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Euclid closes the Elements by proving that\nthose conditions admit exactly five shapes and no sixth, which makes this the oldest complete\ncatalogue in mathematics. This bundle is all of it.\n\n### The set\n\n- **Tetrahedron** · {3,3} · 4 faces · 8.165 x 7.071 x 6.667 mm · 8.1650 mm edge · 64.15 mm³ · 1.129 ct in diamond\n- **Hexahedron** · {4,3} · 6 faces · 5.774 x 5.774 x 5.774 mm · 5.7735 mm edge · 192.45 mm³ · 3.387 ct in diamond\n- **Octahedron** · {3,4} · 8 faces · 8.165 x 7.071 x 5.774 mm · 7.0711 mm edge · 166.67 mm³ · 2.933 ct in diamond\n- **Dodecahedron** · {5,3} · 12 faces · 9.822 x 9.342 x 7.947 mm · 3.5682 mm edge · 348.15 mm³ · 6.127 ct in diamond\n- **Icosahedron** · {3,5} · 20 faces · 9.822 x 8.507 x 7.947 mm · 5.2573 mm edge · 317.02 mm³ · 5.580 ct in diamond\n\nAll five share one **10 mm circumsphere**, so they nest inside a single sphere and stand\ntogether in a scene at their true relative sizes. That is why the tetrahedron looks small: it\nis small. On a common sphere it encloses 12.3% of that sphere's volume against the\ndodecahedron's 66.5%.\n\n### Built to the mathematics, and checked against it\n\nEach solid is generated from its exact vertex coordinates. Every face plane is then derived as\na supporting plane of that vertex set, so the faceting is a property of the solid rather than a\nsecond table that could disagree with it. Before release each one is measured back against the\nclosed forms:\n\n| Solid | Schläfli | Faces | Vertices | Edges | Edge (mm) | Dihedral | Volume (mm³) |\n|---|---|---|---|---|---|---|---|\n| Tetrahedron | {3,3} | 4 equilateral triangles | 4 | 6 | 8.1650 | 70.529° | 64.15 |\n| Hexahedron | {4,3} | 6 squares | 8 | 12 | 5.7735 | 90.000° | 192.45 |\n| Octahedron | {3,4} | 8 equilateral triangles | 6 | 12 | 7.0711 | 109.471° | 166.67 |\n| Dodecahedron | {5,3} | 12 regular pentagons | 20 | 30 | 3.5682 | 116.565° | 348.15 |\n| Icosahedron | {3,5} | 20 equilateral triangles | 12 | 30 | 5.2573 | 138.190° | 317.02 |\n\nEvery edge within one solid is the same length to one part in 1e15. Every dihedral angle is the\nexact value, held on every edge. Every volume matches its closed form to one part in 1e9. The\nfiles are then read back one reader per format: Open CASCADE for the STEP, assimp for the FBX\nand the GLB, rhino3dm for the 3DM, and the signed volume compared, which only comes out right\nif winding, normals and scale all survived the write.\n\n### Three of these are real crystals. Two of them cannot be.\n\nThe **octahedron** is the shape rough gem diamond actually grows in, and the early diamond cuts\nare that crystal worked down: the single cut is an octahedron with the top point sawn off for a\ntable. The **cube** is the commonest habit in the mineral kingdom, fluorite, pyrite, halite and\ngalena. The **tetrahedron** is sphalerite and tetrahedrite, which is where the latter gets its\nname.\n\nThe **dodecahedron** and the **icosahedron** cannot occur as ordinary crystals at all. Both\ncarry fivefold symmetry, and the crystallographic restriction theorem rules fivefold rotation\nout of any periodic lattice. The two minerals usually mistaken for the dodecahedron are not it:\npyrite's pyritohedron has twelve pentagonal faces, but irregular ones, and garnet's rhombic\ndodecahedron has twelve faces that are rhombi. Icosahedral symmetry does turn up in\nquasicrystals, which are ordered without being periodic, and the natural one, icosahedrite, was\nconfirmed in 2009.\n\n### Formats\n\nSix per solid, thirty files.\n\n- **`.step`** AP214 solid, one planar face per face of the solid, closed shell. Opens in Rhino,\n  SolidWorks, Fusion, FreeCAD and MatrixGold, with the faces selectable, filletable and\n  booleanable.\n- **`.stl`** binary mesh, watertight, outward normals. Print ready.\n- **`.3dm`** Rhino 7: the closed mesh on a Brilliani Labs layer, model units millimetres.\n- **`.obj`** the same mesh in a text format, for renderers and DCC tools.