Notes on Rhombic Polyhedra Puzzles
George Bell
May 2019
This document contains the following sections:
1) Puzzle Geometry
a) The vertices of the triacontahedron are summarized in three groups
b) Various geometric facts about rhombs, triacontahedra and icosahedra are listed
2) Rhombic Triacontahedron puzzles
a) The vertices of the triacontahedron are numbered (external and internal vertices)
b) Rhombs are numbered
c) Triacontahedron puzzles are described, using these rhomb numbers
3) Rhombic Icosahedron puzzles
a) The vertices of the icoshardon are numbered (including triacontahedron vertices)
b) Different rhomb numbers are identified
c) Icosahedron puzzles are described, using these rhomb numbers
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Part 1: Puzzle Geometry
# Powers of the Golden Ratio, needed below
p = 0.5*(sqrt(5.0)+1.0) ~ 1.6180...
p2 = p+1 # p**2
p3 = 2p+1 # p**3
p4 = 3p+2 # p**4
p5 = 5p+3 # p**5
p6 = 8p+5 # p**6
Triacontahedron vertices in three groups:
Group 1: use 0, p2, p3 (12 vertices): [0, +/-p3, +/-p2], [+/-p2, 0, +/-p3], [+/-p3, +/-p2, 0]
[0,p3,p2], [0,p3,-p2], [0,-p3,p2], [0,-p3,-p2]
[p2,0,p3], [p2,0,-p3], [-p2,0,p3], [-p2,0,-p3]
[p3,p2,0], [p3,-p2,0], [-p3,p2,0], [-p3,-p2,0]
norm((p2,0,p3)) ~ 4.980...
Group 1 forms the 12 vertices of a regular icosahedron.
Group 2: use 0, p, p3 (12 vertices): [0, +/-p, +/-p3], [+/-p3, 0, +/-p], [+/-p, +/-p3, 0]
[0,p,p3], [0,p,-p3], [0,-p,p3], [0,-p,-p3]
[p3,0,p], [p3,0,-p], [-p3,0,p], [-p3,0,-p]
[p,p3,0], [p,-p3,0], [-p,p3,0], [-p,-p3,0]
norm((0,p,p3)) = sqr(3)*p**2 ~ 4.535...
Group 3: use p2, p2, p2 (8 vertices): [+/-p2, +/-p2, +/-p2]
[p2,p2,p2] , [p2,p2,-p2], [p2,-p2,p2], [p2,-p2,-p2]
[-p2,p2,p2] , [-p2,p2,-p2], [-p2,-p2,p2], [-p2,-p2,-p2]
norm((p2,p2,p2)) = sqrt(3)*p**2 ~ 4.535...
Groups 2 & 3 (12+8=20) form the vertices of a regular dodecahedron.
edgelength = p*sqrt(p+2.0) ~ 3.07768...
area of each face = 2p3 = 4p+2
fat rhomb volume = 2p5 = 10p+6
fat rhomb face to face distance (F) = p2 = p+1
thin rhomb volume = 2p4 = 6p+4
thin rhomb face to face distance (T) = p
triacontahedron inradius = p3 = 2p+1
triacontahedron circumradius = p*edgelength
triacontahedron volume = 20p**6 = 20(8p+5)
triacontahedron face to opposite face distance: (2F+2T) 4p+2 ~ 8.4721...
icosahedron volume = 10p**6 = 10(8p+5)
icosahedron face to opposite face distance: (2F+1T) 3p+2 ~ 6.8541... or (1F+2T) 3p+1 ~ 5.8541...
