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 --------------------------------------------------------------------------------- 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) --------------------------------------------------------------------------------- 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 --------------------------------------------------------------------------------- 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]