Unit V — Timaeus: Elementary Right Triangles & Construction of the Cosmic Polyhedra — Structural Synopsis

Argumentative Architecture & Interactive Diagram Guide · Timaeus 53c–56c

Timaeus 54d-55a: Six Elementary 1:sqrt(3):2 Scalene Right Triangles Forming the Cosmic Face ( G M1 ): (x - (0))^2 + (y - (0))^2 - (1) = 0 ( G A ): (x - (0))^2 + (y - (0))^2 - (4) = 0 [ M1 C ]: (0)*x + (1)*y + (1) = 0 [ C A ]: (-3)*x + (-sqrt(3))*y + (2*sqrt(3)) = 0 [ A B ]: (3)*x + (-sqrt(3))*y + (2*sqrt(3)) = 0 [ M1 G ]: (1)*x + (0)*y + (0) = 0 [ B G ]: (-1)*x + (sqrt(3))*y + (0) = 0 [ C G ]: (-1)*x + (-sqrt(3))*y + (0) = 0 A = (0, 2) B = (-sqrt(3), -1) M1 = (0, -1) M2 = (sqrt(3)/2, 1/2) M3 = (-sqrt(3)/2, 1/2) G = (0, 0) A B M1 M2 M3 G
Cosmic Equilateral Face Assembly (Timaeus 54d–55a): six elementary 1 : sqrt(3) : 2 right-angled scalene triangles (quadrances 1, 3, 4; spreads 1/4, 3/4, 1) meet thrice along their diagonals and altitudes at centroid G = X(2) to compose equilateral face < A B C >.

Architectural Thesis

In Timaeus 53c–56c, Plato executes the boldest geometrical reduction in ancient natural philosophy: fire, air, water, and earth are not ultimate stoicheia (‘not even syllables’, 48b), because all three-dimensional bodies are bounded by planar surfaces, every rectilinear plane surface resolves into triangles, and all triangles derive from two elementary right-angled triangles (stoicheia):

  1. The Two Elementary Right Triangles (53c–54b): - The isosceles right triangle (\(1 : 1 : \sqrt{2}\), quadrances \(1, 1, 2\), spreads \(\frac{1}{2}, \frac{1}{2}, 1\)), unique of its kind; four assemble around a central right-angle vertex to form each square face of the Cube (Earth). - The fairest scalene right triangle (hēmitrigōnon, \(1 : \sqrt{3} : 2\), quadrances \(1, 3, 4\)—‘having the greater leg triple in power of the lesser’, 54b5—and rational spreads \(\frac{1}{4}, \frac{3}{4}, 1\)), formed by bisecting an equilateral triangle ABC along its altitude C M3.

  2. Six-Triangle Assembly of the Cosmic Equilateral Face (54d–55a): Six elementary \(1 : \sqrt{3} : 2\) right triangles—such as A M3 G and B M3 G—converge along their diagonals and short legs at centroid G (\(X(2)\)) to compose a single equilateral triangle ABC.

  3. Generation & Transmutation of the Five Cosmic Polyhedra (54e–56c): - Tetrahedron (4 equilateral faces = 24 scalenes) \(\rightarrow\) Fire - Octahedron (8 equilateral faces = 48 scalenes) \(\rightarrow\) Air - Icosahedron (20 equilateral faces = 120 scalenes) \(\rightarrow\) Water - Cube (6 square faces = 24 isosceles right triangles) \(\rightarrow\) Earth (incommensurable with the \(\sqrt{3}\) family and therefore non-transmutable into fire/air/water) - Dodecahedron (12 regular pentagonal faces, requiring \(\mathbb{Q}(\phi)\)) \(\rightarrow\) The Whole Cosmos (55c4–6).

Stephanus Conspectus

Section

Stephanus Span

Mathematical & Philosophical Argument

Tracks

V.1 Elementary Triangles

53c–54d

Stereometric genesis; \(1:1:\sqrt{2}\) isosceles vs. \(1:\sqrt{3}:2\) half-equilateral (triple in power).

EN · EL

V.2 Five Cosmic Solids

54d–56c

6-triangle equilateral face, 4-triangle square face, 5 regular solids, and transmutation equations.

EN · EL