Pieces of Pi?Polyhedra, Orthoschemes and Dihedral Kaleidoscopes

14th Annual Bridges Conference

Curtis Palmer M. Des. ; B. Sc.

Others  :  http://bridgesmathart.org/2011/cdrom/proceedings/73/paper_73.pdf
PID  :  53365

来源: CEUR

【 摘 要 】

Studying polyhedral forms is essential for mathematicians, architects, scientists, biologists, even artists, and forchildren it can be a lot of creative fun.This workshop willshow that dihedralkaleidoscopes are useful toolsforteaching mathematical conceptsto a range of age groups.Workshop participants will experience creating a paperorthoscheme (alsocalled: simplex, plug, quantum of shape, symmetry unit) and discover that polyhedra can beunderstood as products of kaleidoscopic reflections and rotations of such a simplex, see Coxeter [3].The workshopwillconclude with the collective creation of a paper polyhedra out of individualized, i.e. decorated simplexes. Thistransient sculpture will serve as visceral proof of the polyhedral consequences of symmetry operations. Introduction. This workshop will provide educators with a hands on, experiential method for teachingmathematical concepts to students of varying age and receptivity.The use of cost effective classroommanipulatives: dihedral kaleidoscopes and paper models of orthoschemes will be demonstrated andencouraged. Workshop Activity Simplex really:A set of kaleidoscopes and orthoscheme plugs will be available for immediate viewingof the 27 polyhedra.Each workshop participant will be given a template of a plug, a Symmetry Unit Net(see Figure 3) of a truncated tetrahedron; to assemble and view in a kaleidoscope.Of these, twenty fourof the individually decorated quanta of shape will be collected and conjoined to make the wholepolyhedron.The floor will be open to discussion on classroom use of these manipulatives for varyingages and sophistication of students. Pieces of Pi or Pieces of 8? How 3 sets of 3 become 27.Each kaleidoscope’s three mirrors are cut from a rational subdivision of acircle.Three kaleidoscopes are made with 3 sets of 3 mirrors intersecting at angles defined by the centreangles of the tetrahedron, octahedron and icosahedron, Figure 1.The tetrahedron’s mirror set is cut from4/8 of a circle, the octahedron’s from 3/8 and the icosahedron’s from 2/8 for a total of: 9/8 of a circle.(Isthere a connection to be made with the whole tone of music? To Pirates?)