Includes 195 precision START HERE! Plato’s Solids The Platonic Solids are named Zometool components, Instructions according to the number of faces 50 foam dual pieces, (F) they possess. For example, and detailed instructions You can build the five Platonic Solids, by Dr. Robert Fathauer “octahedron” means “8-faces.” The or polyhedra, and their duals. number of Faces (F), Edges (E) and Vertices (V) for each solid are shown A polyhedron is a solid whose faces are Why are there only 5 perfect 3D shapes? This secret was below. An edge is a line where two polygons. Only fiveconvex regular polyhe- closely guarded by ancient Greeks, and is the mathematical faces meet, and a vertex is a point dra exist (i.e., each face is the same type basis of nearly all natural and human-built structures. where three or more faces meet. of regular polygon—a triangle, square or Build all five of Plato’s solids in relation to their duals, and see pentagon—and there are the same num- Each Platonic Solid has another how they represent the 5 elements: ber of faces around every corner.) Platonic Solid as its dual. The dual • the Tetrahedron (4-faces) = fire of the tetrahedron (“4-faces”) is again If you put a point in the center of each face • the Cube (6-faces) = earth a tetrahedron; the dual of the cube is • the Octahedron (8-faces) = water of a polyhedron, and connect those points the octahedron (“8-faces”), and vice • the Icosahedron (20-faces) = air to their nearest neighbors, you get its dual. versa. The dual of the dodecahedron • the Dodecahedron (12-faces) is the shape reserved for the cosmos! These center points are now the (“12-faces”) is the icosahedron corners (or vertices) of the dual. (“20-faces”), and vice versa. US Patents RE 33,785; 6,840,699 B2 Zometool is a registered trademark of Zometool Inc. Based on the 31- zone system discovered by Steve Baer, Zomeworks Corp., USA © 2009 Tetrahedron Cube (Hexahedron) Octahedron Dodecahedron Icosahedron F = 4 F = 6 F = 8 F = 12 F = 20 E = 6 E = 12 E = 12 E = 30 E = 30 CONTAINS SMALL PARTS that V = 4 V = 8 V = 6 V = 20 V = 12 WARNING: are NOT suitable for children 888-966-3386 3 What do you notice about the number of faces and vertices for the solids and their duals? the edges? 1 O- ! Choking Hazard under 3 years of age. Plane tessellations without “real” gaps or overlaps. A regu- If you bring these close the gap left by the removed square. must meet at the vertex of a polyhedron, The Platonic Solids and and polyhedra: lar tessellation is like a two-dimensional together to close the This forms half a cube. there are no regular solids with hexago- other polyhedra were Platonic Solid. gap, you get a pyra- nal faces. Any regular polygon with more widely studied dur- Finally, look at the penta- Before building the solids, look at how the mid with an equilateral Now look at a point (vertex) where six trian- than six sides will have larger angles than ing the Renaissance. foam pieces fit together on a flat surface. gon pieces. If you fit three gles come together. Remove one of the six to triangle base. This is a a hexagon, so there are no regular solids Johannes Kepler was a First try the small triangles. Notice the shape of these together on a flat leave a gap. If you bring tetrahedron. whose faces have more than five sides. German mathematician of a piece is different when it’s flipped over. surface, there is a small the two edges of the gap and astronomer who Be sure the same side is facing up (in a flat gap. Regular pentagons Solid history together, you get a three- lived from 1571 – 1630. arrangement) or outward (in a polyhedron) for will not tessellate; i.e., they will not cover the dimensional pyramid with Plato was a Greek He discovered the laws of planetary motion all of them. This is also true for the pentagons. plane without gaps or overlaps. a regular pentagon as its philosopher, that for the first time explained the orbits base. This pyramid is part If you bring the two mathematician, and of the planets in our solar system. Earlier, of an icosahedron. edges in the gap teacher who lived he incorrectly associated the Next look at the square pieces. Notice they together, you will form Next, lay them flat around 429/423 – orbits of the six known planets aren’t all the same. Arrange four of them as part of a dodeca- again and remove 348/347 BCE. His with a nesting of the five Pla- shown. They don’t quite lay flat, because hedron. You’ve now another triangle, so philosophical writings were a major tonic Solids within a sphere. the notches don’t line up right in a flat ar- formed parts of each of only four are left. If influence in Western thought. rangement. Regular squares do fit together Renaissance artists also Ignoring holes and small gaps, they cover a you bring these together the five Platonic Solids. flat surface. Equilateral triangles form one of without gaps or overlaps, and form the While Plato didn’t discover the Platonic studied the Platonics and to close the gap, you The final regular tessellation is made up the three regular tessellations (tilings). second regular tessellation. solids, he and his followers associated other polyhedra: Albrecht get a pyramid with a of regular hexagons. Three hexagons square base. This is half them with the four classical elements – Dürer (1471 – 1528) This means that each piece, or tile, is a Remove the upper right fit together on a flat of an octahedron. earth with the cube, air with the octahe- prominently featured a regular polygon of the same type, and square as shown, surface without a gap dron, water with the icosahedron, and polyhedron in one of his when put together they cover a flat surface Finally, lay them flat again and remove an- and bring the two or overlap. Since at fire with the tetrahedron. The dodecahe- most famous engravings, (theoretically the infinite mathematical plane) other triangle, so only three are left. edges together to least three polygons dron was reserved for the heavens. Melancolia I. 2 3 Tetrahedron Cube/Octahedron Octahedron/Cube Dodeca/Icosahedron Icosa/Dodecahedron ZOMETOOL RULES! 1 If it works, it works perfectly. 1 1 1 1 …and if it doesn’t work, it doesn’t work at all. Don’t force 1 Zometool components. You can bend a strut to fit it into a tight spot, but struts in finished models are always straight, never under tension. Hint: you can tell which strut fits between two balls in a model by lining up the balls and looking through the holes. The holes show you the shape of the strut that fits! 2 Don’t break it apart; take it apart! Take Zometool models apart by grasping a strut with your fingers and pushing the ball straight off with your thumb. Twisting balls, 2 2 2 2 2 pulling models apart or crushing them can cause parts to break!* 3 Leave the place cleaner than you found it. It’s always a good idea to clean up when you’re done. If we work together, we can make the world better. *We replace accidentally broken parts for free: visit www.zometool.com/warranty. 3 3 3 3 3 zometool ® Whether you want to ask better questions or learn better answers, Zometool is your ticket to discovery and fun. From numeracy to nanotechnology, quasicrystals to quantum 4 4 4 4 4 mechanics, the destination is always the same: understanding our amazing universe. Our mission: • make learning fun • create value • build a better world Discover more! Please visit zometool.com or call 888-966-3386 or 303-297-3387. Zometool Plato’s Solids Project — thanks to Dr. Robert Fathauer, concept and copywriting; Dr. T.W. Hildebrandt, editing; Dr. Scott Vorthmann, vZome software used for renderings; Anni Wildung and Tara Brouwer, graphic design; Paul Hildebrandt, project management etc. Contact [email protected]. Based on the 31-zone system, discovered by Steve Baer, Zomeworks Corp., USA. © 2009 Zometool Inc. .