(the icosahedron has two face to face distances, you can measure both with calipers)
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Part 2: Rhombic Triacontahedron puzzles
Main vertex order (0-31, obtained from labeling a geometrical model):
vertices = [[p2,0,p3], [0,p,p3], [-p2,0,p3], [0,-p,p3],
[p2,p2,p2], [0,p3,p2], [-p2,p2,p2],[-p2,-p2,p2],
[0,-p3,p2], [p2,-p2,p2], [p3,0,p], [-p3,0,p],
[p3,p2,0], [p,p3,0], [-p,p3,0], [-p3,p2,0],
[-p3,-p2,0], [-p,-p3,0], [p,-p3,0], [p3,-p2,0],
[p3,0,-p], [-p3,0,-p], [p2,p2,-p2],[0,p3,-p2],
[-p2,p2,-p2],[-p2,-p2,-p2],[0,-p3,-p2],[p2,-p2,-p2],
[p2,0,-p3], [0,p,-p3], [-p2,0,-p3],[0,-p,-p3]]
# The ten internal vertices (32-41, for the symmetrical dissection)
vertices += [[p,-p,p], [-p,-p,p], [p,p,p], [p,-p,-p],
[-1,0,p], [-1,p2,0], [p,1,0], [-p2,0,-1],
[0,-p,-1], [0,1,-p2]]
North Pole: [p2,-p2,p2] (9)
South Pole: [-p2,p2,-p2] (24)
Core vertex between N and S poles: [p,-p,p] (32)
Rhomb numbering (0-19):
rhombs = [[29,28,22,23, 30,31,41,24], [30,31,41,24, 25,26,40,39],
[25,26,40,39, 16,17,33,11], [16,17,33,11, 7,8,3,2],
[21,16,25,30, 15,11,39,24], [15,11,39,24, 6,2,36,37],
[6,2,36,37, 5,1,34,13], [5,1,34,13, 4,0,10,12],
[14,5,6,15, 23,13,37,24], [23,13,37,24, 22,12,38,41],
[22,12,38,41, 28,20,35,31], [28,20,35,31, 27,19,18,26],
[2,3,33,11, 36,32,40,39], [12,10,34,13, 38,32,36,37],
[26,18,35,31, 40,32,38,41], [8,18,26,17, 3,32,40,33],
[0,3,2,1, 10,32,36,34], [19,10,12,20, 18,32,38,35],
[9,0,10,19, 8,3,32,18], [24,39,40,41, 37,36,32,38]]
Rhombs [0,1,2,3] make a "Batman piece", as do [4,5,6,7] and [8,9,10,11]
Rhomb #18 is the North Pole rhomb
Rhomb #19 is the core
Rhombs [2,3,12,15] make a Bilinski dodecahedron, as do [6,7,13,16] and [10,11,14,17]
Internal rhombs: [12,13,14, 19]
# This array records the type of each rhomb
# 1 = A thin rhomb (oblate)
# 0 = A fat rhomb (prolate)
rhombType = [1,0,0,1, 1,0,0,1, 1,0,0,1, 1,1,1, 0,0,0, 1,0]
fatRhombs = [1,2,5,6,9,10,15,16,17,19]
thinRhombs = [0,3,4,7,8,11,12,13,14,18]
# Rhomb connection matrix (20x20)
adjMatrix=[[0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0],
[1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1],
[0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0],
[0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0],
[0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,0,0,0,1],
[0,0,0,0,0,1,0,1,1,0,0,0,0,1,0,0,1,0,0,0],
[0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0],
[0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1],
[1,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,1,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0],
[0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,1],
[0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,0,1,1,0,1],
[0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,1],
[0,0,1,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0],
[0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0],
[0,1,0,0,0,1,0,0,0,1,0,0,1,1,1,0,0,0,0,0]]
# Symmetry partners (18 and 19 are alone)
symPart = [[0,4,8], [1,5,9], [2,6,10], [3,7,11], [12,13,14], [15,16,17]]
# There are 22 (connected) pieces made from 3T and 3F such that 3 copies
# of each piece make a triacontahedron minus the N pole (18) and core (19).
# These pieces are:
#1 [0,1, 2, 3,12,15] 8 connections # Modified spoon
#2 [0,1, 2, 3,12,16] 6 connections
#3 [0,1, 2, 3,14,15] 7 connections # Spoon piece
#4 [0,1, 2, 3,14,17] 5 connections # Interesting (see GBRT04)
#5 [0,1, 2, 7,12,16] 5 connections # Snake
#6 [0,1, 2,11,14,15] 6 connections
#7 [0,1, 2,11,14,17] 6 connections
#8 [0,1,10, 3,14,15] 6 connections # A different spoon
#9 [0,1,10, 7,13,17] 6 connections # Weird
#10 [0,1,10,11,13,17] 6 connections
#11 [0,1,10,11,14,15] 7 connections
#12 [0,1,10,11,14,17] 9 connections # Dodecahedron plus (last 4 make a dodecahedron)
#13 [0,9, 6, 7,13,16] 8 connections
#14 [0,9, 6, 7,13,17] 6 connections # Not coordinate motion
#15 [0,9, 6,11,13,17] 5 connections # Hole piece (see GBRT05)
#16 [0,9,10, 3,14,15] 6 connections
#17 [0,9,10, 7,13,16] 7 connections # Interesting symmetry
#18 [0,9,10, 7,13,17] 7 connections
#19 [0,9,10,11,13,16] 6 connections # Dinosaur, related to #4
#20 [0,9,10,11,13,17] 8 connections
#21 [0,9,10,11,14,15] 7 connections
#22 [0,9,10,11,14,17] 9 connections # Dodecahedron plus (last 4 make a dodecahedron)
!!!!!!!!!!!!!! PUZZLE DEFINITIONS !!!!!!!!!!!
Puzzle ID: AGRT01
Puzzle Name: Mateos
Year: 1989
Designer: Albert Gubeli
Pieces: 5
Coordinate motion: No
piece1 = [0,1,2,3] # First Batman piece
piece2 = [4,5,6,7] # Second Batman piece (identical)
piece3 = [8,9,10,11] # Third Batman piece (identical)
piece4 = [12,13,16,19]
piece5 = [14,15,17,18] # Mirror of previous
Puzzle ID: RBRT01
Puzzle Name: Triakon
Year: 2006
Designer: Rik Brouwer
Pieces: 4
Coordinate motion: Yes
piece1 = [0,1,2,3,12]
piece2 = [4,5,6,7,19]
piece3 = [8,9,10,11,13]
piece4 = [14,15,16,17,18] # The remaining "propeller" piece
Puzzle ID: AGRT02
Puzzle Name: Soccerit
Year: 2015
Designer: Albert Gubeli
Pieces: 4
Coordinate motion: Yes
piece1 = [7,8,12, 5,16] # Has a hole!
piece2 = [0,10,14,15,18]
piece3 = [1,2,4,19,3]
piece4 = [6,13,9,17,11]
SCRT01 uses the non-symmetrical decomposition and will be found
under the Icosahedron puzzles below.
Puzzle ID: SCRT02
Puzzle Name: SC2
Year: 2016
Designer: Stephen Chin
Pieces: 4
Coordinate motion: No
piece1 = [0,1,2,3] # Batman piece
piece2 = [4,5,6,7,12,15] # Modified Batman piece
piece3 = [8,9,10,11,19] # Modified Batman piece
piece4 = [13,14,16,17,18] # Remaining pieces
Puzzle ID: GBRT01
Puzzle Name: None
Year: 2016
Designer: George Bell
Pieces: 3
Coordinate motion: Yes
piece1 = [0,1,2,3,14,15] # The basic "Spoon" piece
piece2 = [4,5,6,7,12,16,18] # Adds the N pole
piece3 = [8,9,10,11,13,17,19] # Adds the core
GBRT02 cannot be described because it adds rhombs cut in thirds to the basic "Spoon" piece.
