My initial intention was to create just the dodecahedron with its bottom face sitting flat on the horizontal surface of a Minecraft Flat Map. If you type, "Dodecahedron in Minecraft" into a Search Engine a You Tube Video should appear of a bloke who'd built a wire frame Dodec on his main normal terrain map. In the video he intonates he will finish it later as it is incomplete...but no subsequent video ever appeared so I assume he never did finish it...it was similar size to what I ultimately created. The only other listing under those search criteria is this page here as far as I am aware...although there might be a few others now.
The way I created the first Dodec was actually pretty easy...for the one with its edges parallel to the game's axies ....I started with the "Golden Mean", an unusual number that fits the following equation...
(x + 1) = 1/x So "the number plus one" is equal to its reciprocal (where x = 1.618 specifically).
See this Wiki page... https://en.wikipedia.org/wiki/Golden_ratio
If, in Minecraft you place a block up in the air, then along one axis from that block make a string of blocks "1 unit" long....let's say 100 blocks just to make the maths easy. Then IF you make strings of blocks out from that same starting block along the other two axies that are "x long" and "1/x long" so in this case 62 blocks and 162 blocks. Now you have three edges of a rectangular prism. This is where you have to approximate things and the bigger you go the smaller the approximation error, but also the fact the finished shape won't fit under the 256 block high build limit...so there is a compromise between making it fit in the build space or having it small but full of approximation errors as the irrational number of the golden mean HAS to be approximated to an integral whole-number of Minecraft blocks.
Next stage is to complete the prism out to all six edges and eight verteces. Now comes the hardest bit, this is the bit that "creates" the dodecahedron...you have to have this "wire frame" rectangular prism way up in the sky, well away from any obstacles and now you make a DIAGONAL from one corner to the opposite corner...not along one face but right through the middle of the 3D shape. You have to get way back from it to "sight" along your rather "voxelated" diagonal to see if you are still heading for the opposite corner and remove any blocks that "stick out too far". Once this is done and you are satisfied you have gotten the "straightest" approximation you can achieve, you then remove the rest of the prism to leave just that diagonal.....which is just ONE edge of the final dodecahedron! I used netherrak blocks to do this part, then I exited Minecraft.
I then switched to MC Edit and placed the diagonal into its "working prism clone tool". I then cloned it, translated it and rotated it to create a second edge...I continued to do this until I had the whole Dodecahedron as a wire frame. I then went back to the game and painstakingly filled in the faces with glowstone. So that gave me the "edge oriented model" within a few hours. Looking at the Wikipaedia pages for the five perfect solids of Pythagoras and Plato showed me how one shape related to another...i.e, how a cube fits inside the dodecahedron or the doedcahedron fits inside the icosahedron. These relationships allowed me to create the other shapes from the vertices of the dodecahedron I had already made.
The second attempt was to make a dodecahedron with one pentagonal face lying flat on the game's landscape...like I'd seen in the You Tube Video I mentioned above. And, Honestly, I can't clearly remember how I did it..it was difficult and took a few days of game time even using MC Edit at times. The result was imperfect because I had to approximate the Golden Mean from 1.618 to 1.62..that is why they are the size they are...to get the best approximation of the Golden Mean. This second model had a noticeable imperfection. When I finished it, the edges zig-zagging around its "equator" were all wrong, some vertices were higher or lower than their neighbors and it looked TERRIBLE! I found that if I slid the whole top half ...(Oh, I forgot to mention that MC Edit will make hollow spheres, I used these as circumspheres to check to see of all vertices of the polyhedron touched the sphere.) ..a few blocks along the x or z axis of the Game, the zig-zag "equator" came right..it looked good and the vertices still all touched the sphere...but the top and bottom pentagons centres are not directly one above the other. Now I know why that bloke in the You Tube Video "got bogged"...he started with a "difficult" orientation and expended all his mental "oompf" on it and ran out of "oompf" before he finished it! If he'd started with the edge-oriented model first (the easiest), presumably it would have spurned him on to tackle the "face oriented model"...which has to be distorted a bit vertically to look good horizontally.
At this point I then realized there remained one more Dodec model...the VERTEX oriented model, i.e, standing up on one corner....this one was very difficult, it took months of mucking around to achieve and I can't remember clearly how I did it...I do recall using the corner-oriented cube as a starting point and somehow built the dodec around it...and, again I used the functions of MC Edit to create any "mirrored" or "rotated" versions of faces and edges to cut down the time.
Some of the other shapes, the tetrahedron, cubes and octahedron were absolutely trivial after the corner oriented dodec!
Two more of my builds here on MC Schematics use these polyhdera, one is the "truncoctagon", which is simply the octagon with each vertex lopped off and made of blue and red glass. The other is "The Cubagon"...a not-very-effective stack of eight edge oriented Docecs in a trivial cubic frame. I have a third structure here in my main map made from the corner oriented Dodec standing upon three lags like a tripod...but the legs are curved and I have placed three rings around it...sort of like the rings around Rotwag's robot in Fritz Lang's 1928 Movie, "Metropolis" if you'e ever seen that? This structure straddles x=0 and z=0 with a beacon up through it. It has a teleport lounge inside so I can get out to places on the 20K x 20K map quickly...it is also full of other stuff to control the time of day, weather, creeper and enderman damage etc. I may post it up here someday but it is full of chests and command blocks which make its file big and complex.
I will be building more stuff in the future, but life crowds in, I have to replace the roof on the house and that is more important than mucking about in Minecraft at present. I am attempting to construct the General Assembly of Sarawak building in Kuching, Malaysia..this is a nine-sided structure...use google maps to go and look at it...as it is the seat of the Sarawak State Government it is easy to find..on the northern side of the river opposite the CBD of Kuching. Like the polyhedra, every wall of this structure has to be hand made and can't be cloned with an editor because of the non-right angles. I have made the nine sided foundation...but the roof looks terrible and voxellated..so I am waiting for Mojang to raise the game's ceiling from 256 to 512 so I can make it higher and consequently wider (with eight times the volume) and thus eight times the detail.
Icosahedron, Dodecahedron, Octahedron, Hexahedron and Tetrahedron on April 28th, 2020 08:09 PM EST