Platonic Polyhedra 1: Dodecahedron & Parts Thereof

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"These Polyhedra are Just Good Friends!" "It is related to Hippasus that he was a Pythagorean, and that, owing to his being the first to publish and describe the sphere from the twelve pentagons, he perished at sea for his impiety, but he received credit for the discovery, though really it all belonged to HIM ( for in this way they refer to Pythagoras, and they do not call him by his name)." Let's hope I don't get drowned for showing you how to construct a dodecahedron! Yes, Greeks used to care about maths so much that they killed people for publicly disclosing the existence of a dodecahedron. Thank goodness Intellectual Property has moved along since then and we have patents, copyright, and creative commons. You'd never hear of someone dying for disclosing the existence of a new iPhone (sorry, I shouldn't joke about Foxcon employees being bullied to death). This is my standard CAD System test drive. I first studied the geometry of the dodecahedron when Uwe Meffert asked me to write a solution to his Skewb Ultimate. Uwe Meffert is the owner of Meffett's puzzles in Hong Kong, he invented the Pyraminx about two or three years before Ernõ Rubik invented the Büvös Kocka (which became known as "The Rubik Cube" after patent difficulties). Meffert asked me to write a solution for the Junior Pyraminx with David Singmaster after reading an academic paper I coauthored about the Masterball with David Joyner, group theory expert at USNA. I agreed to write the "Junior Pyraminx" solution, met with Zingmaster, but asked Meffert to rename the JPM to "Pyramorphix". He accepted. Even today, Uwe Meffert's site still has the solution to the Pyramorphix on a page called "jpmsol.HTML" http://www.mefferts.com/puzzles/jpmsol.html The artwork I did for the JPM was incorrectly copied for the Japanese solution leaflet, so before I agreed to do the Skewb Ultimate for Uwe, I first sought approval for my artwork to be used. Hence I worked out how to draw a dodecahedron from basics, and how to draw one by subtracting from a Cube. Results are still available today, though I do really want to rework the end images. They look a bit 90's! http://www.mefferts.com/puzzles/solution-skewbultimate1.html I see how easy it is to create a Dodecahedron to learn about the systems handling of planes, reflection, rotation, and other basic operations. Just so happens Dear Daughter needs a 12 sided dice. I just hope she's not starting to play Dungeons & Dragons(!). Basic files included showing how I make this. From a Circle, 1 line and an angle for the second line, the rest of it is good old fashioned Descriptive Geometry as taught at Imperial College, London, in 1995. Once you form the Pentagon from basic principles, rotate it greater than the angle required around two edges. This then has 1 intersect point at the common corner not on the orginial face plane. Form two new planes from the axis of rotation and this intersect point. Draw a LARGE circle on those planes and cut outwards to remove all excess material. You have arrived at the "first three faces" - model one above. Next form a mirror plane from the original sketches and the top, centre intersect point. Mirror the solid. Repeat to form all five L2 Pentagons. Merge solid. This is the "First Half" (6 faces formed) - the next part model I've uploaded. Mirror or rotate into position to form the 2nd 6 sides, then merge - voila - you've got the finished Dodecahedron. The sort of thing we should teach more often in Universities these days. This has no numbers on it, so you can write your own. NEW: 10-sided added, and rounded and squished. also blank. Simply pull the faces 1 & 12 until they are points (alternatively extrude those faces larger and then cut back from them to form the new faces). Define the points you've made as two point of an axis, define that as "Z-direction" and scale x as 1, y as 1, and z as 0.707 NEW: Octahedron, constructed in ProE. Parts thereof to show the way for maths and CAD jockeys. Start with a Square, draw four equilateral triangles, revolve them to form new volumes. Form planes from intersection points and the axis of rotation. Cut away from those planes. Voilà. Octahedron. NEW NEW NEW! Don't forget our Footy design process, you can see how to make a footy from a Dodecahedron, other methods work, but this was just a quick discussion with #BathRoboticsDaughter...https://www.thingiverse.com/thing:3004042 Peace out EUR ING Andy Southern aka BathRobotics