SOMA Crystal
SOMA News 9 Feb 2000
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Twisting SOMA figures:

by Bob Allen (wb0fvk@arrl.net)

Bob Allen have found the 25 figures 201-225 by using a new technique on some of the already existing figures.

While studying some figures with two of the 8-fold symmetries, Bob noticed that the figures could be changed from one symmetry into the other by a "45 degree rotation". We call the technique TWISTING because it rotates cubes around an axis.

We Twist a figure by looking at it as being made of concentric "cylinders", which just look like squares, and "twisting them around".
The general case for a 5x5 rotation as seen from above is something like this:

     J K L M N             X Y J K L
     Y b c d O    twist    W i b c M
     X i A e P     ==>     V h A d N
     W h g f Q    45 CW    U g f e O
     V U T S R             T S R Q P

(CW means Clockwise)
Each letter is representing a "1" or a "." in the figure.
The outer "ring" consists of 16 positions, J to Y, and we "rotate" Twist them two positions, or 45 degrees of 360.
The inner 3x3 ring, b to i, is rotated one of its 8 position "circumference";
The central A is unaffected.
Notice that the Diagonal "JbAfR" becomes a Vertical, and the Horizontal "XiAeP" becomes a Diagonal.

Bob had to "cheat" to rotate the 3x5 Bathtub (Fig A223), and had little luck playing with 4x4's.

Bob only played with figures with the related Symmetries A & B, and C & E, at first to see if it was possible, then out of the fun of seeing what shapes might emerge.
This Twisting is not like the mathematically "real Rotation Transforms" of figures in the SOMA Space, and often produces figures with disconnected regions.


During the Twisting, Diagonal Reflections and Rotations are converted into Planar Reflections and Rotations, and vice versa.
Bob then noticed that the old 4-fold symmetries "C" & "E" could also be often changed into each other in this manner. In general, the two dimensions perpendicular to the rotation axis of the figure must be equal.
And, 3x3 squares and 5x5 squares both have perimeters that are evenly divisible by 8:
360 degrees divided by 8 is 45, and so the Twisting is accomplished by "shifting the perimeters".
It seems that all Class E figures can be converted into Class C, and some Class C figures can be rotated into Class E. Further investigation showed the same is true with 2-fold Classes "A" and "B".

For example, rotating the Class E figure A017 "The Steamer" produces this figure with Class C symmetry :

   |...--|...--|111--           |.....|.....|11111
   |....-|.11.-|1111-           |-...-|-111-|-111-
   |..1..|.111.|11111    ==>    |-.1.-|-.1.-|-111-
   |-....|-.11.|-1111   45 CW   |-...-|-111-|-111-
   |--...|--...|--111           |.....|.....|11111

It is nice looking, but has no solutions.!
And, rotating the Class E figure A058, which Bob refer to by the "Classic" name "Almost Impossible", also produces a Class C figure :

   |11-|11-|11-|11-             |111|111|111|111
   |1.1|111|111|111     ==>     |-.-|-1-|-1-|-1-
   |-11|-11|-11|-11    45 CW    |111|111|111|111

Which also unfortunately has no solutions.

Some figures rotate into Nonominoes :

   A021 "Cornerhouse 1"         Nonomino # 102
   |111|111|111|111             |111|111|111|111
   |11.|11.|11.|111     ==>     |111|111|111|111
   |1..|1..|1..|111    45 CW    |...|...|...|111


   A022 "Cornerhouse 2"           Nonomino # 38
   |11.|111|111|111|111           |1.|11|11|11|11
   |1.-|11-|11-|11-|11-    ==>    |1.|11|11|11|11
   |.--|1--|1--|1--|1--    CCW    |1.|11|11|11|11

Perhaps some Nonominoes will rotate into solvable figures.?


Now, here are some figures that are actually solvable.

A "Cornerstone" becomes a Fireplace
   /SOMA203
   ;Rotated A015 by Bob Allen
   /..1../.411./44766/45776/35522
   /.-.-./.-.-./.-.-./3-7-6/3-5-2
   /.---./.---./.---./.---./3---2


One "Monument" turns into another
   /SOMA204
   ;Rotated A019 by Bob Allen
   /..2../..2../44211/34471
   /...../...../36275/36775
   /...../...../...../36655


   /SOMA212
   ;Rotated A151 by Bob Allen
   /222/266/446/544/557/177
   /.../.3./336/.3./.5./117


   /SOMA213
   ;Rotated A176 by Bob Allen
   /...../11223
   /6..../61233
   /.7.../66243
   /-77../-7544
   /--.5./--554


This "Teddy Bear" surprisingly hid a Cross, looking very much like A054 after the rotation
   /SOMA214
   ;Rotated A194 by Bob Allen
   /.4./644/654/.5./113
   /.../.../662/553/173
   /.../.../222/.7./773


These are also "Twisters":
"Chair" becomes Building A009 => C3A05 Sym=1
"Castle II" becumes Castle A011 => CP218 Sym=1
Rotated B025 => LF006/F130 Sym=3
Diagonal Bow Tie => CP1/12 Sym=1
Planar Bow Tie => LF006/F135 Sym=1
Stealth 1 => CO5/10 Sym=7
Stealth 2 => C5C/05 Sym=1

Bob and I trust you will have fun with this.


- submitted by Bob Allen.

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