Title page for ETD etd-0501102-141322

Document Type thesis

Author Name Eshaq, Hassan

URN etd-0501102-141322

Title Polyhedral Models

Degree MS

Department Mathematical Sciences

Advisors
Brigitte Servatius, Advisor

Keywords
Polyhedron
rigid motion
animation

Date of Presentation/Defense 2002-04-30

Availability unrestricted

Abstract
Consider a polyhedral surface in three-space that has the property that it can change its shape while keeping all its polygonal faces congruent. Adjacent faces are allowed to rotate along common edges. Mathematically exact flexible surfaces were found by Connelly in 1978. But the question remained as to whether the volume bounded by such surfaces was necessarily constant during the flex. In other words, is there a mathematically perfect bellows that actually will exhale and inhale as it flexes? For the known examples, the volume did remain constant. Following an idea of Sabitov, but using the theory of places in algebraic geometry (suggested by Steve Chase), Connelly et al. showed that there is no perfect mathematical bellows. All flexible surfaces must flex with constant volume.

We built several models to illustrate the above theory, in particular, we built a model of the cubeoctahedron after a suggestion by Walser. This model is cut at a line of symmetry, pops up to minimize its energy stored by 4 rubber bands in its interior, and in doing so also maximizes its volume. Three MATLAB programs were written to illustrate how the cuboctahedron is obtained by truncation, how the physical cuboctahedron moves and how the motion of the cubeoctahedron can be described if self-intersection is possible.

Files
hassan.tar

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Document Type	thesis
Author Name	Eshaq, Hassan
URN	etd-0501102-141322
Title	Polyhedral Models
Degree	MS
Department	Mathematical Sciences
Advisors	Brigitte Servatius, Advisor
Keywords	Polyhedron rigid motion animation
Date of Presentation/Defense	2002-04-30
Availability	unrestricted
Abstract Consider a polyhedral surface in three-space that has the property that it can change its shape while keeping all its polygonal faces congruent. Adjacent faces are allowed to rotate along common edges. Mathematically exact flexible surfaces were found by Connelly in 1978. But the question remained as to whether the volume bounded by such surfaces was necessarily constant during the flex. In other words, is there a mathematically perfect bellows that actually will exhale and inhale as it flexes? For the known examples, the volume did remain constant. Following an idea of Sabitov, but using the theory of places in algebraic geometry (suggested by Steve Chase), Connelly et al. showed that there is no perfect mathematical bellows. All flexible surfaces must flex with constant volume. We built several models to illustrate the above theory, in particular, we built a model of the cubeoctahedron after a suggestion by Walser. This model is cut at a line of symmetry, pops up to minimize its energy stored by 4 rubber bands in its interior, and in doing so also maximizes its volume. Three MATLAB programs were written to illustrate how the cuboctahedron is obtained by truncation, how the physical cuboctahedron moves and how the motion of the cubeoctahedron can be described if self-intersection is possible.
Files	hassan.tar