Chapter 7 - Exploration of (R)(A) Toroids

Our next diversion from (Q) models is this chapter of (R)(A) toroids. These are conglomerations of convex polyhedra that together form a ring, or multiple rings, often interconnected. One area of exploration here is generating models that are comprised of specified n-gons, and finding models of arbitrarily high genus.

Many of the initial models consist of rings that are arranged in a planar fashion, but some other combinations of rings that are of special interest are those that wrap around to create models of tetrahedral, octahedral and icosahedral symmetry. Many of these models are based on the discoveries of Kurt Schmucker that 8 octahedra form a ring with special properties relating it to the rhombic dodecahedron, and 8 dodecahedra form a similar ring with properties relating it to the rhombic triacontahedron. I would propose calling the models based on these discoveries Stewart/Schmucker toroids.

Model .Stel Genus Properties Pages with Diagram Pages with Description Notes 36P5+45P6=genus10 .Stel 10 (R)(A)(D) 47   pentagonal rings with hex-prism spacers 7(Q3P6 / P3Q3)=linear .Stel 7 (R)(A)(D) 48   linear arrangement 7(Q3P6 / P3Q3)=two_row .Stel 7 (R)(A)(D) 48   two row arrangement 6(Q3P6 / P3Q3)=ring .Stel 7 (R)(A)(D) 48   ring arrangement 13P4=coplanar .Stel 2 (R)(D) 49   coplanar adjacent faces 35P4 .Stel 2 (R)(A)(D) 49   no coplanar adjacent faces 44P4 .Stel 5 (R)(A)(D) 49   cubical sponge Q32Y36AY46A / S3S3 .Stel 1 (R)(A)(D) 50   all triangles (Q32Y36AY46A / S3S3)(Q32Y34AY46A / S3S3) .Stel 2 (R)(A)(D) 50   all triangles 8S3 .Stel 1 (R)(A)(D) 51   all triangles 29S3 .Stel 6 (R)(A)(D) 51   planar rings 20S3 .Stel 5 (R)(A)(D)   59 hexahedral 21S3 .Stel 6 (R)(A)(D) 51   hexahedral with body diagonal 47S3 .Stel 18 (R)(A)(D)   52 rhomb dodeca with body diagonals 38S3 .Stel 11 (R)(A)(D)   52 rhomb dodeca formation 96Y3=Cheat .Stel 1 (R)(A)(D)   53 last two tetrahedra don't meet 96Y3=Quarter_Cheat .Stel 0 (R)(A)(D) 53   angle = 88.35 deg instead of 90 12Y5=quasihelix .Stel 0 (R)(A)(D) 54   almost periodic 8I5 .Stel 1 (R)(A)(D) 55 54 same as 8S3 not 8D5 8I5 in 8S3 .Stel 1 (R)(A)(D) 55     20I5 .Stel 5 (R)(A)(D)     not in "Adventures" 21I5 .Stel 6 (R)(A)(D)   54 hexahedral with body diagonal 38I5 .Stel 11 (R)(A)(D)   54 rhomb dodeca formation 47I5 .Stel 18 (R)(A)(D)   54 rhomb dodeca with body diagonals 8D5 .Stel 1 (R)(A)(D) 57   rhombic ring 29D5 .Stel 6 (R)(A)(D)   56 planar rings 92D5 .Stel 29 (R)(A)(D) 58   rhomb triacontahedron Z30 62D5 .Stel 19 (R)(A)(D) 58   rhomb icosahedron Z20 38D5 .Stel 11 (R)(A)(D) 58   rhomb dodecahedron Z12 20D5=cheat .Stel 5 (R)(A)(D) 58   rhomb hexahedron Z6 interference 20D5* .Stel 5 (R)(A)(D) 58, 60   rhomb hexahedron Z6* no interference 16D5 .Stel 3 (R)(A)(D)   59 3 rhombs 70D5 .Stel 23 (R)(A)(D)   62 3x2x2 195D5 .Stel 70 (R)(A)(D)   62 20 Z6* on central D5 4D54S3 .Stel 1 (R)(A)(D)   62 rhomb with S5 edges 6(47S3) .Stel 96 (R)(A)(D)   62   4K3 .Stel 1 (R)(A)(D)   62   C10 .Stel 0 (R)(A)(D)   63 n-gons 3-10 78P14 .Stel 3 (R)(A)(D) 64     14P714P6 .Stel 1 (R)(A)(D) 65     38P740P6 .Stel 3 (R)(A)(D)   65   Y4P4Y4+P3Y4P3+P3P4Y4 .Stel 0 (R)(A)(D) 65   alternate links 154P14 .Stel 15 (R)(A)(D) 66   ring of rings 154P14=detail .Stel NA (R)(A)(D) 66   close-up detail 31P6 .Stel 6 (R)(A)(D) 66     3P11 .Stel 0 (R)(A)(D) 67   parallel faces top and bottom 6P76P6v1 .Stel 1 (R)(A)(D) 67     6P76P6v2 .Stel 1 (R)(A)(D) 67   heptagons would overlap without hexagons 30P538P6 .Stel 9 (R)(A)(D) 68   tiling of rings 12P516P8 .Stel 5 (R)(A)(D) 68   portico fashion 20P8 .Stel 5 (R)(A)(D)   68 portico fashion 40P5 .Stel 6 (R)(A)(D) 69   links missing - coplanar 36P5 .Stel 10 (R)(A)(D) 69   links missing - coplanar 84P7119P6 .Stel 35 (R)(A)(D) 69     P22 / P11 .Stel NA (R) 70   coplanar - math practice P22 / P11P11 .Stel NA (R) 70   coplanar - math practice P6 / 6P3 .Stel NA (R)   70 coplanar - math practice P10 / P5P5P5 .Stel NA (R) 70   coplanar - math practice P16 / P8P8 .Stel NA (R) 71   coplanar - math practice P8 / P4P4+P8 .Stel NA (R) 71   coplanar - math practice P12 / P6P4P6 .Stel NA (R) 71   coplanar - math practice P14 / 14P7 .Stel NA (R) 71   coplanar - math practice Prev - Chapter 6 - Simplest (R)(A)(Q)(T) Toroids of genus p=2,3 Next - Chapter 8 - Limiting Properties Back to Adventures Among the Toroids - Index Disclaimer: These pages are dedicated to B.M. Stewart. They are intended to be used by readers of his book to better illustrate his ideas in three dimensions. His book has tons of information that I do not intend to represent here, and I highly encourage anyone interested in this subject to buy the book. If you like my illustrations, please let me know. And if you are interested in a particular Stewart Toroid being modeled, please let me know and I will do what I can. Thanks also to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things). Back to the main Polyhedron Page. Link to this page as http://Polyhedra.Doskey.com/Stewart07.html Alexander's Polyhedra, (c) 1998-2006, Alex Doskey