Suture Mechanics is a unique system for generating polyhedra that evolved from years of studying patterns in nature. The form generating system is very simple in concept but yet very powerful and significant in terms of potential applications to the natural sciences and engineering disciplines at all scales. It is based on the spatial relationship and patterns that emerge between the components/parts that make up all things in nature.
During years of comparative analysis and research of natural patterns by the author, there was a realization that the information about the assembly of patterns was not to be found by focusing upon the solid material but by focusing on the spaces between the material parts. This research strongly suggest that the patterns found in nature are not in the materials but that the materials are in the patterns. The gaps between the parts of material pattern systems are where all assembly and separation takes place. In the Suture Mechanics System, these spaces between the components of materials are called "sutures". The spatial arrangements of the sutures exhibit repeating and recognizable patterns that are ubiquitous in nature at all scales of observation. The concept is illustrated in the photographs of natural patterns below showing some examples of the corresponding suture patterns.
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With this conclusion, new assembly patterns were recognized whose minimal components are comprised of a few simple interactive lines and their junctions called "foundation sutures". The foundation sutures represent the patterns that emerge from between the parts of material components. They comprise the prime structural components in the Suture Mechanics System for generating polyhedral lattices. Through the replication and junctions of these foundation sutures, the Suture Mechanics System can be used to generate and analyze hundreds of regular and hybrid three-dimensional polyhedra. The most significant, unique, and powerful attribute of the system is that it can be used to generate polyhedra and their lattices in a systematic progression from very simple to more complex.The process of generating the forms is accomplished through a simple replication of the foundation sutures. An example of the polyhedra that can be generated in a progression from the simplest to more complex is presented below.
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The Suture Mechanics System of generating polyhedra and lattices is different from the conventional process of constructing classical polyhedra because polyhedra are not constructed from preformed polygons but are generated from lines and their junctions (foundation sutures). The conventional faces, vertices, and edges in the anatomy of classical polyhedra do not exist in the Suture Mechanics polyhedra generating process. It is the lines (sutures) interacting with each other that institute the process of polyhedra lattice assembly with the polygons in the bodies of the polyhedra being the last to form.
As an example of the difference in the conventional approach to constructing polyhedra compared to that of the Suture Mechanics System, the cube and twist octahedron are shown below. The drawings on the top represent the conventional structural nets for construction of the cube and the twist octahedron. The drawings on the bottom show the Suture Mechanics line increment net (or lattice) for construction of the cube and twist octahedron. The structural lattice for the polyhedra are constructed in a flat plain and then wrapped in a spherical form to produce the polygons that make-up the polyhedra.
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Suture Mechanics requires a shift in focus from the flat facial polygons of classical polyhedra to the space lattices of suture space. This shift in focus provides instant flexibility for glide/reflection symmetry operations and lattices that generate naturally in evolving family groups. The structural components in Suture Mechanics are line units and their junctions. Facial polygons become open cells, vertices become junctions, and edges become line increments.
As the approach for the Suture Mechanics form-generating system was developed, a unique terminology had to be created by the author to describe the components, concepts, and process of the generating system. Some of the terminology used in the Suture Mechanics System has already been briefly introduced above. A more detailed discussion of the basic terminology and concepts of the Suture Mechanics System is described in Part I of this site (Return to Table of Contents). For a clear understanding of the system and how it works, it is recommended that the sections on Suture Mechanics and Terminology and Concepts be reviewed first, following the remainder of the introduction. For additional information on the author and contact information for the author, see About the Author.
Please provide comments on the website at the the following address wiggspol@wiggspolysutures.com.
1992 Comments on the Suture Mechanics System by Dr. Cyril Stanley Smith (1904-1992)
"Sculptor Bob Wiggs' approach to form generation is a fine example of the way in which aesthetic intuition about structure can provide a signpost directing thought toward better ways of treating relationships mathematically. His unique view of the inherent structure in the natural world illustrates the principles underlying both quantum theory and particle physics. His form generation system will be of interest to physicist studying amorphous solids and strain and imperfections in polycrystalline matter as well as to the molecular biologist. All hot topics today. The closed-wall, plane-faced straight-line approach to spatial problems has dominated thought about 3-dimensional structures for too long, even in the minds of such masters as Bucky Fuller ." (1)
(1) Dr. Smith's academic career earned him the title of Institute Professor at the Massachusetts Institute of Technology, which distinguished him as a faculty member whose work transcended individual departments and disciplines. His analysis of the boundary patterns between metal grains had inspired his general theory of structure, which led him to found the Philomorph Society, a group of Harvard and MIT professors studying form and structure. He was the author of more than 200 articles and books. Among his awards was the Presidential Medal of Merit (presented with colleagues) for his work at Los Alamos, NM, directing preparation of the fissionable metal for the atomic bomb.
