Unit IV · Executive Geometric Summary

Core Architectonic Theorems (Harmonices Mundi Book II, 1619)

  1. First Complete Classification of the 11 Uniform Planar Tilings: Solving the rational interior-angle equation

    $$\sum_{i=1}^{k} \frac{n_i - 2}{2 n_i} = 1 \qquad (3 \le k \le 6,\; n_i \ge 3)$$

    and filtering local solutions by global edge-alternation consistency around odd-sided faces, geometor.kepler.harmonices_geometry.enumerate_archimedean_planar_tilings reproduces Kepler's exact 11 uniform vertex-transitive planar tessellations:

The 11 Uniform Archimedean Planar Tilings (Book II)

Vertex Symbol

Valence $k$

Interior Angles ($= 360^\circ$)

Symmetry & Classification

3.3.3.3.3.3

$6$

$60^\circ \times 6$

Pure regular triangular lattice ($p6m$)

4.4.4.4

$4$

$90^\circ \times 4$

Pure regular square lattice ($p4m$)

6.6.6

$3$

$120^\circ \times 3$

Pure regular hexagonal honeycomb ($p6m$)

3.12.12

$3$

$60^\circ + 150^\circ + 150^\circ$

Truncated hexagonal ($p6m$)

4.6.12

$3$

$90^\circ + 120^\circ + 150^\circ$

Great rhombitrihexagonal ($p6m$)

4.8.8

$3$

$90^\circ + 135^\circ + 135^\circ$

Truncated square ($p4m$)

3.4.6.4

$4$

$60^\circ + 90^\circ + 120^\circ + 90^\circ$

Small rhombitrihexagonal ($p6m$)

3.6.3.6

$4$

$60^\circ + 120^\circ + 60^\circ + 120^\circ$

Trihexagonal / Kagome ($p6m$)

3.3.3.3.6

$5$

$60^\circ \times 4 + 120^\circ$

Chiral snub hexagonal ($p6$, enantiomorphic)

3.3.3.4.4

$5$

$60^\circ \times 3 + 90^\circ \times 2$

Elongated triangular ($cmm$)

3.3.4.3.4

$5$

$60^\circ + 60^\circ + 90^\circ + 60^\circ + 90^\circ$

Snub square ($p4g$)

  1. Kepler–Poinsot Star Polyhedra & Non-Periodic Tiling Precursor: Book II introduces the Small Stellated Dodecahedron $\{5/2, 5\}$ and Great Stellated Dodecahedron $\{5/2, 3\}$ (Poinsot 1810) alongside the Figure Aa pentagram–decagon tiling that directly inspired Roger Penrose's aperiodic P1 tiling (1974).

  2. Completeness of the 13 Archimedean Convex Solids: By separating prisms ($4.4.n$) and antiprisms ($3.3.3.n$) into infinite families, Kepler gave the first complete modern proof that exactly 13 convex vertex-transitive semi-regular solids exist.