\n- **`.fbx`** binary FBX 7.4 with a material and per-face normals, for Blender, 3ds Max, Maya,\n  Cinema 4D, Unity and Unreal. Binary rather than ASCII, so Blender opens it.\n- **`.glb`** binary glTF, flat shaded with the material embedded, for web viewers and AR.\n\n### Conventions\n\nMillimetres, real-world scale. Y up, one face resting on the ground plane at y = 0, the\ncentroid on the Y axis, and the midpoint of one edge of that resting face on the +X axis, which\nis what leaves the cube axis-aligned rather than standing on a plan diagonal. Meshes are welded\nand closed: no duplicate vertices, every edge shared by exactly two triangles, Euler\ncharacteristic 2. Scale uniformly for any other size.\n\nEach solid ships with a `SPEC.txt` carrying its edge length, circumsphere, midsphere, insphere,\ndihedral angle, surface area, volume and the weight it would carry cut in diamond or moissanite.\n\nGenerated by Brilliani Labs. No third-party model was converted, retopologised or derived from\nat any stage.\n","imageAlt":"Platonic Solids - All Five - Print and Render ...","url":"https://www.cgtrader.com/3d-model-packs/platonic-solids-all-five-print-and-render-ready","isCLDApplicable":false,"isSaleOffApplicable":false,"gaAttributes":{},"primaryImage":{"gridFallbackUrl":"https://media.cgtrader.com/variants/j9sr6wg4966bbhefvjxdli0lf3bx/a26e47dab5f2d22c43d6c5ce4b4b46ecc30c70918878397cba1a10c1e35d7bfc/platonic-solids-all-five-print-and-render-ready-3d-model-7f604f84e4.jpg","gridUrl":"https://media.cgtrader.com/variants/j9sr6wg4966bbhefvjxdli0lf3bx/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/platonic-solids-all-five-print-and-render-ready-3d-model-7f604f84e4.webp","gridCandidates":[{"url":"https://media.cgtrader.com/variants/j9sr6wg4966bbhefvjxdli0lf3bx/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/platonic-solids-all-five-print-and-render-ready-3d-model-7f604f84e4.webp","width":612}],"isAdultContent":false},"modelInfo":{"modelUrl":"https://www.cgtrader.com/3d-model-packs/platonic-solids-all-five-print-and-render-ready","isCgtVerified":false,"types":{"printReady":false,"animated":false,"pbr":false,"rigged":false,"lowPoly":false}},"metaverseFormatsList":[],"categorySlug":"various","subcategorySlug":"various-models","categoryTitle":"Various","subcategoryTitle":"Various models","schemaImageUrl":"https://img-new.cgtrader.com/items/7553684/7f604f84e4/platonic-solids-all-five-print-and-render-ready-3d-model-7f604f84e4.webp","saleOffDiscount":0,"eligibleForSubscription":true,"subscriptionSubscribed":false,"subscriptionPotential":false,"subscriptionTierBand":"premium"}},{"id":"3139261","type":"listingItem","attributes":{"id":3139261,"title":"Simple platonic","price":0.0,"description":"Simple platonic","imageAlt":"3D model Simple platonic","url":"https://www.cgtrader.com/free-3d-models/various/various-models/simple-platonic","isCLDApplicable":false,"isSaleOffApplicable":false,"gaAttributes":{},"primaryImage":{"gridFallbackUrl":"https://media.cgtrader.com/variants/zZUxCmwVWFTD3h8NadSxBHUD/a26e47dab5f2d22c43d6c5ce4b4b46ecc30c70918878397cba1a10c1e35d7bfc/Platonic.jpg","gridUrl":"https://media.cgtrader.com/variants/zZUxCmwVWFTD3h8NadSxBHUD/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/Platonic.webp","gridCandidates":[{"url":"https://media.cgtrader.com/variants/zZUxCmwVWFTD3h8NadSxBHUD/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/Platonic.webp","width":612}],"isAdultContent":false},"modelInfo":{"modelUrl":"https://www.cgtrader.com/free-3d-models/various/various-models/simple-platonic","isCgtVerified":false,"types":{"printReady":false,"animated":false,"pbr":false,"rigged":false,"lowPoly":false}},"metaverseFormatsList":[{"id":6,"name":".c4d","is_native":false}],"categorySlug":"various","subcategorySlug":"various-models","categoryTitle":"Various","subcategoryTitle":"Various models","schemaImageUrl":"https://img-new.cgtrader.com/items/3139261/6acfb29cd8/simple-platonic-3d-model-6acfb29cd8.webp","saleOffDiscount":0,"eligibleForSubscription":false,"subscriptionSubscribed":false,"subscriptionPotential":false,"subscriptionTierBand":null}},{"id":"7544138","type":"listingItem","attributes":{"id":7544138,"title":"Octahedron - Platonic Solid - Print and Render Ready","price":3.0,"description":"Eight equilateral triangles, and the crystal habit rough diamond actually grows in.