I'm archiving this here as a specimen of how Python is so useful in an educational setting.
You could think of this as pre Python Notebooks, which it is (I'll do a Notebook version later), and quite spreadsheet-like, as is MathCAD (a competitor to Mathematica). The top to bottom flow of a Python script is like MathCAD's: any name mentioned must already be defined higher up, unless just being introduced now: enter stage left as in player = .... What I'm doing is feeding six-edge objects (PlaneNets), i.e. various tetrahedrons' edges identified in a certain order (fan out from a corner, then mention opposite triangle edges in the same order), to a volume-maker that takes edges as inputs (not invented here), and tabulating various results of investigations, mainly by a guy named Koski in Minnesota, using the Bucky Fuller stuff (see links below) as a springboard. Koski and I visited Magnus Wenninger recently, a big name among polyhedralists, a priest, age 92 or so, check Wikipedia. I got to see Popko's Divided Spheres in manuscript, well before I got my Kindle edition (my name goes by in the footnotes brag brag). You may have heard of a conference named 'Bridges', about bridging art and math. Koski goes to that sometimes, and Popko. Next one in Korea, last one in Holland. The vocabulary / terminology used below would have some currency / truck with that crowd (George and Vi Hart also go, Vi being a star on Youtube, the author of those whimsical math videos many of you well know and love). Anyway, I use the script below to confirm / explore / clarify in my own mind, what on earth Koski is yakking about. If you're not into polyhedrons, no need to dive in. If you are, and want to see a relevant video, here's one of Dave's using vZome, a Java application for ZomeTool afficionados: http://worldgame.blogspot.com/2014/03/e-module-mensuration.html One wrinkle I've noticed but did not show off here, is that the greek letter phi is standard in Unicode and Python being UTF-8 now accepts that symbol no problem, as a Python name. I'm not sure how happy mailman and/or your email client will be with ΓΈ = (sqrt(5)+1)/2 but Python handles it just fine. Calling all greek variable names: the floodgates have opened; use judiciously and with some caution. Kirby ============== """ Includes: PlaneNets from Synergetics for A, B, E, T, S Koski breakdowns of some shapes in E and S vols with phi scaling Euler volume, modified by Gerald de Jong http://www.grunch.net/synergetics/quadvols.html Kirby Urner (c) MIT License """ from math import sqrt, hypot class PlaneNet: """Any six edge tet in pattern described in API notes""" def __init__(self, oa, ob, oc, ab, bc, ca): self.oa = oa self.ob = ob self.oc = oc self.ab = ab self.bc = bc self.ca = ca class Tetrahedron: """ Takes six edges of tetrahedron with faces (a,b,d)(b,c,e)(c,a,f)(d,e,f) -- returns volume if ivm and xyz """ def __init__(self, a,b,c,d,e,f): self.a, self.a2 = a, a**2 self.b, self.b2 = b, b**2 self.c, self.c2 = c, c**2 self.d, self.d2 = d, d**2 self.e, self.e2 = e, e**2 self.f, self.f2 = f, f**2 def ivm_volume(self): return ((self._addopen()- self._addclosed() - self._addopposite())/2) ** 0.5 def xyz_volume(self): return sqrt(8/9) * self.ivm_volume() def _addopen(self): a2,b2,c2,d2,e2,f2 = self.a2, self.b2, self.c2, self.d2, self.e2, self.f2 sumval = f2*a2*b2 sumval += d2 * a2 * c2 sumval += a2 * b2 * e2 sumval += c2 * b2 * d2 sumval += e2 * c2 * a2 sumval += f2 * c2 * b2 sumval += e2 * d2 * a2 sumval += b2 * d2 * f2 sumval += b2 * e2 * f2 sumval += d2 * e2 * c2 sumval += a2 * f2 * e2 sumval += d2 * f2 * c2 return sumval def _addclosed(self): a2,b2,c2,d2,e2,f2 = self.a2, self.b2, self.c2, self.d2, self.e2, self.f2 sumval = a2 * b2 * d2 sumval += d2 * e2 * f2 sumval += b2 * c2 * e2 sumval += a2 * c2 * f2 return sumval def _addopposite(self): a2,b2,c2,d2,e2,f2 = self.a2, self.b2, self.c2, self.d2, self.e2, self.f2 sumval = a2 * e2 * (a2 + e2) sumval += b2 * f2 * (b2 + f2) sumval += c2 * d2 * (c2 + d2) return sumval PHI = sqrt(5)/2 + 0.5 D = 1.0 R = D/2 def volume(net): return Tetrahedron(net.oa, net.ob, net.oc, net.ab, net.bc, net.ca ).ivm_volume() # Fig. 913.01 A Quanta Module # http://www.rwgrayprojects.com/synergetics/s09/figs/f1301.html a = D EF = a * sqrt(6) / 12 EC = a * sqrt(6) / 4 ED = a * sqrt(2) / 4 FC = a * sqrt(3) / 3 CD = a/2 DF = a * sqrt(3) / 6 Amod = PlaneNet(EF, EC, ED, FC, CD, DF) Avol = volume(Amod) print("Amod volume = :", Avol) # Fig. 916.01 B Quanta Module # http://www.rwgrayprojects.com/synergetics/s09/figs/f1601.html a = D EA = a * sqrt(2) / 2 EC = a/2 EB = a * sqrt(6) / 12 AC = a/2 CB = a * sqrt(2) / 4 BA = a * sqrt(6) / 4 Bmod = PlaneNet(EA, EC, EB, AC, CB, BA) Bvol = volume(Bmod) print("Bmod volume = :", Bvol) # Fig. 986.411A T & E Module # http://www.rwgrayprojects.com/synergetics/s09/figs/f86411a.html h = R OC = h OA = h * sqrt((5 - sqrt(5))/2) OB = h * sqrt((9 - 3 * sqrt(5))/2 ) CA = (h/2) * (sqrt(5) - 1) AB = h * sqrt(5 - 2 * sqrt(5)) BC = (h/2) * (3 - sqrt(5)) Emod = PlaneNet(OC, OA, OB, CA, AB, BC) Evol = volume(Emod) print("Emod volume = :", Evol) # Fig. 986.411A T & E Module # http://www.rwgrayprojects.com/synergetics/s09/figs/f86411a.html h = R * pow(2/3,1/3) * (PHI / sqrt(2)) OC = h OA = h * sqrt((5 - sqrt(5))/2) OB = h * sqrt((9 - 3 * sqrt(5))/2 ) CA = (h/2) * (sqrt(5) - 1) AB = h * sqrt(5 - 2 * sqrt(5)) BC = (h/2) * (3 - sqrt(5)) Tmod = PlaneNet(OC, OA, OB, CA, AB, BC) Tvol = volume(Tmod) print("Tmod volume = :", Tvol) # Fig. 988.13A S Quanta Module Edge Lengths # http://www.rwgrayprojects.com/synergetics/s09/figs/f8813a.html a = D FG = (a/2) * sqrt(3) * sqrt(7-3*sqrt(5)) FE = a * sqrt(7 - 3*sqrt(5)) FH = (a/2) * (sqrt(5)-1) GE = (a/2) * sqrt(7 - 3*sqrt(5)) EH = (a/2) * (3 - sqrt(5)) HG = (a/2) * sqrt (7 - 3*sqrt(5)) Smod = PlaneNet(FG, FE, FH, GE, EH, HG) Svol = volume(Smod) print("Smod volume = :", Svol) print("================") sFactor = Evol / Svol s3 = Svol * pow(PHI, -3) e3 = Evol * pow(PHI, -3) print("VE: ", 20, 420 * Svol + 100 * s3) print("Icosa: ", 20 * sFactor ** 1, 420 * Evol + 100 * e3) print("BizzaroTet: ", 20 * sFactor ** 2, 360 * Svol + 85 * s3) print("Small Guy: ", 20 * sFactor ** 3, 360 * Evol + 85 * e3) import unittest class Test_Tetrahedron(unittest.TestCase): def test_unit_volume(self): tet = Tetrahedron(D, D, D, D, D, D).ivm_volume() self.assertAlmostEqual(tet, 1.0) def test_unit_volume2(self): tet = Tetrahedron(R, R, R, R, R, R).xyz_volume() self.assertAlmostEqual(tet, 0.1178511) def test_phi_edge_tetra(self): tet = Tetrahedron(D, D, D, D, D, PHI) self.assertAlmostEqual(tet.ivm_volume(), 0.70710678) def test_right_tetra(self): e = hypot(sqrt(3)/2, sqrt(3)/2) # right tetrahedron tet = Tetrahedron(D, D, D, D, D, e).xyz_volume() self.assertAlmostEqual(tet, 1.0) if __name__ == "__main__": unittest.main()
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