The Four Kepler
Discovery A: The Star Polygon
Given an image of a regular pentagon & star-shaped pentagram. Answer the questions below.
Question 1: Is the pentagram a polygon?
Question 2: Is the pentagram a regular polygon?
Question 3: If we can have star-shaped polygons (2D), can we have star-shaped solids (3D)?
Investigation B: The Big Four
Observe the properties / characteristics of the four solids below.
*Focus: face shape, face regularity, vertex regularity, convexity, limit, vertex cross-section
Solid I
Describe the properties observed in Solid I.
Solid II
Describe the properties observed in Solid II.
Solid III
Describe the properties observed in Solid III.
Solid IV
Describe the properties observed in Solid IV.
Regularity Analysis
Do these four solids deserve the title of 'Regular Polyhedra'?
Guiding Questions
Question 1: What are the requirements for a regular (Platonic) Solid?
Question 2: Do the four solids above meet these requirements?
Question 3: Are the four solids above semi-regular (Archimedean)?
Conclusion:
- Because they satisfy the symmetry and face-vertex consistency of Platonic solids, they are classified as Regular Non-Convex Polyhedra.
- They are just as regular as a cube or tetrahedron, but they have a non-convex space.
- By allowing faces to be stars or to pass through one another, we expanded the Platonic family from 5 members to 9.
- These additional four solids are considered special Platonic Solids.
- They are called Kepler-Poinsot Solids: Small Stellated Dodecahedron, Great Stellated Dodecahedron, Great Dodecahedron, & Great Icosahedron.
Kepler vs. Poinsot
The Kepler-Poinsot solids are the only four regular non-convex polyhedra in existence. While the five Platonic solids are simple and convex, these four are star-like shape and intersect themselves.
Why are they regular?
The Poinsot Solids (Inward Stars)
Summary:
- They follow the same strict rules as the Platonic solids.
- All their faces are identical, regular polygons (pentagon / pentagram)
- The same number of faces meet at each vertex
- The faces self-intersect / cut through one another.
- The Line Segment Test: If you select two points, a straight line segment connecting them may pass through the empty space outside the solid.
- They are formed by stellation - imagine taking a regular dodecahedron and extending its faces outward until they meet to form points (analogy: how you draw a 2D star by extending the sides of a pentagon).
- Example 1: Small Stellated Dodecahedron - formed by pentagram faces. It looks like a dodecahedron with a five-sided pyramid on each face.
- Example 2: Great Stellated Dodecahedron - made of pentagrams, but it is pointier and based on a different arrangement of the triangular points.
The Poinsot Solids (Inward Stars)
- They are formed by 'greatening' / expanding. Instead of extending edges outward, these solids use large, overlapping faces that pass through the center of the shape inward.
- Example 1: Great Dodecahedron - It has the same outer shell as a regular dodecahedron (20 pentagons), but its faces are five-pointed stars that intersect deep inside the shape.
- Example 2: Great Icosahedron - It has the same outer shell as a regular icosahedron (20 triangles), but it is composed of large, intersecting triangles that create a complex star shape.
Summary: