The Magic Encyclopedia ™

Projection cubes order 4
(by Aale de Winkel)

<Peter Manyakhin> investigates intensively the projection cubes, below is a full listing of his results order according to the least representative, which is the smallest of all possible projections. The example is one uploaded by <Harvey Heinz> on his "unusual cubes page" shows how to derive the projection cube itself as well as the other projections.
<Peter Manyakhin> and <Günter Sterten> arrived at a total of 41472 cubes, sorting through this list I found
<PRJ(035F72D1CB98E64A)> to be the smallest number. The mayor part of the table below forms a full runthrough through 12 monagonal permutation which avoid the obvious aspectial variants by eliminating the mirrorred permutations from the possible 24 permutations. For each projection cube all 48 aspectial variants are determined, and iff those were not obtained previously all 48 aspectial variants are listed as hexadecimal numbers between '<' and '>' brackets. If the aspectial varant was already found, the listing mentionns it and mentions the first projection square (of 8 possible) identifier, for manual verification. (A future upload might hold the actual cube) I thus found a total of 888 cubes without aspectial variants, a number also found by Günther and Peter, but now with explanation of that curious number which is due to the existence of self-aspectial projection cubes (easily spotted in the listing, those that start with "known..."'s halveway the identifier listing). Those cubes showing only "known..."'s are simply aspectial variants of other cubes.
(the numbers of the "known..."'s are indices to the search array and are not of real use, as said a future upload might hole the actual cubes monagonal permuters, currently I don't see the implementation of this.)

Projection Cubes order 4
The projection cube can be defined as a binairy cube where the bits on all monagonal directions form the full range of numbers 0 .. m2-1, it
thus is fully describable by a single square listing of the binairy squares front square
representation projection cube Identification of this sample needs yet to be done by spotting
the projection square in the listings below

All projection cubes currently know are derived from a single one
The listing below starts from the single one shown below.
<1,0,0>

07 02 03 05
01 12 15 04
11 14 00 06
08 10 09 13

0 1 1 1
0 0 1 0
0 0 1 1
0 1 0 1
0 0 0 1
1 1 0 0
1 1 1 1
0 1 0 0
1 0 1 1
1 1 1 0
0 0 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 0 0 1
1 1 0 1
resulting other projections
<0,1,0>

00 09 14 11
06 07 02 10
12 05 13 08
15 01 04 03
<0,0,1>

03 08 10 14
07 06 11 00
05 04 12 13
01 15 02 09
<PRJ(035F72D1CB98E64A)>
projection cube <1,0,0>

00 03 05 15
07 02 13 01
12 11 09 08
14 06 04 10
<-1,0,0>

00 12 10 15
14 04 11 08
03 13 09 01
07 06 02 05
<0,1,0>

01 02 15 09
03 10 08 14
05 12 04 13
07 11 06 00
<0,-1,0>

08 04 15 09
12 05 01 07
10 03 02 11
14 13 06 00
<0,0,1>

12 14 10 02
04 08 15 05
06 11 00 07
13 01 09 03
<0,0,-1>

03 07 05 04
02 01 15 10
06 13 00 14
11 08 09 12
<PRJ(035F72D1CB98E64A>
<PRJ(037EA92BCD64F158>
<PRJ(067B9F12D45CE83A>
<PRJ(06DE715C9F48B23A>
<PRJ(067B9D31AC2EF458>
<PRJ(059F63B2D781E4CA>
0 0 0 0
0 0 1 1
0 1 0 1
1 1 1 1
0 1 1 1
0 0 1 0
1 1 0 1
0 0 0 1
1 1 0 0
1 0 1 1
1 0 0 1
1 0 0 0
1 1 1 0
0 1 1 0
0 1 0 0
1 0 1 0
!4[0 1 2 3]
!2[..] !1[0 1 2 3] !1[0 1 3 2] !1[0 2 1 3] !1[0 2 3 1] !1[0 3 1 2] !1[0 3 2 1] !1[1 0 2 3] !1[1 0 3 2] !1[1 2 0 3] !1[1 3 0 2] !1[2 0 1 3] !1[2 1 0 3]
0
1
2
3
00 03 05 15
07 02 13 01
12 11 09 08
14 06 04 10
00 03 15 05
07 02 01 13
12 11 08 09
14 06 10 04
00 05 03 15
07 13 02 01
12 09 11 08
14 04 06 10
00 05 15 03
07 13 01 02
12 09 08 11
14 04 10 06
00 15 03 05
07 01 02 13
12 08 11 09
14 10 06 04
00 15 05 03
07 01 13 02
12 08 09 11
14 10 04 06
03 00 05 15
02 07 13 01
11 12 09 08
06 14 04 10
03 00 15 05
02 07 01 13
11 12 08 09
06 14 10 04
03 05 00 15
02 13 07 01
11 09 12 08
06 04 14 10
03 15 00 05
02 01 07 13
11 08 12 09
06 10 14 04
05 00 03 15
13 07 02 01
09 12 11 08
04 14 06 10
05 03 00 15
13 02 07 01
09 11 12 08
04 06 14 10
<035F72D1CB98E64A>
<07CE32B65D94F18A>
<E64ACB9872D1035F>
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<F5301D2789BCA46E>
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