Unit V — Timaeus: Elementary Right Triangles & Construction of the Cosmic Polyhedra — Critical Commentary & Notes¶
Exact Symbolic Reconstruction, Greek Mathematical Philology & Canon Reception · Timaeus 53c–56c
I. Exact Geometric & Symbolic Reconstruction¶
In geometor.plato.constructions.construct_timaeus_triangles and construct_timaeus_cube_face (see Demonstrations), Plato’s two elementary right triangles (53c–55c) are constructed purely by compass and straightedge and audited via Rational Trigonometry invariants:
The Fairest Scalene Half-Triangle (`1 : √3 : 2`, `54a–b`):
Constructing equilateral triangle ABC of side \(2\sqrt{3}\) (\(Q(A,B) = 12\), Elements I.1) and drawing its three medians/altitudes through midpoints M1, M2, M3 meeting at centroid G (\(X(2)\)) partitions the face into six congruent half-triangles (e.g., A M3 G and B M3 G).
Each elementary triangle \(\triangle G M_3 A\) has exact rational quadrances and spreads over \(\mathbb{Q}\):
\[Q(G, M_3) = 1, \quad Q(A, M_3) = 3, \quad Q(G, A) = 4, \qquad s_A = \frac{1}{4}, \quad s_G = \frac{3}{4}, \quad s_{M_3} = 1.\]
Why Six Triangles Instead of Two? (54d–e): A geometric reader naturally asks why Timaeus composes each equilateral cosmic face out of six \(1:\sqrt{3}:2\) triangles meeting at G (ἐκ ἓξ τοιούτων συμπαγέντων, 54e1) rather than just two back-to-back along altitude C M3. As Cornford (1937, pp. 214–218) and Taylor (1928, p. 374) point out, bisecting an equilateral triangle along one altitude breaks its threefold rotational symmetry, whereas drawing all three altitudes (A M1, B M2, C M3) restores complete \(D_3\) symmetry around \(X(2)\)!
Exact Triangle Accounting & Chemical Transmutation Equations (`56a–c`):
Letting \(T\) denote one equilateral cosmic face (6 scalenes), the three transmutable regular solids satisfy exact integer conservation of faces and elementary triangles:
1 Water (icosahedron, \(20T = 120\) scalenes) \(\longrightarrow\) 1 Fire (\(4T = 24\)) \(+\) 2 Air (\(2 \times 8T = 16T = 96\)): \(20 = 4 + 2(8)\).
2.5 Air (\(20T\)) \(\Longleftrightarrow\) 1 Water (\(20T\)); 1 Air (\(8T\)) \(\Longleftrightarrow\) 2 Fire (\(2 \times 4T = 8T\)).
Conversely, Earth (cube, 6 square faces \(\times 4\) isosceles \(1:1:\sqrt{2}\) triangles \(= 24\) isosceles triangles) belongs to the \(\mathbb{Q}(\sqrt{2})\) quadratic field, algebraically disjoint from \(\mathbb{Q}(\sqrt{3})\), so Earth particles can disintegrate and recombine only into Earth (56d)!
II. Literal Greek Mathematical Philology (Contra Paraphrasim)¶
54b4–5 — οὗ ἡ ὑποτείνουσα τῆς ἐλάττονος πλευρᾶς διπλῆ μήκει· τίθεμεν δὴ τοῦτο… τὴν μείζω πλευρὰν τῆς ἐλάττονος τριπλῆν δυνάμει ἔχον (54b5 / 54b): Notice Plato’s razor-sharp contrast between μήκει (in length) and δυνάμει (in square/power): the hypotenuse (2) is double in length of the lesser leg (1), whereas the greater leg (\(\sqrt{3}\)) is triple in power (\(Q = 3 = 3 \times 1\)) of the lesser leg! Jowett paraphrases at 54b ‘having the square of the greater side three times the square of the lesser’, which is mathematically sound, though in Greek Plato writes compactly τριπλῆν δυνάμει (‘triple in power’), applying Theaetetus's exact vocabulary from Unit II.
55c4–6 — ἔτι δὲ οὔσης συστάσεως μιᾶς πέμπτης, ἐπὶ τὸ πᾶν ὁ θεὸς αὐτῇ κατεχρήσατο ἐκεῖνο διαζωγραφῶν: The fifth regular construction—the regular dodecahedron bounded by twelve regular pentagons—cannot be tiled by \(1:1:\sqrt{2}\) or \(1:\sqrt{3}:2\) triangles because the pentagon diagonal-to-side ratio is the Golden Ratio \(\phi = \frac{1+\sqrt{5}}{2}\) (Elements XIII.8 & XIII.17). Plato therefore reserves it for embroidering the spherical zodiacal cosmos itself (διαζωγραφῶν).
III. Reception in Euclid Book XIII, Aristotle & Pacioli¶
Euclid Book XIII & Proclus: Proclus (In Euclidem, Friedlein 68.20–23) famously declares that Euclid was a Platonist who composed the entire thirteen-book Elements with the construction and exhaustive uniqueness proof of the five cosmic Platonic figures (XIII.13–18) as its supreme architectural telos.
Aristotle (De Caelo III.7–8, 306a1–307b18;
plato:timaeus): Mounts a sustained physical and geometric critique—arguing that suspending division at two-dimensional weightless planes violates mathematical continuity, and that neither pyramids nor octahedra/icosahedra tile three-dimensional space without leaving voids (anticipating the honeycomb theorem that among regular Platonic solids only the cube tiles \(\mathbb{E}^3\), though combinations of regular tetrahedra and octahedra do tile \(\mathbb{E}^3\)!).Luca Pacioli (De Divina Proportione, 9 occurrences of
plato:timaeus:53c-56c): Grounds his entire Renaissance treatise on the Golden Ratio and Leonardo da Vinci’s sixty polyhedral plates in Timaeus 53c–56c.
IV. Executable Symbolic Verification¶
In accordance with GEOMETOR’s strict content/code separation for historical editions, executable geometor.model SymPy constructions and Rational Trigonometry verification routines for Unit V live in src/geometor/plato/ and the companion walkthrough Demonstrations.