Supercircles: Expanding Buckminster Fuller’s Foldable Circle Models
Year: 2004 Authors: Chris Fearnley; Jeannie Moberly
Core claim
Supercircles provide foldable modules that overcome Fuller’s 360° limit and enable construction of greater great-circle sets such as the 25-circle VE model.
Topics
great circle polyhedra, spherical geometry, foldable modular construction, symmetry and Eulerian circuits
Domains
spherical trigonometry, graph theory, geodesics, spherical polyhedra, paper folding, modular sculpture, geometric design, Buckminster Fuller
Methods
scored paper folding, sector insertion, symmetry-based calculation, laws of sines and cosines
Media
paper, tyvek, scored circles, folded modules
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
BRIDGES Mathematical Connections in Art, Music, and Science
Supercircles: Expanding Buckminster Fuller’s Foldable Circle Models
Chris Fearnley cjf@CJFearnley.com
Jeannie Moberly moberlypj@yahoo.com
240 Copley Road Upper Darby, PA 19082, USA http://www.CJFearnley.com/supercircles.pdf
What would you call a circle with more than ? Supercircles are our solution to a folded circle problem that involves building modules to form great circle polyhedra.
Bucky Fuller demonstrated 7 folded circle models in his book Synergetics [1]. He was fascinated by the great circles of spherical polyhedra, probably from his sailing experiences, where navigation on the globe finds the shortest distance between two points is a circle whose plane cuts through the center. Land based geometrical thinking by contrast defines the shortest distance between two points as a straight line. Fuller’s folded circle models show sets of great circles from the icosahedron and cuboctahedron (Fuller’s Vector Equilibrium or VE). The great circles are the equators given by diametrically opposite poles. Each set of great circles is given by either pairs of opposite vertices as poles (6 great circles for both the icosahedron and VE), or centers of faces (10 for the icosahedron, 7 for the VE: 4 for triangular faces and 3 for squares), or centers of edges (15 for the icosahedron and 12 for the VE). Fuller’s modules are made by folding a circle along scored lines into a conical form. Join the modules together to build the whole sphere. In Fuller’s models it is important to note that the number of folded circle modules exactly equals the number of great circles in the set. The spherical surface of each conical module (except the 3 great circle model of the VE) forms a path known in graph theory as an eulerian closed circuit (each edge is traversed once and only once). The bowtie motif predominates which Fuller likens to wave interference, energy along an arc bounces or is shunted at the great circle intersections.
Figure 1 shows one of 24 supercircle modules for the spherical cuboctahedron. This supercircle contains when a sector is inserted at the dotted line. Score asterisk () lines below and fold outward. Score non lines on the topside and fold inward. Attach A to A, to , and so on for like labeled vertices.
Although Fuller examined the spherical polyhedron that results from combining the 4 sets of the VE , he did not demonstrate a folded circle module for its 25 great circles (totalling ). Since each unique arc in Fuller’s 48 Lowest Common Denominator (LCD) triangles of the VE [1] occurs exactly 24 or 48 times, there is no way to partition these arcs into 25 identical modules (pigeonhole principle). Could we fold 24 modules formed from an LCD triangle and its image by reflection in an edge? By cutting a circle along a radius and splicing in a sector (one possibility is shown in figure 1), we discovered a supercircle of which was foldable into the
2004 Bridges Proceedings
shape in figure 2 (24 of which make the 25 great circle model). A whole new realm is opened for exploration when the supercircle breaks away from the limitation of 360°. For example, the 6+10+15 = 31 great circles of the icosahedron can be built with 30 supercircles of 372°. The 3+4+6 = 13 great circles of the octahedron (or cube) can be built with 12 supercircles of 390°. Surprisingly the number of great circles in an aggregate set seems to have exactly one more great circle than the model’s symmetry can accommodate.
Figure 2 shows the result of folding the supercircle in figure 1. Connect inwardly within the module B (midpoint of edge), C₁, C₂, E and F₂. Connect F₁ outwardly within the module. Connect A (center of triangle), D (vertex), G₁, G₂, and H (center of square) to adjoining modules. Notice how in this symmetrical module spaces on one side are inside (shaded) and outside (unshaded) on the other, positive and negative space. 24 of these folded supercircle modules composes the foldable 25 great circle model of the spherical cuboctahedron (or VE).
Models of spherical polyhedra built from folded circle-like modules, besides being beautiful, offer some interesting details of stable 3D construction from a 2D medium such as paper or tyvek. One aspect is partitioning the sphere into bowties by joining two points of the edge of the supercircle to form the intersections of great circles. The 31 great circles of the icosahedron are totally triangulated, but the 25 great circles of the VE have a quadrilateral shape amid many various triangles. When a supercircle module forms an eulerian closed circuit, the whole model will use minimal material. This means that some triangulated (or quadrilateral) shapes will be enclosed by the module, positive spaces. Others will be open with some walls provided by an adjoining module, negative spaces. A folded module creates inside and outside space at the same time.
In spherical geometry arc lengths are measured in degrees, 360° on every great circle, 360° around each vertex. Napier’s rule helps calculate all arcs and angles, given two knowns in a right spherical triangle. Spherical triangles with no 90° angles require three knowns. Calculations are often aided by identifying symmetries. Four equal angles at a vertex must be 90°. An angle at a pole is equal to the arc at the equator cut by the angle [2]. After calculating angles and arcs using symmetry and arithmetic, the rest are determined using the laws of sines and cosines [3].
Supercircles as a tool to build complex great circle models may have applications to spherical trigonometry education, graph theory, geodesics, electron orbits [4] and cell growth.
Now can you guess, what is a super-duper circle? A supercircle with a large number of degrees added will approach a sphere.
[1] Buckminster Fuller, Synergetics, Macmillan, Vol. 1, pp. 164-189, 1975. The on-line account is at http://www.rwgrayprojects.com/synergetics/s04/p5000.html. [2] Kaj Nielsen and John Vanlonkhuyzen, Plane and Spherical Trigonometry, Barnes & Noble, pp. 103-151, 1946. [3] Robert W. Gray, Notes to R. B. Fuller’s Synergetics: Appendices, 1993. Additional notes at http://www.rgrayprojects.com/rbfnotes/greatc/greatc1.html. [4] Edward Suzuki Hoerdt, Atommetrics: Another View of Atomic Structure Based on Electron Orbital Geometry, Forma, Vol 17, No 4, pp. 275-350, 2002.