Proceedings of the First Conference
Transformables 2013.In the Honor of Emilio Perez Piñero
18th-20th September 2013, School of Architecture Seville, Spain
EDITORIAL STARBOOKS. Felix Escrig and Jose Sanchez (eds.)
Stability of Quasicrystal Frameworks in 2D and 3D
Eliana M. Duarte Gelvez1, George K. Francis2
1
Mathematics Department, University of Illinois, Urbana, Illinois, USA, emduart2@illinois.edu
Prof., Mathematics Department, University of Illinois, Urbana, Illinois, USA, gfrancis@illinois.edu
2
Summary: The solution to the problem of how to make planar Penrose rod and pinionframeworks rigid was
conjectured by Ture Wester \cite{WESTER}. In his paper, Wester explains how to make a given framework rigid in a
minimal way. He states that the proof of his method can be achieved byfollowing the exposition of Baglivo and Graver \
cite{BAGLIVO}. They dealt with the case where the framework is a grid of squares. The
interesting case of 3D quasicrystal frameworks was proposed to us by
Tony Robbin \cite{ROBBIN} in a lecture on 26 November 2012.
Our goal in this paper is to explain the essence of the Baglivo-Graver and
Wester approach and prove a modest generalization of both. Their solution
applies to any planar rod and pinion framework composed of quadrilaterals
with equal length rods. We also present a real-time interactive computer
animation designed to explore distortions of 2D Penrose frameworks.
The understanding of such distortions naturally leads to a better
understanding of the rigidity problem of quasicrystal frameworks in 3D.
The question of how to make an equilateral rod and pinion 3D
quasicrystal framework rigid by introducing plates bracing certain faces of
the rhombohedra is still under discussion.
We have studied 3D quasicrystal frameworks and their
rigidity. However, their possible distortions can be
so severe as to make them hard to analyse. For one,
an equilateral quadrilateral
no long need to be planar rhombi. Moreover, rhombohedra cease to have
parallel faces under twisting distortion. In our exposition of the
2D case we will indicate which properties do not generalize to 3D.
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Table 1. Bar lengths for the San Pablo deployable roof
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Fig. 02 San Pablo roofing of swimming pool. Seville.
References
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[1] Calatrava, S.; Escrig, F. & Valcarcel, J.P.
«Arquitectura Transformable».E.T.S.A. de Sevilla.
1993. ISBN 84-600-8583-X
[2] Escrig F.: Sánchez J. “Mobile and Rapidly assembled
architecture”. STAR Structural Architecture n1.
University of Seville. 1996.
[3] Escrig,F. “Modular, ligero, transformable. Un paseo
por la arquitectura ligera móvil” Universidad de
Sevilla. 2012. ISBN 978-84-472-1427-3.
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Proceedings of the First Conference
Transformables 2013.In the Honor of Emilio Perez Piñero
18th-20th September 2013, School of Architecture Seville, Spain
EDITORIAL STARBOOKS. Felix Escrig and Jose Sanchez (eds.)
[4] Gantes Ch.J. Deployable structures. WIT Press. 2001.
ISBN: 978-1-85312-660-4.
[5] Gantes, Ch.J. “Structural Analysis and Design of
Deployable Structures” Computer & Structures Vol.
32 nº 3/4. 1989 PP 661-669
[6] Baverel, O., Nooshin, H. « Nexorades Based on
Regular Polyedra. Nexus Network Journal 2007.
Kim Williams Books, Turin. 2007. Pp 281-298 ISSN
1522-4600.
[7] Pellegrino S. Deployable Structures”. Springer. 2001
ISBN 3-211.83685.3.
[8] Popovich, O. “Reciprocal Frame Architecture”
Architectural Press. Amsterdam 2008. ISBN 978-07506-8263-3.
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