The Holey Monster (with 934 faces) - Numberphile

Key Concepts:

• Platonic Solids
• Archimedian Solids
• Johnson Solids
• Convexity
• Quasi-convexity
• Stewart Toroids
• Genus (of a shape)
• Regular Polygons
• Truncation
• Planer
• Tunnled

1. Introduction to Bonnie Stewart's "Adventures Among the Toroids"

• The video introduces Bonnie Stewart's mathematics book, "Adventures Among the Toroids," specifically the second edition revised in 1980, originally published in 1970.
• The book is notable for being written, illustrated, and hand-lettered by Bonnie Stewart, showcasing calligraphy and hand-drawn pictures.
• The book explores Toroids, a particular type of polyhedra that Stewart investigated.

2. Platonic Solids: The Foundation

• Platonic solids are the starting point for understanding polyhedra.
• A cube is used as an example to illustrate the fundamental properties of Platonic solids:
  • Every face is a regular polygon (e.g., squares for a cube).
  • All faces are identical or interchangeable due to symmetry.
  • Edges are interchangeable.
  • Corners are interchangeable.
• The theory of polyhedra develops by relaxing or tweaking these properties.

3. Relaxing the Conditions: Cuboctahedron and Icosidodecahedron

• By relaxing the condition that all faces must be identical, we can derive new shapes.
• Cuboctahedron:
  • Faces are regular polygons (triangles and squares).
  • Every edge is identical (connecting a triangle and a square).
  • Every corner is identical.
  • It has six square faces (like a cube) and eight triangular faces (like an octahedron).
• Icosidodecahedron:
  • Faces are pentagons and triangles.
  • Satisfies all conditions for Platonic solids except that not all faces are identical.
  • Combination of a dodecahedron (12 pentagonal faces) and an icosahedron (20 triangular faces).

4. Further Relaxation: Truncated Cube and Archimedean Solids

• Relaxing the constraint on identical edges leads to further shapes.
• Truncated Cube:
  • All edges are the same length.
  • Edges are not in identical positions (some are between an octagon and a triangle, others between two octagons).
  • Every corner is identical.
• Archimedean Solids:
  • Faces are regular polygons.
  • Every corner is identical.
  • There are 13 Archimedean solids.
• Prisms and Anti-prisms:
  • Infinitely many prisms can be created by joining two n-gons with a ring of squares.
  • Infinitely many anti-prisms can be created by joining two n-gons with a ring of equilateral triangles.
• Rhombicosidodecahedron:
  • Faces are regular polygons.
  • Faces and edges are not identical.
  • Every corner is identical.
• Truncated Icosidodecahedron:
  • Created by truncating (chopping off corners) of an icosidodecahedron.

5. Johnson Solids: No Overall Symmetry Required

• In 1966, Norman Johnson explored shapes made of regular polygons without requiring any overall symmetry.
• An extra condition of convexity is imposed.
• Convexity: A shape is convex if, for any two points inside the shape, the straight line connecting them is also entirely inside the shape.
• Johnson focused on convex shapes built from regular polygons.
• Examples of Johnson solids:
  • Square-based pyramid
  • Pentagonal pyramid
  • Square cupola (slice of a rhombicuboctahedron)
  • Pentagonal rotunda (half of an icosidodecahedron)
  • Triangular hebesphenorotunda (Johnson solid #92, featuring triangles, squares, pentagons, and a hexagon)
• Norman Johnson listed 92 solids, and in 1969, Victor Zagal proved that the list was complete.

6. Stewart Toroids: Introducing Holes and Quasi-Convexity

• Stewart explored Toroids, polyhedra with holes.
• Toroids violate the property of convexity.
• Quasi-convexity: A softening of convexity that allows for holes.
  • Imagine filling in all the holes and gaps in the shape to create a convex hull.
  • If all the edges of the convex hull are already present as edges in the original shape, the shape is quasi-convex.
  • If new edges are created during the filling process, the shape is not quasi-convex.

7. Defining Stewart Toroids: Four Criteria

• Stewart defined Toroids based on four criteria:
  1. Regular Faces: All faces are regular polygons.
  2. Quasi-Convexity: Filling in holes doesn't create new edges.
  3. A Planer: Neighboring faces should not lie flat against each other (angle greater than zero).
  4. Being Tunnled: The shape has clear, definite tunnels going through it, not just pinholes or slits.
• Genus: Relates to the number of holes in a shape (e.g., a ball or cube has genus 0, a torus has genus 1).

8. Examples of Stewart Toroids

• Rotunda-Drilled Truncated Icosidodecahedron:
  • Derived from a truncated icosidodecahedron (an Archimedean solid).
  • Each decagonal face is drilled out with a rotunda (half of an icosidodecahedron).
  • The internal polyhedron is a rhombicosidodecahedron.
• The Holy Monster:
  • A Stewart Toroid with genus 46.
  • Has 934 faces.
• The Webb Toroid:
  • Discovered by Robert Webb.
  • Has genus 41.
  • Large enough to fit the Holy Monster inside.
• Record Holder:
  • Created by combining the Webb Toroid and the Holy Monster.
  • Has genus 87.
  • Has 2,298 faces.

9. Conjecture on the Number of Stewart Toroids

• It is conjectured that the number of Stewart Toroids satisfying the defined criteria is finite, although enormously large.
• The possibilities for creating tunnels and variations within shapes are vast.

10. Conclusion

• The video explores the fascinating world of polyhedra, starting with Platonic solids and progressing to more complex shapes like Archimedean solids, Johnson solids, and Stewart Toroids.
• Bonnie Stewart's "Adventures Among the Toroids" introduced the concept of quasi-convexity and the study of polyhedra with holes.
• The quest to find Toroids with the highest genus continues, with the current record held by a combination of the Webb Toroid and the Holy Monster.