From: Tommy the Terrorist Subject: All perfect shapes 3D - all platforms Date: 2000/03/21 Message-ID: <8b72it0feu@enews3.newsguy.com> Distribution: world Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=ISO-8859-1 X-XXMessage-ID: Organization: Dis Mime-Version: 1.0 Newsgroups: rec.arts.sf.science I think you guys will like this... it's not exactly sci-fi, but I still think it's the sort of thing that gets our sort of people curious, and there's no alt.textfiles.erotica.geometry :) If the geometric relationships between perfect shapes, pentagrams, the Golden Mean, etc. are at all interesting to you --- if the elegance of design of the universe, so to speak, argues to you for the Platonic-ideal existence of God --- or if you just like REALLY COOL 3-d objects you can manipulate... (Such as the data set at the end of this post, which depicts the inscription of five cubes, ten tetrahedrons, five octahedrons and an icosahedron within a dodecahedron, in a pattern which can then be repeated at larger and smaller scales...) ....then take 2 minutes to download the freeware viewer, which allows the data below to be visualized on PC, Linux, Mac, Unix, etc. platforms. You can also turn and twist the pattern as you will. That program, which incidentally is called "MAGE" by ... coincidence... is a popular viewer for crystal structures - in other words, it is supported and it is available for many computers for actual professional reasons. So it serves as a single, common forum for exchanging interesting sets of 3-dimensional data of many sorts. [note: I have nothing to do with this program... except enjoying it] For an explanation of MAGE (it is by intent a structure viewer for protein crystal structures, but I found it to be particularly amenable to accepting text files of coordinates for arbitrary lines) look briefly at: http://www.ncbi.nlm.nih.gov/Structure/CN3D/mage.html (you don't really have to understand this; it's plug-and-play) Now go to: ftp://suna.biochem.duke.edu/ and download the appropriate version; for instance, the Mac power PC version is at: ftp://suna.biochem.duke.edu/MACprograms/MAGEPPC_5_70.bin Once you've downloaded it, save the data I've appended at the end of this post as a text file, open the text file in MAGE, and voila! Enjoy! (Note that you can turn off and on the different shapes so as to be able to make more sense out of what you see) (I also provide what I term a "Druidic pentagram"; the argument is basically that the Druids cast dodecahedrons of bronze; Plato, a contemporary, analogized the dodecahedron to the "limiting sphere" of a pentagram (which the Druids stamped on coins); and because it is very difficult without a computer to work out which combination of lines won't turn back on itself before you make a 20-sided, 3-dimensional star inscribed within the dodecahedron (i.e. "a complete circle without stopping or recrossing your line..."); not to mention because it is chiral - there are left and right handed versions, though I have no clue if this one is widdershins or (the other way...) myself... and because it has this disturbing feature of symmetry breaking... well I have this suspicion that the Druids worked it out and that the smaller version is just an ideogram or analogy. Oh well, but have a look at it. :) ) ---- SAVE FILE BELOW AS TEXT AND OPEN IN MAGE VIEWER --- @kinemage -1 @caption Relationship of the perfect shapes and Druidic pentagram From outermost to innermost: (G = golden mean; G = 1/G+1) Vertex coordinates (+/-; coordinates in cube frame) Outer Dodecahedron: G 1 1/G Cube #1/Tetrahedrons #1: 1 Other Cubes/Tetrahedrons: G 1 1/G 0 Octahedron #1: 1 0 Other Octahedrons: 1/2 x G 1 1/G Icosahedron: 1/G, 1/G^2 Inner Dodecahedron: 1/3 x G 1 1/G Note the possible repetition of inscription with 3x scaling factor between sets @group {dodecahedron} dominant @vectorlist {dodec} color= blue master= {Perfect} {DOD 1 } P 1, 1, 1 {DOD 2 } L 0, 0.618, 1.618 {DOD 1 } P 1, 1, 1 {DOD 5 } L 0.618, 1.618, 0 {DOD 1 } P 1, 1, 1 {DOD 6 } L 1.618, 0, 0.618 {DOD 2 } P 0, 0.618, 1.618 {DOD 7 } L 0, -0.618, 1.618 {DOD 5 } P 0.618, 1.618, 0 {DOD 4 } L -0.618, 1.618, 0 {DOD 6 } P 1.618, 0, 0.618 {DOD 8'} L 1.618, 0, -0.618 {DOD 2 } P 0, 0.618, 1.618 {DOD 3 } L -1, 1, 1 {DOD 5 } P 0.618, 1.618, 0 {DOD 10} L 1, 1, -1 {DOD 6 } P 1.618, 0, 0.618 {DOD 9'} L 1, -1, 1 {DOD 7 } P 0, -0.618, 1.618 {DOD 9'} L 1, -1, 1 {DOD 4 } P -0.618, 1.618, 0 {DOD 3 } L -1, 1, 1 {DOD 8'} P 1.618, 0, -0.618 {DOD 10} L 1, 1, -1 {DOD 7 } P 0, -0.618, 1.618 {DOD10'} L -1, -1, 1 {DOD 4 } P -0.618, 1.618, 0 {DOD 9 } L -1, 1, -1 {DOD 8'} P 1.618, 0, -0.618 {DOD 3'} L 1, -1, -1 {DOD 8 } P -1.618, 0, 0.618 {DOD 3 } L -1, 1, 1 {DOD 4'} P 0.618, -1.618, 0 {DOD 9'} L 1, -1, 1 {DOD 7'} P 0, 0.618, -1.618 {DOD 10} L 1, 1, -1 {DOD 8 } P -1.618, 0, 0.618 {DOD10'} L -1, -1, 1 {DOD 4'} P 0.618, -1.618, 0 {DOD 3'} L 1, -1, -1 {DOD 7'} P 0, 0.618, -1.618 {DOD 9 } L -1, 1, -1 {DOD 6'} P -1.618, 0, -0.618 {DOD 9 } L -1, 1, -1 {DOD 5'} P -0.618, -1.618, 0 {DOD10'} L -1, -1, 1 {DOD 2'} P 0, -0.618, -1.618 {DOD 3'} L 1, -1, -1 {DOD 6'} P -1.618, 0, -0.618 {DOD 8 } L -1.618, 0, 0.618 {DOD 5'} P -0.618, -1.618, 0 {DOD 4'} L 0.618, -1.618, 0 {DOD 2'} P 0, -0.618, -1.618 {DOD 7'} L 0, 0.618, -1.618 {DOD 1'} P -1, -1, -1 {DOD 6'} L -1.618, 0, -0.618 {DOD 1'} P -1, -1, -1 {DOD 5'} L -0.618, -1.618, 0 {DOD 1'} P -1, -1, -1 {DOD 2'} L 0, -0.618, -1.618 @group {cube} @vectorlist {cube 1} color= yellow master= {Perfect} {CUBE 1 } P 1, 1, 1 {CUBE 2 } L 1, 1, -1 {CUBE 1 } P 1, 1, 1 {CUBE 3 } L 1, -1, 1 {CUBE 1 } P 1, 1, 1 {CUBE 4 } L -1, 1, 1 {CUBE 2 } P 1, 1, -1 {CUBE 3'} L -1, 1, -1 {CUBE 2 } P 1, 1, -1 {CUBE 4'} L 1, -1, -1 {CUBE 3 } P 1, -1, 1 {CUBE 4'} L 1, -1, -1 {CUBE 3'} P -1, 1, -1 {CUBE 4 } L -1, 1, 1 {CUBE 2'} P -1, -1, 1 {CUBE 4 } L -1, 1, 1 {CUBE 2'} P -1, -1, 1 {CUBE 3 } L 1, -1, 1 {CUBE 1'} P -1, -1, -1 {CUBE 4'} L 1, -1, -1 {CUBE 1'} P -1, -1, -1 {CUBE 3'} L -1, 1, -1 {CUBE 1'} P -1, -1, -1 {CUBE 2'} L -1, -1, 1 @vectorlist {cube 2} color= yellow master= {Perfect} {CUBE 1 } P 1, 1, 1 {CUBE 2 } L 1.618, 0, -0.618 {CUBE 1 } P 1, 1, 1 {CUBE 3 } L -0.618, 1.618, 0 {CUBE 1 } P 1, 1, 1 {CUBE 4 } L 0, -0.618, 1.618 {CUBE 2 } P 1.618, 0, -0.618 {CUBE 3'} L 0.618, -1.618, 0 {CUBE 2 } P 1.618, 0, -0.618 {CUBE 4'} L 0, 0.618, -1.618 {CUBE 3 } P -0.618, 1.618, 0 {CUBE 4'} L 0, 0.618, -1.618 {CUBE 3'} P 0.618, -1.618, 0 {CUBE 4 } L 0, -0.618, 1.618 {CUBE 2'} P -1.618, 0, 0.618 {CUBE 4 } L 0, -0.618, 1.618 {CUBE 2'} P -1.618, 0, 0.618 {CUBE 3 } L -0.618, 1.618, 0 {CUBE 1'} P -1, -1, -1 {CUBE 4'} L 0, 0.618, -1.618 {CUBE 1'} P -1, -1, -1 {CUBE 3'} L 0.618, -1.618, 0 {CUBE 1'} P -1, -1, -1 {CUBE 2'} L -1.618, 0, 0.618 @vectorlist {cube 3} color= yellow master= {Perfect} {CUBE 1 } P 0, 0.618, 1.618 {CUBE 2 } L 0.618, 1.618, 0 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 3 } L 1, -1, 1 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 4 } L -1.618, 0, 0.618 {CUBE 2 } P 0.618, 1.618, 0 {CUBE 3'} L -1, 1, -1 {CUBE 2 } P 0.618, 1.618, 0 {CUBE 4'} L 1.618, 0, -0.618 {CUBE 3 } P 1, -1, 1 {CUBE 4'} L 1.618, 0, -0.618 {CUBE 3'} P -1, 1, -1 {CUBE 4 } L -1.618, 0, 0.618 {CUBE 2'} P -0.618, -1.618, 0 {CUBE 4 } L -1.618, 0, 0.618 {CUBE 2'} P -0.618, -1.618, 0 {CUBE 3 } L 1, -1, 1 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 4'} L 1.618, 0, -0.618 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 3'} L -1, 1, -1 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 2'} L -0.618, -1.618, 0 @vectorlist {cube 4} color= yellow master= {Perfect} {CUBE 1 } P 0, 0.618, 1.618 {CUBE 2 } L -0.618, 1.618, 0 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 3 } L 1.618, 0, 0.618 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 4 } L -1, -1, 1 {CUBE 2 } P -0.618, 1.618, 0 {CUBE 3'} L -1.618, 0, -0.618 {CUBE 2 } P -0.618, 1.618, 0 {CUBE 4'} L 1, 1, -1 {CUBE 3 } P 1.618, 0, 0.618 {CUBE 4'} L 1, 1, -1 {CUBE 3'} P -1.618, 0, -0.618 {CUBE 4 } L -1, -1, 1 {CUBE 2'} P 0.618, -1.618, 0 {CUBE 4 } L -1, -1, 1 {CUBE 2'} P 0.618, -1.618, 0 {CUBE 3 } L 1.618, 0, 0.618 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 4'} L 1, 1, -1 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 3'} L -1.618, 0, -0.618 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 2'} L 0.618, -1.618, 0 @vectorlist {cube 5} color= yellow master= {Perfect} {CUBE 1 } P -1, 1, 1 {CUBE 2 } L 0.618, 1.618, 0 {CUBE 1 } P -1, 1, 1 {CUBE 3 } L 0, -0.618, 1.618 {CUBE 1 } P -1, 1, 1 {CUBE 4 } L -1.618, 0, -0.618 {CUBE 2 } P 0.618, 1.618, 0 {CUBE 3'} L 0, 0.618, -1.618 {CUBE 2 } P 0.618, 1.618, 0 {CUBE 4'} L 1.618, 0, 0.618 {CUBE 3 } P 0, -0.618, 1.618 {CUBE 4'} L 1.618, 0, 0.618 {CUBE 3'} P 0, 0.618, -1.618 {CUBE 4 } L -1.618, 0, -0.618 {CUBE 2'} P -0.618, -1.618, 0 {CUBE 4 } L -1.618, 0, -0.618 {CUBE 2'} P -0.618, -1.618, 0 {CUBE 3 } L 0, -0.618, 1.618 {CUBE 1'} P 1, -1, -1 {CUBE 4'} L 1.618, 0, 0.618 {CUBE 1'} P 1, -1, -1 {CUBE 3'} L 0, 0.618, -1.618 {CUBE 1'} P 1, -1, -1 {CUBE 2'} L -0.618, -1.618, 0 @group {tetrahedron} @vectorlist {tetra 1 } color= red master= {Perfect} {CUBE 1 } P 1, 1, 1 {CUBE 2'} L -1, -1, 1 {CUBE 1 } P 1, 1, 1 {CUBE 3'} L -1, 1, -1 {CUBE 1 } P 1, 1, 1 {CUBE 4'} L 1, -1, -1 {CUBE 2'} P -1, -1, 1 {CUBE 3'} L -1, 1, -1 {CUBE 3'} P -1, 1, -1 {CUBE 4'} L 1, -1, -1 {CUBE 4'} P 1, -1, -1 {CUBE 2'} L -1, -1, 1 @vectorlist {tetra 1'} color= red master= {Perfect} {CUBE 1'} P -1, -1, -1 {CUBE 2 } L 1, 1, -1 {CUBE 1'} P -1, -1, -1 {CUBE 3 } L 1, -1, 1 {CUBE 1'} P -1, -1, -1 {CUBE 4 } L -1, 1, 1 {CUBE 2 } P 1, 1, -1 {CUBE 3 } L 1, -1, 1 {CUBE 3 } P 1, -1, 1 {CUBE 4 } L -1, 1, 1 {CUBE 4 } P -1, 1, 1 {CUBE 2 } L 1, 1, -1 @vectorlist {tetra 2} color= red master= {Perfect} {CUBE 1 } P 1, 1, 1 {CUBE 2'} L -1.618, 0, 0.618 {CUBE 1 } P 1, 1, 1 {CUBE 3'} L 0.618, -1.618, 0 {CUBE 1 } P 1, 1, 1 {CUBE 4'} L 0, 0.618, -1.618 {CUBE 2'} P -1.618, 0, 0.618 {CUBE 3'} L 0.618, -1.618, 0 {CUBE 3'} P 0.618, -1.618, 0 {CUBE 4'} L 0, 0.618, -1.618 {CUBE 4'} P 0, 0.618, -1.618 {CUBE 2'} L -1.618, 0, 0.618 @vectorlist {tetra 2'} color= red master= {Perfect} {CUBE 1'} P -1, -1, -1 {CUBE 2 } L 1.618, 0, -0.618 {CUBE 1'} P -1, -1, -1 {CUBE 3 } L -0.618, 1.618, 0 {CUBE 1'} P -1, -1, -1 {CUBE 4 } L 0, -0.618, 1.618 {CUBE 2 } P 1.618, 0, -0.618 {CUBE 3 } L -0.618, 1.618, 0 {CUBE 3 } P -0.618, 1.618, 0 {CUBE 4 } L 0, -0.618, 1.618 {CUBE 4 } P 0, -0.618, 1.618 {CUBE 2 } L 1.618, 0, -0.618 @vectorlist {tetra 3} color= red master= {Perfect} {CUBE 1 } P 0, 0.618, 1.618 {CUBE 2'} L -0.618, -1.618, 0 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 3'} L -1, 1, -1 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 4'} L 1.618, 0, -0.618 {CUBE 2'} P -0.618, -1.618, 0 {CUBE 3'} L -1, 1, -1 {CUBE 3'} P -1, 1, -1 {CUBE 4'} L 1.618, 0, -0.618 {CUBE 4'} P 1.618, 0, -0.618 {CUBE 2'} L -0.618, -1.618, 0 @vectorlist {tetra 3'} color= red master= {Perfect} {CUBE 1'} P 0, -0.618, -1.618 {CUBE 2 } L 0.618, 1.618, 0 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 3 } L 1, -1, 1 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 4 } L -1.618, 0, 0.618 {CUBE 2 } P 0.618, 1.618, 0 {CUBE 3 } L 1, -1, 1 {CUBE 3 } P 1, -1, 1 {CUBE 4 } L -1.618, 0, 0.618 {CUBE 4 } P -1.618, 0, 0.618 {CUBE 2 } L 0.618, 1.618, 0 @vectorlist {tetra 4} color= red master= {Perfect} {CUBE 1 } P 0, 0.618, 1.618 {CUBE 2'} L 0.618, -1.618, 0 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 3'} L -1.618, 0, -0.618 {CUBE 1 } P 0, 0.618, 1.618 {CUBE 4'} L 1, 1, -1 {CUBE 2'} P 0.618, -1.618, 0 {CUBE 3'} L -1.618, 0, -0.618 {CUBE 3'} P -1.618, 0, -0.618 {CUBE 4'} L 1, 1, -1 {CUBE 4'} P 1, 1, -1 {CUBE 2'} L 0.618, -1.618, 0 @vectorlist {tetra 4'} color= red master= {Perfect} {CUBE 1'} P 