Unit IV · Harmonices Mundi Liber II: De Congruentia Figurarum (1619)

Original GEOMETOR English translation from Johannes Kepler’s Harmonices Mundi Liber II: Architectonicus, seu ex Geometria Figurata, de Figurarum Regularium Congruentia in Plano vel Solido (Linz: Tampach/Plancus, 1619, pp. 47–66; Frisch V, pp. 126–144).

1. Definitions — Planar and Solid Congruence

In the First Book we contemplated regular plane figures singly, examining which of them are knowable by inscription in a circle; now in the Second Book—which may rightly be called Architectonic—we consider their sociability and mutual fitting together (congruentia), inquiring which regular and star polygons join edge-to-edge either to pave an entire plane continuously without gaps, or to close up in three dimensions into a uniform solid body.

  • Definitions I–III: Regular polygons exhibit planar congruence when the interior angles meeting around every common vertex equal four right angles ($360^\circ$), and when the surrounding rings propagate consistently across every edge to infinity. They exhibit solid congruence when three or more plane angles (or star-polygon points) meet around a vertex with sum strictly less than four right angles ($< 360^\circ$), folding upward into a closed polyhedron.

2. Propositions I–XX — The Eleven Uniform Planar Tilings

First, three and only three species of pure regular plane congruence fill the plane using a single polygon species:

  1. Six Equilateral Triangles around each vertex (3.3.3.3.3.3, $6 \times 60^\circ = 360^\circ$);

  2. Four Squares around each vertex (4.4.4.4, $4 \times 90^\circ = 360^\circ$);

  3. Three Regular Hexagons around each vertex (6.6.6, $3 \times 120^\circ = 360^\circ$).

Next, when two or three species of regular polygons meet identically at every vertex, arithmetic alone permits seventeen local angle sums equal to $360^\circ$—including deceptive combinations such as a triangle, heptagon, and forty-two-gon ($60^\circ + 128\frac{4}{7}^\circ + 171\frac{3}{7}^\circ = 360^\circ$) or two pentagons with a decagon ($108^\circ + 108^\circ + 144^\circ = 360^\circ$). Yet local angle sum does not suffice! For around any odd-sided polygon—such as a triangle ($n = 3$) or pentagon ($n = 5$)—two distinct neighboring polygon species cannot alternate evenly around an odd number of edges without colliding! Eliminating all self-contradictory vertex figures leaves eight and strictly eight semi-regular uniform tilings—making eleven uniform planar congruences in all:

  • Three-valent vertices ($k = 3$): 3.12.12 (triangles and dodecagons), 4.6.12 (square, hexagon, dodecagon), and 4.8.8 (V488, squares and octagons).

  • Four-valent vertices ($k = 4$): 3.6.3.6 (V3636, alternating triangles and hexagons), and 3.4.6.4 (V3464, triangle, square, hexagon, square).

  • Five-valent vertices ($k = 5$): 3.3.3.4.4 (adjacent squares), 3.3.4.3.4 (separated squares), and the chiral snub tiling 3.3.3.3.6 (V33336, four triangles and one hexagon, existing in left- and right-handed mirror forms).

3.6.3.6 (Trihexagonal) 4.8.8 (Truncated Square) 3.4.6.4 (Rhombitrihexagonal) 3.3.3.3.6 (Chiral Snub)

3. Pentagonal & Star-Pentagram Congruence (Figure Aa)

Although regular pentagons ($108^\circ$) and regular decagons ($144^\circ$) cannot form a uniform vertex-transitive tiling of convex polygons alone, if we admit the five-pointed star (stella quinquangula, $\{5/2\}$, tip angle $36^\circ$, re-entrant pocket $72^\circ$), marvelous fivefold flower-like congruences blossom across the plane! Around a central decagon DEC0 (D0 in the left panel below), five regular pentagons and five star pentagrams interlock seamlessly because:

$$144^\circ + 108^\circ + 108^\circ = 360^\circ, \qquad 144^\circ + 144^\circ + 72^\circ = 360^\circ, \qquad 3 \times 108^\circ + 36^\circ = 360^\circ,$$

generating radiating rings of decagons, stars, and twin-pentagon monsters that expand outward indefinitely according to a hidden law of the Divine Section!

D_0 I. Congruentia Stellata Aa (Decagons, Pentagons & {5/2}) II. Triacontaedron Rhombicum (30 Golden Rhombi, phi : 1)

4. Propositions XXV–XXVIII — Star Solids, Rhombic Solids & 13 Archimedeans

In three-dimensional solid congruence, Kepler completes the inventory of symmetric polyhedra across four families:

  1. The Five Convex Platonic Solids (corpora quinque regularia): Cube, Tetrahedron, Dodecahedron, Icosahedron, Octahedron.

  2. Two Regular Non-Convex Star Polyhedra (corpora stellata): Admitting twelve regular pentagrams $\{5/2\}$ as faces produces two utterly regular augmented star solids—the thorny echinus or Small Stellated Dodecahedron ($\{5/2, 5\}$, twelve 5-pointed vertices) and the Great Stellated Dodecahedron ($\{5/2, 3\}$, twenty trihedral vertices)!

  3. Two Isohedral Rhombic Zonohedra: The 12-faced Rhombic Dodecahedron ($\sqrt{2}:1$) and the 30-faced Golden Rhombic Triacontahedron (PQRS in the right panel above, long diagonal PRgold to short diagonal QSgold in ratio $\phi : 1$).

  4. The Thirteen Convex Archimedean Semi-Regular Solids (corpora semiregularia Archimedis, Prop. XXVIII): Cleanly setting aside the two infinite families of regular prisms and antiprisms, Kepler reconstructs, names, and proves the completeness of all thirteen semi-regular convex solids recorded by Pappus (Synagoge V.19): Truncated Tetrahedron, Cuboctahedron, Truncated Cube, Truncated Octahedron, Rhombicuboctahedron, Great Rhombicuboctahedron (Truncated Cuboctahedron), Snub Cube (Cubus simus), Icosidodecahedron, Truncated Dodecahedron, Truncated Icosahedron, Rhombicosidodecahedron, Great Rhombicosidodecahedron (Truncated Icosidodecahedron), and Snub Dodecahedron (Dodecaedron simum).