Puzzle ID: GBRT03 (This is a variation of Triakon)
Puzzle Name: None
Year: 2016
Designer: George Bell
Pieces: 4
Coordinate motion: Yes
piece1 = [0,1,2,3,14] # The basic piece
piece2 = [4,5,6,7,12] # Same
piece3 = [8,9,10,11,13,19] # Same plus core
piece4 = [15,16,17,18] # Symmetrical locking end piece
Puzzle ID: GBRT04
Puzzle Name: None
Year: 2019
Designer: George Bell
Pieces: 3
Coordinate motion: Yes
piece1 = [0,1,2,3,14,17] # The basic piece #4, same as SCRT02 piece2
piece2 = [4,5,6,7,12,15, 8] # 8 moved here from piece3
piece3 = [9,10,11,13,16, 18] # N pole added, 8 removed (now has a hole)
Puzzle ID: GBRT05
Puzzle Name: None
Year: 2019
Designer: George Bell
Pieces: 3
Coordinate motion: Yes
piece1 = [0,9,6,11,13,17] # The basic piece #15 with hole
piece2 = [4,1,10,3,14,2] # 2 moved from piece3, no more hole
piece3 = [8,5,7,12,16,18,15] # Added N pole, 15 moved from piece2
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Part 3: Rhombic Icosahedron puzzles
Vertices are identical, with the exception of vertex #35
Vertex #35 [p,-p,-p] is replaced by [p2,0,-1]
The definition of the rhombs then changes around this vertex.
Rhombs 10,11,14 and 17 are now different:
rhombs[10] = [18,32,10,19, 26,40,35,27] (Fat)
rhombs[11] = [19,20,28,27, 10,12,22,35] (Thin)
rhombs[14] = [10,12,22,35, 32,38,41,40] (Thin)
rhombs[17] = [26,40,35,27, 31,41,22,28] (Fat)
The classification of fat and thin rhombs is unchanged, however adjMatrix is (slightly) different.
# Rhomb connection matrix (20x20)
adjMatrix=[[0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0],
[1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1],
[0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0],
[0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0],
[0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,0,0,0,1],
[0,0,0,0,0,1,0,1,1,0,0,0,0,1,0,0,1,0,0,0],
[0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0],
[0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1],
[0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1,1,0],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0],
[0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,1],
[0,0,0,0,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1],
[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,1],
[0,0,1,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0],
[0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,0],
[1,1,0,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0],
[0,1,0,0,0,1,0,0,0,1,0,0,1,1,1,0,0,0,0,0]]
The following rhombs form an icosahedron:
piece = [7,8,12,13,14, 5,6,9,16,19] # 5 Thin , 5 Fat
Unlucky Rhomb#13 is the only internal rhomb
The remaining rhombs form the end cap:
piece = [0,3,4,11,18, 1,2,10,15,17] # 5 Thin , 5 Fat
!!!!!!!!!!!!!! PUZZLE DEFINITIONS !!!!!!!!!!!
Puzzle ID: AGRI01 (This gives the mirror image of the original puzzle)
Puzzle Name: Aurels
Year: 2004
Designer: Albert Gubeli
Pieces: 3
Coordinate motion: No
piece = [7,13,12,16]
piece = [6,8,9]
piece = [14,5,19]
Puzzle ID: SCRI01
Puzzle Name: SC1
Year: 2016
Designer: Stephen Chin
Pieces: 3
Coordinate motion: No
piece1 = [5,6,16] # 3 Fat
piece2 = [7,13,14] # 3 Thin
piece3 = [8,12,9,19]
Puzzle ID: SCRI02
Puzzle Name: SC2
Year: 2016
Designer: Stephen Chin
Pieces: 3
Coordinate motion: No
piece1 = [6,8,9]
piece2 = [13,14,19]
piece3 = [5,7,12,16]
Puzzle ID: SCRI03
Puzzle Name: SC3 or Golden Rhombic Icosahedron (IPP36 exchange puzzle)
Year: 2016
Designer: Stephen Chin
Pieces: 3
Coordinate motion: No
piece1 = [5,14,19]
piece2 = [7,9,13]
piece3 = [6,8,12,16]
Puzzle ID: SCRT01
Puzzle Name: SC1
Year: 2016
Designer: Stephen Chin
Pieces: 4
Coordinate motion: No
piece1 = [3,6,12,16,18]
piece2 = [4,5,7,13,19]
piece3 = [0,8,9,10,14]
piece4 = [1,2,11,15,17]