    1
40    1    1    1    0    0    1
41    0    0    0    1    0    1
42    1    0    0    1    0    1
43    0    1    0    1    0    1
44    1    1    0    1    0    1
45    0    0    1    1    0    1
46    1    0    1    1    0    1
47    0    1    1    1    0    1
48    1    1    1    1    0    1
49    0    0    0    0    1    1
50    1    0    0    0    1    1
51    0    1    0    0    1    1
52    1    1    0    0    1    1
53    0    0    1    0    1    1
54    1    0    1    0    1    1
55    0    1    1    0    1    1
56    1    1    1    0    1    1
57    0    0    0    1    1    1
58    1    0    0    1    1    1
59    0    1    0    1    1    1
60    1    1    0    1    1    1
61    0    0    1    1    1    1
62    1    0    1    1    1    1
63    0    1    1    1    1    1
64    1    1    1    1    1    1

And there you have it: an electronic realisation of a secteract. Just as real as a Polyworld life-form.

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