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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. 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This is used in aerospace technology to save on fuel while flying and still maintaining structural integrity and strength of the parts that are optimized.\n\nNowadays these structures can be achieved with metal 3D printing (laser sintering, laser melting and the like). But these structures are also interesting from a design perspective. The pure beauty of them inspires to use these structures in architecture, fashion or other fields.\n\nHere are my tests with different topologies, printed on a Formlabs Form2 (SLA printing) with clear UV resin. If you want to create your own structures head over to NTopolgy (https://www.ntopology.com) they have a beautiful free tool for that.\n\nYou will find more footage next week on my Instagram, Facebook and Youtube accounts.\n\nI hope you got inspired like me and get hands-on designing. 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