Chapter 8 - Limiting Properties

  • This chapter discusses the main criteria for their exploration of toroidal polyhedra. If we only limit the search to (R)(A)(D) polyhedra, the possibilities are limitless (as seen in the previous chapter). Requiring (Q) and (T) will limit the scope to a size that may be reasonably explorable. Other properties (Q') - weak quasiconvexity, and (Q") - regular-faced quasiconvexity are also discussed. (see the index page for definitions).
    Model.Stel GenusProperties Pages with Diagram Pages with DescriptionNotes
    P34Y46A .Stel0 (R)(A)(D)73 the hole for the next figure
    64P3Y4-24EY3-12E / P34Y46A .Stel1 (R)(A)(T)(D)(Q')(Q'')73 not (Q)
    K5(Q32 / S3S3)-E .Stel1 (R)(A)(Q)(D) 75not tunneled - with rod
    K5(Q32 / S3S3)-30E .Stel30 (R)(A)(Q)(D) 75not tunneled - with 30 rods
    D(K5(Q32 / S3S3)-30E) .StelNA NA 75Difference Set H(P)-P
    D(P6Y4-6E) .StelNA NA 75Difference Set H(P)-P
    P6P6 .Stel0 (R)(A)(D) 76not convex
    P3Q3 .Stel0 (R)(A)(D)77 not convex
    H(P3Q3) .Stel0 (A)(D)77 convex hull
    2(Q3P6Q3 / P6) .Stel2 (R)(A)(D)77 not (Q)
    (343P3Y3-54EY4-126E) / 3(P37Y415A) .Stel4 (R)(A)(D)(Q')(Q'')78 not (Q)
    (64P3Y4-21EY3-12E) / 3(2P3Y3-EY3A)(P34Y43A) .Stel4 (R)(A)(D)(Q')(Q'')79 not (Q)
    (27P3Y4-13EY3-6E) / (6P3Y3-6E) .Stel2 (R)(A)(D)(Q')(Q'')80 not (Q)
    (P1013Y4-60E) / ((P52P3-2EY4-E)(P52P3-2E)8(P52P3-2EY4-E)) .Stel9 (R)(A)(D)(Q')(Q'')8170spiralling hole
    (P149Y4-53E) / (P73Y4-5EP3-3E)3 .Stel2 (R)(A)(D)(Q')(Q'')82 spiralling hole
    H(P6P6) .Stel0 (A)(D) 83convex hull
    4Diff(P3Q3) .Stel0 each (R)(A)(D) 834 different P3Q3 arrangements
    20Diff(Q3P6 / P3Q3) .Stel2 or 1 each (R)(A)(D) 8320 different QP6/P3_Q3 arrangements
    P62Y4-12E .Stel0 (R)(A)(D)(Q') 83Weak quasiconvex
    P52Y4-10E=cheat .Stel0 (R)(A)(Q') 83slight interference between pyramids
    P3P3Y3-2E .Stel0 (R)(A)(Q)(D) 83not (Q'')
    (P3Y3-E)(P4Y4-E) .Stel0 (R)(A)(Q)(D) 83not (Q'')
    (P3Y3-E)(P5Y5-E) .Stel0 (R)(A)(Q)(D) 83not (Q'')
    (P129Y4-60E) / 2P12 .Stel2 (R)(A)(D)(Q')83 vertical prism tunnels
    (P3011Y4-150E) / (P30Y4-4E) .Stel1 (R)(A)(D)(Q')83 excavations needed within tunnel

    Prev - Chapter 7 - Exploration of (R)(A) Toroids
    Next - Chapter 9 - (R)(A)(Q)(T) Models Based on Prisms
    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).


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