Humanitext Reader

Archimedes · Fragments §1.1#2

Calculation of Vertices and Edges in Polyhedra

Passage 2 of 4 · Greek

Summary

For each of the 13 Archimedean polyhedra, the number of solid angles and edges is calculated based on the types and numbers of their faces, applying the formula presented in the previous section.

§1.1#2Τὸ μὲν οὖν πρῶτον τῶν ἀνομοιογενῶν ιγ πολυέδρων, ἐπεὶ περιέχετοι τριγώνοις △Ζ καὶ ἑξαγώνοις δ, γωνίας μὲν ἔχει στερεὰς ιβ, πλευρὰς δὲ ιη.
\nTherefore, the first of the 13 polyhedra of different kinds, since it is contained by 6 triangles (△) and 4 hexagons, has 12 solid angles and 18 edges.
Τῶν μὲν γὰρ τεσσάρων τριγώνων αἵ τε γωνίαι ιβ εἰσὶν καὶ αἱ πλευραὶ ιβ, τῶν δὲ δ ἑξαγώνων αἵ τε γωνίαι κδ εἰσὶν καὶ αἱ πλευραὶ κδ· γενομένου δὴ τοῦ ἀριθμοῦ παντὸς λϚ΄ ἀναγκαῖόν ἐστιν τὸν μὲν τῶν στερεῶν γωνιῶν ἀριθμὸν τρίτον μέρος εἶναι τοῦ προειρημένου ἀριθμοῦ, ἐπεὶ καὶ ἑκάστη τῶν στερεῶν αὐτοῦ γωνιῶν ἐπιπέδοις γωνίαις περιέχεται γ, τὸ δὲ τῶν πλευρῶν πλῆθος τὸ ἥμισυ τοῦ ἀριθμοῦ, τουτέστιν τοῦ λϚ΄, ὥστε εἶναι πλευρὰς ιη. Τῶν δὲ τετρακαιδεκαέδρων τὸ πρῶτον περιέχεται τργώνοις η καὶ τετραγώνοις Ϛ΄, ὥστε ἔχειν στερεὰς μὲν γωνίας ιβ ἑκάστη γὰρ αὐτοῦ γωνία ὑπὸ τεσσάρων ἐπιπέδων γωνιῶν περιέχεται·
For of the four triangles, the angles are 12 and the edges are 12, and of the 4 hexagons, the angles are 24 and the edges are 24; so when the whole number becomes 36, it is necessary that the number of the solid angles is one-third of the aforementioned number, since indeed each of its solid angles is contained by 3 plane angles, and the multitude of the edges is half of the number, that is, of 36, so that there are 18 edges.\nAnd of the tetradecahedra, the first is contained by 8 triangles and 6 squares, so that it has 12 solid angles (for each of its angles is contained by four plane angles) and has 24 edges.
πλευρὰς δὲ ἔχει κδ, Τὸ δὲ δεύτερον τῶν τετρακαιδεκαέδρων, ἐπεὶ περιέχεται τετραγώνοις Ϛ΄ καὶ ἑξαγώνοις η, ἕξει στερεὰς μὲν γωνίας κδ ἑκάστη γὰρ τῶν γωνιῶν αὐτοῦ περιέχεται ὑπὸ γ γωνιῶν ἐπιπέδων πλευρὰς δὲ ἔχει λϚ΄, Τὸ δὲ τρίτον τῶν τετρακαιδεκαέδρων, ἐπεὶ περιέχετοι τριγώνοις η καὶ ὁκταγώνοις Ϛ΄, ἕξει στερεὰς μὲν γωνίας κδ, πλευρὰς δὲ λς΄1. Tῶν δὲ ἑκκαιεικοσαέδρων τὸ μὲν πρῶτον, ἐπεὶ περιέχεται τριγώνοις τε η καὶ τετραγώνοις ιη, ἕξει στερεὰς μὲν γωνίας κδ, πλευρὰς δὲ μη, Τὸ δὲ δεύτερον τῶν ἑκκαιεικοσαέδρων, ἐπεὶ περιέχεται τετραγώνοις ιβ καὶ ἑξαγώνοις η καὶ ὀκταγώνοις Ϛ΄, ἕξει στερεὰς μὲν γωνίας μη, πλευρὰς δὲ οβ. Τῶν δὲ δυοκαιτριακονταέδρων τὸ μὲν πρῶτον, ἐπεὶ περιέχεται τριγώνοις τε κ καὶ πενταγώνοις ιβ, ἕξει στερεὰς μὲν γωνίας λ, πλευρὰς δὲ ξ.
And the second of the tetradecahedra, since it is contained by 6 squares and 8 hexagons, will have 24 solid angles (for each of its angles is contained by 3 plane angles) and has 36 edges. And the third of the tetradecahedra, since it is contained by 8 triangles and 6 octagons, will have 24 solid angles and 36 edges1.\nAnd of the icosihexahedra, the first, since it is contained by 8 triangles and 18 squares, will have 24 solid angles and 48 edges. And the second of the icosihexahedra, since it is contained by 12 squares, 8 hexagons, and 6 octagons, will have 48 solid angles and 72 edges.\nAnd of the triacontadihedra, the first, since it is contained by 20 triangles and 12 pentagons, will have 30 solid angles and 60 edges.
Τὸ δὲ δεύτερον τῶν δυοκαιτριακονταέδρων, ἐπεὶ περιέχεται πενταγώνοις ιβ καὶ ἑξαγώνοις κ, ἕξει στερεὰς μὲν γωνίας ξ, πλευρὰς δὲ 𝔮.
And the second of the triacontadihedra, since it is contained by 12 pentagons and 20 hexagons, will have 60 solid angles and 90 edges.
Τὸ δὲ τρίτον τῶν δυοκαιτριακονταέδρων, ἐπεὶ περιέχεται τριγώνοις τε κ καὶ δεκαγώνοις ιβ, ἕξει στερεὰς μὲν γωνίας ξ, πλευρὰς δὲ (??). Τὸ δὲ ὀκτωκαιτριακοντάεδρον, ἐπεὶ περιέχεται τριγώνοις τε λβ καὶ τετραγώνοις ἕξ, ἕξει στερεὰς μὲν γωνίας κδ, πλευρὰς δὲ ξ. Tῶν δὲ δυοκαιεξηκονταέδρων τὸ μὲν πρῶτον, ἐπεὶ περιέχεται τριγώνοις τε κ καὶ τετραγώνοις λ καὶ πενταγώνοις ιβ, ἕξει στερεὰς μὲν γωνίας ξ, πλευρὰς δὲ ρκ.
And the third of the triacontadihedra, since it is contained by 20 triangles and 12 decagons, will have 60 solid angles and (??) edges.\n And the triacontaoctahedron, since it is contained by 32 triangles and six squares, will have 24 solid angles and 60 edges.\nAnd of the hexacontadihedra, the first, since it is contained by 20 triangles, 30 squares, and 12 pentagons, will have 60 solid angles and 120 edges.
Τὸ δὲ λοιπὸν τῶν δυοκαιεξηκονταέδρων, ἐπεὶ περιέχεται τετραγώνοις λ καὶ ἑξαγωνοις καὶ δεκαγώνοις ιβ, ἕξει στερεὰς μὲν γωνίας ρκ, πλευρὰς δὲ ρπ. Τὸ δὲ δυοκαιενενηκοντάεδρον, ἐπεὶ περιέχεται τριγώνοις τε π καὶ πενταγώνοις ιβ, ἕξει στερεὰς μὲν γωνίας ξ, πλευρὰς δὲ ρν.
And the remaining one of the hexacontadihedra, since it is contained by 30 squares, hexagons, and 12 decagons, will have 120 solid angles and 180 edges.\nAnd the nonacontadihedron, since it is contained by 80 triangles and 12 pentagons, will have 60 solid angles and 150 edges.

