## Creating Polyhedral Models by Computer (1997)

Venue: | Journal of Computers in Mathematics and Science Teaching |

Citations: | 15 - 7 self |

### BibTeX

@INPROCEEDINGS{Eisenberg97creatingpolyhedral,

author = {Michael Eisenberg and Ann Nishioka},

title = {Creating Polyhedral Models by Computer},

booktitle = {Journal of Computers in Mathematics and Science Teaching},

year = {1997},

pages = {477--512}

}

### OpenURL

### Abstract

This paper describes a computer application named HyperGami that permits users to design, explore, decorate, and study a rich variety of paper polyhedral models. In structure, HyperGami is a "programmable design environment", including both a direct manipulation interface as well as a domain-enriched programming environment based on the Scheme language; the application is thus designed to be accessible to students of geometry while providing challenging projects for long-term or expert users (such as professional mathematicians and designers). In the course of this paper, we describe the HyperGami interface and language; illustrate the construction of "customized polyhedra" of various sorts; discuss the results of our initial experiences using the system in working with middle-school students; and argue for the utility of embedding programming languages in educational design environments such as this one. 1. Introduction Over the centuries, human beings have been fascinated by polyhe...

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Citation Context ... may be employed within HyperGami for decorative purposes. The program includes procedures for experimenting with dynamical systems (e.g., for creating fractal patterns via iterated function systems (=-=Barnsley, 1988)); f-=-or creating new color objects via hue/saturation/value or RGB representations; for designing custom gradients and patterns (much like those found in commercial paint programs); and for "reading i... |

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Citation Context ... of soapstone (ca. 500 B.C.), and regular icosahedral objects—dating perhaps from the Roman period—that have been found in Egypt and have since been displayed at the British Museum (Malkevitch, 19=-=88; Coxeter, 1973-=-, p. 13). Renaissance artists were attracted to polyhedra: for instance, da Vinci depicted the regular solids (among others) in his illustrations for Fra Luca Pacioli's book Divina Proportione, while ... |

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Citation Context ...he user simultaneously sees the corresponding line drawn on the solid shape in 4 Indeed, paper-folding tasks have been studied as psychological measures of mental imagery (cf. the earlier citation of =-=Piaget and Inhelder, 1948; -=-see also Shepard and Feng, 1972). Creating Polyhedral Models by Computer— 8sthe ThreeD window. This technique allows the user to get a sense of how polygons in the net correspond to the faces of the... |

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Citation Context ...t reflects (and, we hope, fruitfully extends) Creating Polyhedral Models by Computer— 1sthis venerable tradition of polyhedral modelling. In structure, HyperGami is a programmable design environment=-= (Eisenberg and Fischer, 1994-=-): a computational system that integrates direct manipulation interface tools and application-specific programming elements. The purpose of the HyperGami system is to provide an environment that is bo... |

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Citation Context ...agons: (define (octagon side) (repeat 8 (forward side) (right 45))) (define (octagon-pattern side) (repeat 8 (octagon side) (right 45))) These procedures are based on standard Logo turtle primitives (=-=Abelson and diSessa, 1980); the oct-=-agon-pattern procedure, when called on a numeric argument, instructs the programmable pen (or "turtle") to draw eight octagons of the given side-length, with a rotation of 45 degrees between... |

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Citation Context ...read view among mathematicians of the 1 For an eloquent exception, see the final section in the essay by Senechal (1990). Creating Polyhedral Models by Computer— 2simportance of mathematical imagery=-= (Hadamard, 1949; Fom-=-enko, 1994). Hilbert and Cohn-Vossen's well-known text on geometry opens with the first author's observation: "In mathematics, as in any scientific research, we find two tendencies present. On th... |

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Citation Context ...esponding line drawn on the solid shape in 4 Indeed, paper-folding tasks have been studied as psychological measures of mental imagery (cf. the earlier citation of Piaget and Inhelder, 1948; see also =-=Shepard and Feng, 1972).-=- Creating Polyhedral Models by Computer— 8sthe ThreeD window. This technique allows the user to get a sense of how polygons in the net correspond to the faces of the folded solid; and it gives rise ... |

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Citation Context ...mathematicians of the 1 For an eloquent exception, see the final section in the essay by Senechal (1990). Creating Polyhedral Models by Computer— 2simportance of mathematical imagery (Hadamard, 1949=-=; Fomenko, 1994). Hi-=-lbert and Cohn-Vossen's well-known text on geometry opens with the first author's observation: "In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the... |

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Citation Context ...ilot Project Activities Not all of the children's HyperGami projects involved polyhedral models; as the name of the program suggests, it may also be used to decorate folding nets for origami figures (=-=Eisenberg and Nishioka, 1994-=-), and some of the projects that the children undertook involved the creation of (e.g.) decorated origami frogs and turtles. On the other hand, all of the children did choose, at some point, to undert... |

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Citation Context ...cially lovely illustration of this practice—a set of cardboard "classroom demonstration" models, now over two centuries old—is still on display in a cabinet at the University of Götting=-=en in Germany (Mühlhausen, 1993-=-). In our own century, a variety of books have been published that include instructions for the creation of paper polyhedra: some examples are Holden (1971), Wenninger (1971), Schattschneider and Walk... |

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Citation Context ... dodecahedron made of soapstone (ca. 500 B.C.), and regular icosahedral objects—dating perhaps from the Roman period—that have been found in Egypt and have since been displayed at the British Muse=-=um (Malkevitch, 1988-=-; Coxeter, 1973, p. 13). Renaissance artists were attracted to polyhedra: for instance, da Vinci depicted the regular solids (among others) in his illustrations for Fra Luca Pacioli's book Divina Prop... |

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Citation Context ... Luca Pacioli's book Divina Proportione, while a famous portrait of Pacioli—rendered by Jacopo de Barbari—shows the mathematician with two solid polyhedral models, one on either side (Marinoni, 19=-=74; Coxeter, 1988; -=-Senechal, 1988). Polyhedra needn't always be modelled in stone, like the pyramids: the more homely material of paper will also serve quite nicely. The invention of paper polyhedral models— generated... |

1 |
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Citation Context ... truncated octahedron (at right) and its corresponding folding net. 10 We are currently preparing an internal report including a more thorough technical description of the algorithm and its variants (=-=Eisenberg 1996).-=- Creating Polyhedral Models by Computer— 15s4.2 Operations on Solids:Vertex Truncation, Capping of Faces, and Application of Linear Maps Once a starting solid object has been created in HyperGami, t... |

1 | Creating Polyhedral Models by Computer— 30 - Fischer, Lemke, et al. - 1991 |

1 |
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(Show Context)
Citation Context ...trations for Fra Luca Pacioli's book Divina Proportione, while a famous portrait of Pacioli—rendered by Jacopo de Barbari—shows the mathematician with two solid polyhedral models, one on either si=-=de (Marinoni, 1974-=-; Coxeter, 1988; Senechal, 1988). Polyhedra needn't always be modelled in stone, like the pyramids: the more homely material of paper will also serve quite nicely. The invention of paper polyhedral mo... |

1 |
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(Show Context)
Citation Context ... book Divina Proportione, while a famous portrait of Pacioli—rendered by Jacopo de Barbari—shows the mathematician with two solid polyhedral models, one on either side (Marinoni, 1974; Coxeter, 19=-=88; Senechal, 1988).-=- Polyhedra needn't always be modelled in stone, like the pyramids: the more homely material of paper will also serve quite nicely. The invention of paper polyhedral models— generated from two-dimens... |