\n\nA Platonic solid from the Brilliani Labs library, generated from its exact coordinates rather\nthan modelled by hand or traced from a scan. It is one of only five solids that can exist:\nevery face the same regular polygon, every vertex the same, every edge the same, and Euclid\nproved in the last book of the Elements that there is no sixth.\n\nBuilt to the mathematics, and checked against it before release:\n\n- 8 equilateral triangles, 12 edges, 6 vertices, Schlafli symbol {3,4}\n- Every edge 7.071068 mm, equal to one part in 1e15\n- Dihedral angle 109.471221°, the exact value, held on all 12 edges\n- Volume 166.6667 mm³, matching the closed form to one part in 1e9\n- 8 triangles, 6 vertices in the delivered mesh\n- 8.165 x 7.071 x 5.774 mm resting on a face, on a 10 mm circumsphere\n- Millimetre units, real-world scale, Y up, sitting on the ground plane at y = 0\n\nAll five solids in the set share that 10 mm circumsphere, so they nest in one sphere and\nsit together in a scene at their true relative sizes.\n\nSix formats are included. STEP carries the true solid body for CAD, so the 8 faces stay as\nplanar faces you can select, fillet and boolean. 3DM opens in Rhino as a closed mesh on its own\nlayer; reach for the STEP when you want the solid. FBX is the binary 7.4 file for Blender, 3ds\nMax, Maya, Cinema 4D, Unity and Unreal, carrying the mesh flat shaded with a material, so every\nface reads as a face on import. STL and OBJ are the watertight mesh for printing and rendering.\nGLB is a compact web and AR preview with the material embedded.\n\nThe mesh is closed and manifold with outward normals and no duplicate vertices, so it prints,\ncasts and renders without repair. Scale it freely: for a 10 mm edge, scale by 1.414214.\n\nFor a gem render, use a refractive index of 2.417 for diamond or 2.65 for moissanite; the SPEC\ngives the weight this solid would carry in each.\n","imageAlt":"Octahedron - Platonic Solid - Print and Render...","url":"https://www.cgtrader.com/3d-models/various/various-models/octahedron-platonic-solid-print-and-render-ready","isCLDApplicable":false,"isSaleOffApplicable":false,"gaAttributes":{},"primaryImage":{"gridFallbackUrl":"https://media.cgtrader.com/variants/0951xk0pyz4cq7xvqee06cz1kw03/a26e47dab5f2d22c43d6c5ce4b4b46ecc30c70918878397cba1a10c1e35d7bfc/02_wireframe.jpg","gridUrl":"https://media.cgtrader.com/variants/0951xk0pyz4cq7xvqee06cz1kw03/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/02_wireframe.webp","gridCandidates":[{"url":"https://media.cgtrader.com/variants/0951xk0pyz4cq7xvqee06cz1kw03/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/02_wireframe.webp","width":612}],"isAdultContent":false},"modelInfo":{"modelUrl":"https://www.cgtrader.com/3d-models/various/various-models/octahedron-platonic-solid-print-and-render-ready","isCgtVerified":false,"types":{"printReady":false,"animated":false,"pbr":false,"rigged":false,"lowPoly":false}},"metaverseFormatsList":[{"id":12,"name":".obj","is_native":false},{"id":13,"name":".3dm","is_native":false},{"id":5,"name":".fbx","is_native":false},{"id":72,"name":".stp","is_native":false},{"id":51,"name":".stl","is_native":false},{"id":117,"name":".gltf","is_native":false},{"id":null,"name":null,"is_native":false}],"categorySlug":"various","subcategorySlug":"various-models","categoryTitle":"Various","subcategoryTitle":"Various models","schemaImageUrl":"https://img-new.cgtrader.com/items/7544138/55ef41728f/octahedron-platonic-solid-print-and-render-ready-3d-model-55ef41728f.webp","saleOffDiscount":0,"eligibleForSubscription":true,"subscriptionSubscribed":true,"subscriptionPotential":false,"subscriptionTierBand":"essentials"}},{"id":"83126","type":"listingItem","attributes":{"id":83126,"title":"Customizable Hinged Polyhedra","price":0.0,"description":"\u003cp\u003eHinged models of each of the five Platonic solids, for educational use. Each polyhedral net prints fully assembled with hinges intact and then snaps together to make a Tetrahedron, Cube, Octahedron, Dodecahedron, or Icosahedron.  \r\nThe demo models are optimized for a Replicator 2 wtih .3mm layer resolution without raft or supports. If you have a different printer or want different sizes then use the Customizer to set your own size and tolerance values.  \r\nThis was the last big model series on the print-something-every-day-for-a-year blog MakerHome: http://makerhome.blogspot.com/2014/08/day-364-customizable-hinged-polyhedra.html\u003c/p\u003e","imageAlt":"Customizable Hinged Polyhedra 3D print model","url":"https://www.cgtrader.com/free-3d-print-models/art/mathematical-art/customizable-hinged-polyhedra","isCLDApplicable":false,"isSaleOffApplicable":false,"gaAttributes":{},"primaryImage":{"gridFallbackUrl":"https://media.cgtrader.com/variants/RZwYcG5MkiWpVR12NPp1QQCn/a26e47dab5f2d22c43d6c5ce4b4b46ecc30c70918878397cba1a10c1e35d7bfc/customizable-hinged-polyhedra-3d-model-stl.jpg","gridUrl":"https://media.cgtrader.com/variants/RZwYcG5MkiWpVR12NPp1QQCn/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/customizable-hinged-polyhedra-3d-model-stl.webp","gridCandidates":[{"url":"https://media.cgtrader.com/variants/RZwYcG5MkiWpVR12NPp1QQCn/6a314ee64365e9a50cc847b264c0df438901d553e153606f3b641b8e48cb207e/customizable-hinged-polyhedra-3d-model-stl.webp","width":306},{"url":"https://media.cgtrader.com/variants/RZwYcG5MkiWpVR12NPp1QQCn/78add9c2f02fbd73a43ffb3970be38683c5f15eff6ca849dc78c644f4ff9ce1b/customizable-hinged-polyhedra-3d-model-stl.webp","width":612},{"url":"https://media.cgtrader.com/variants/RZwYcG5MkiWpVR12NPp1QQCn/fc2d80458320aed4ba722745705c02fa7f690497f6e3961b946cdf20c88a8c12/customizable-hinged-polyhedra-3d-model-stl.webp","width":918}],"isAdultContent":false},"modelInfo":{"modelUrl":"https://www.cgtrader.com/free-3d-print-models/art/mathematical-art/customizable-hinged-polyhedra","isCgtVerified":true,"types":{"printReady":true,"animated":false,"pbr":false,"rigged":false,"lowPoly":false}},"metaverseFormatsList":[{"id":51,"name":".stl","is_native":false},{"id":null,"name":null,"is_native":false}],"categorySlug":"art","subcategorySlug":"mathematical-art","categoryTitle":"Art \u0026 Design","subcategoryTitle":"Mathematical art","schemaImageUrl":"https://img-new.cgtrader.com/items/83126/684bcce5a1/customizable-hinged-polyhedra-3d-model-684bcce5a1.webp","saleOffDiscount":0,"eligibleForSubscription":false,"subscriptionSubscribed":false,"subscriptionPotential":false,"subscriptionTierBand":null}},{"id":"7544175","type":"listingItem","attributes":{"id":7544175,"title":" Tetrahedron - Platonic Solid - Print and Render Ready","price":3.0,"description":"The first Platonic solid: four equilateral triangles, and the least a shape can do and still hold a volume.\n\nA Platonic solid from the Brilliani Labs library, generated from its exact coordinates rather\nthan modelled by hand or traced from a scan. It is one of only five solids that can exist:\nevery face the same regular polygon, every vertex the same, every edge the same, and Euclid\nproved in the last book of the Elements that there is no sixth.\n\nBuilt to the mathematics, and checked against it before release:\n\n- 4 equilateral triangles, 6 edges, 4 vertices, Schlafli symbol {3,3}\n- Every edge 8.164966 mm, equal to one part in 1e15\n- Dihedral angle 70.528779°, the exact value, held on all 6 edges\n- Volume 64.1500 mm³, matching the closed form to one part in 1e9\n- 4 triangles, 4 vertices in the delivered mesh\n- 8.165 x 7.071 x 6.667 mm resting on a face, on a 10 mm circumsphere\n- Millimetre units, real-world scale, Y up, sitting on the ground plane at y = 0\n\nAll five solids in the set share that 10 mm circumsphere, so they nest in one sphere and\nsit together in a scene at their true relative sizes.\n\nSix formats are included. 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