Notes

  1. p.204τριγώνοις △Ζ — In the manuscript tradition, the symbol `△` (representing a triangle) is followed by `Ζ` (7) indicating the number, but the actual number of faces should be 4 (`δ`). Although a similar notation was seen in the previous section, the calculation is carried out here with the correct count of 'four triangles' (`τεσσάρων τριγώνων`).
  2. p.204γενομένου δὴ τοῦ ἀριθμοῦ παντὸς λϚ΄ — Genitive absolute construction. `γενομένου` is the aorist middle participle (neuter genitive singular) of `γίγνομαι`, with `τοῦ ἀριθμοῦ παντὸς` as its subject ('when the entire number has become 36').
  3. p.204ἑκάστη γὰρ αὐτοῦ γωνία ὑπὸ τεσσάρων ἐπιπέδων γωνιῶν περιέχεται — A parenthetical clause introduced by `γάρ` explaining the reason, inserted into the main clause (`ὥστε ἔχειν...`). It explains that each vertex of the first type of tetradecahedron (truncated octahedron) is contained by four plane angles (two triangles and two squares).
  4. p.205τετραγώνοις λ καὶ ἑξαγωνοις καὶ δεκαγώνοις ιβ — This passage indicates the face configuration of the second type of hexacontadihedron (great rhombicosidodecahedron). In the text, the number corresponding to hexagons (`ἑξαγωνοις`), which should be 20 (`κ`), has been omitted in the manuscript between '30 squares' (`τετραγώνοις λ`) and '12 decagons' (`δεκαγώνοις ιβ`).
  5. p.204πλευράς δὲ (??) — This indicates the number of edges of the third type of triacontadihedron (truncated dodecahedron). While it should contain `𝔮` (or `ϟ΄`) representing 90, it is lost in the manuscript tradition and represented as `(??)`.

Cite this passage

Archimedes, Fragments §1.1#2. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg0552.tlg013.humanitext-grc1:1.1%232

Please note the AI-draft status of the translation and the date accessed.

Translation, notes and summary are AI-generated drafts, revised through reader feedback.