Chapter 17 - Map Coloring Problems

  • This chapter discusses the coloring of toroidal maps. It is like the 4-color map problem on a flat plane, but the number of colors needed increases with the genus of the figure. For example, for a genus 1 model, the most number of colors needed to color any map would be 7. The interesting exercise here is to find a Stewart Toroid and to take contiguous sets of complete faces to form "regions" where each region will be colored with a single color, then to find such a model that requires the number of colors for the entire map to be 7 (e.g. a model with 7 regions where all 7 regions share at least one edge with each of the other 6 regions). The same is done with genus 2 and 3 models, which require 8 and 9 colors respectively.
    Model.Stel GenusProperties Pages with Diagram Pages with DescriptionNotes
    K3 / Q3T3=color.Stel1(R)(A)(Q)(T)(D)195, 198, 199 7 colored map - misc diagrams and views:
    Diagram by Component Polyhedron
    Diagram by Rotational Symmetry
    Diagram by Colored Region
    Color Schematic Diagram
    Exploded View of 3D model
    K3 / Q3(S3)Q3=color.Stel1(R)(A)(Q)(T)(D)196, 197 7 colored map - misc diagrams and views:
    Diagram by Colored Region
    Color Schematic Diagram
    Exploded View of 3D model
    T5 / 2R5(D5)S5Q5=color.Stel2(R)(A)(Q)(T)(D)200 8 colored map - misc diagrams and views:
    Diagram by Colored Region
    Exploded View of 3D model
    K5 / 3R5(E5)S5Q5=color.Stel3(R)(A)(Q)(T)(D)201 9 colored map - misc diagrams and views:
    Exploded View of 3D model
    SevenCongruentRegions=NotAplanar.Stel1(T)(D)203 Faces are not Aplanar

    Prev - Chapter 16 - Flight into the "K5-Galaxy"
    Next - Chapter 18 - Knotted (R)(A) Toroids
    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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