0, -0.618, -1.618 {CUBE 2 } L -0.618, 1.618, 0 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 3 } L 1.618, 0, 0.618 {CUBE 1'} P 0, -0.618, -1.618 {CUBE 4 } L -1, -1, 1 {CUBE 2 } P -0.618, 1.618, 0 {CUBE 3 } L 1.618, 0, 0.618 {CUBE 3 } P 1.618, 0, 0.618 {CUBE 4 } L -1, -1, 1 {CUBE 4 } P -1, -1, 1 {CUBE 2 } L -0.618, 1.618, 0 @vectorlist {tetra 5} color= red master= {Perfect} {CUBE 1 } P -1, 1, 1 {CUBE 2'} L -0.618, -1.618, 0 {CUBE 1 } P -1, 1, 1 {CUBE 3'} L 0, 0.618, -1.618 {CUBE 1 } P -1, 1, 1 {CUBE 4'} L 1.618, 0, 0.618 {CUBE 2'} P -0.618, -1.618, 0 {CUBE 3'} L 0, 0.618, -1.618 {CUBE 3'} P 0, 0.618, -1.618 {CUBE 4'} L 1.618, 0, 0.618 {CUBE 4'} P 1.618, 0, 0.618 {CUBE 2'} L -0.618, -1.618, 0 @vectorlist {tetra 5'} color= red master= {Perfect} {CUBE 1'} P 1, -1, -1 {CUBE 2 } L 0.618, 1.618, 0 {CUBE 1'} P 1, -1, -1 {CUBE 3 } L 0, -0.618, 1.618 {CUBE 1'} P 1, -1, -1 {CUBE 4 } L -1.618, 0, -0.618 {CUBE 2 } P 0.618, 1.618, 0 {CUBE 3 } L 0, -0.618, 1.618 {CUBE 3 } P 0, -0.618, 1.618 {CUBE 4 } L -1.618, 0, -0.618 {CUBE 4 } P -1.618, 0, -0.618 {CUBE 2 } L 0.618, 1.618, 0 @group {octahedron} @vectorlist {octa 1} color= purple master= {Perfect} {OCTA 1 } P 1, 0, 0 {OCTA 2 } L 0, 1, 0 {OCTA 2 } P 0, 1, 0 {OCTA 3 } L 0, 0, 1 {OCTA 3 } P 0, 0, 1 {OCTA 1 } L 1, 0, 0 {OCTA 1 } P 1, 0, 0 {OCTA 2'} L 0, -1, 0 {OCTA 1 } P 1, 0, 0 {OCTA 3'} L 0, 0, -1 {OCTA 2 } P 0, 1, 0 {OCTA 3'} L 0, 0, -1 {OCTA 2'} P 0, -1, 0 {OCTA 3 } L 0, 0, 1 {OCTA 1'} P -1, 0, 0 {OCTA 3 } L 0, 0, 1 {OCTA 1'} P -1, 0, 0 {OCTA 2 } L 0, 1, 0 {OCTA 3'} P 0, 0, -1 {OCTA 1'} L -1, 0, 0 {OCTA 2'} P 0, -1, 0 {OCTA 3'} L 0, 0, -1 {OCTA 1'} P -1, 0, 0 {OCTA 2'} L 0, -1, 0 @vectorlist {octa 2} color= purple master= {Perfect} {OCTA 1 } P 0.5, 0.809, -0.309 {OCTA 2 } L -0.309, 0.5, 0.809 {OCTA 2 } P -0.309, 0.5, 0.809 {OCTA 3 } L 0.809, -0.309, 0.5 {OCTA 3 } P 0.809, -0.309, 0.5 {OCTA 1 } L 0.5, 0.809, -0.309 {OCTA 1 } P 0.5, 0.809, -0.309 {OCTA 2'} L 0.309, -0.5, -0.809 {OCTA 1 } P 0.5, 0.809, -0.309 {OCTA 3'} L -0.809, 0.309, -0.5 {OCTA 2 } P -0.309, 0.5, 0.809 {OCTA 3'} L -0.809, 0.309, -0.5 {OCTA 2'} P 0.309, -0.5, -0.809 {OCTA 3 } L 0.809, -0.309, 0.5 {OCTA 1'} P -0.5, -0.809, 0.309 {OCTA 3 } L 0.809, -0.309, 0.5 {OCTA 1'} P -0.5, -0.809, 0.309 {OCTA 2 } L -0.309, 0.5, 0.809 {OCTA 3'} P -0.809, 0.309, -0.5 {OCTA 1'} L -0.5, -0.809, 0.309 {OCTA 2'} P 0.309, -0.5, -0.809 {OCTA 3'} L -0.809, 0.309, -0.5 {OCTA 1'} P -0.5, -0.809, 0.309 {OCTA 2'} L 0.309, -0.5, -0.809 @vectorlist {octa 3} color= purple master= {Perfect} {OCTA 1 } P 0.809, 0.309, 0.5 {OCTA 2 } L -0.5, 0.809, 0.309 {OCTA 2 } P -0.5, 0.809, 0.309 {OCTA 3 } L 0.309, 0.5, -0.809 {OCTA 3 } P 0.309, 0.5, -0.809 {OCTA 1 } L 0.809, 0.309, 0.5 {OCTA 1 } P 0.809, 0.309, 0.5 {OCTA 2'} L 0.5, -0.809, -0.309 {OCTA 1 } P 0.809, 0.309, 0.5 {OCTA 3'} L -0.309, -0.5, 0.809 {OCTA 2 } P -0.5, 0.809, 0.309 {OCTA 3'} L -0.309, 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