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Robert A. Wiggs received international recognition following more than 30 years as a art professor with his discovery of the ninth self all-space filling polyhedron. He called his discovery the "Twist Octahedron" and published his findings in 1987 in Leonardo, Journal of the International Society for the Arts, Sciences and Technology. Prior to the Twist Octahedron, eight self all-space filling forms were known. The Greeks discovered five of the eight and the remaining three were discovered by Lord Kelvin, Buckminster Fuller, and Keith Critchlow.
Since 1987, Wiggs has presented papers on his form-generating research, usually illustrated with his sculpture, at conferences in Japan, Hungary, South Africa, Israel, and the United States. He has shown his polyhedral designs and sculpture at numerous exhibitions nationwide. During his academic career, he taught art (drawing and sculpture) at the University of Colorado, University of Kentucky, Southern Illinois University, Louisiana State University, and the University of Louisiana in Lafayette. He has been researching natural patterns and developing his system for generating polyhedra (Suture Mechanics) for almost 40 years.
For those that are interested in additional details about the Suture Mechanics System, the author would be pleased to correspond with you at the addresses listed below. Also, please provide comments on the website at the following address wiggspol@wiggspolysutures.com.
Email: wiggspol@wiggspolysutures.com
Robert A. Wiggs 128 Hugh Wallis Road Lafayette, Louisiana 70806
In Part I of this site, portions of the Suture-Mechanics polyhedra-generating system are described along with drawings/diagrams illustrating the system and its flexibility and transformation properties. Also, analyses of some of the polyhedra generated by the system are discussed and other concepts and features are presented to show the scope and significance of this unique form generating system.
Part II of the site describes the Twisted Loop concept of Suture Mechanics that was inspired by thinking about how the angular polyhedra could be represented as angular and curvilinear loops. The concepts of the Twisted Loop System are discussed along with drawings/diagrams illustrating polyhedra and other forms/lattices represented as angular and curvilinear twisted loops. The loop system also has applications to hyperbolic geometry. Part II is currently under construction.
This site also includes examples of the Sculpture that was created by the author based on the concepts of the Suture Mechanics System. For a sculptor, the Suture Mechanics System is a perfect physical and conceptual vehicle for studying natural systems and for creating forms of art.
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Introduction
Part I - Suture Mechanics
Part II - Suture Mechanics Twisted Loops
This site is in the process of being created so additional contents/topics will be added periodically.
In this section, portions of the Suture-Mechanic polyhedra-generating system are described along with drawings/diagrams illustrating the polyhedra generating system and its flexibility and transformation properties. As an example of the Suture Mechanics form-generating system, the figure below illustrates the structural evolution and generation of the twist octahedron. The sequence starts out with a single line increment or suture (drawing 1) which progresses through the addition of other line increments (drawing 2) that interact (drawing 3) and are conjoined (drawing 4), replicated (drawing 5), wrapped (drawing 6), and capped (drawing 7) to form the twist octahedron (drawing 8). The suture lattice in drawing 5 is constructed in a flat plain and then wrapped in a spherical form to produce the twist octahedron.
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By repeating (or replicating) the foundation suture 2, 3, 4, 5, and 6 times, a series (or family) of polyhedra can be generated that are based on the same suture. The drawings below illustrate the generated family of polyhedra that starts with the simplest form (twist octahedron) and progresses to more complex forms. The suture lattices are shown across the top of the drawing with the corresponding polyhedra below. Remember, the suture lattices are constructed in a flat plain and then wrapped in a spherical form to produce the polyhedra.
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This example shows the basic concepts for generating polyhedra using the Suture Mechanics System. For additional details concerning the concepts and process of generating forms using this system, see the section on Terminology and Concepts of Suture Mechanics.
As the approach for the Suture Mechanics polyhedral-lattice generating system was developed, a unique terminology had to be created by the author to describe the components, concepts, and processes of the form generating system. The terminology is presented below along with diagrams illustrating the process and concepts.
Terminology and Concepts
Suture and Foundation Suture - A suture is defined as the spaces between the parts/components of materials in nature. The simplest suture is represented by a single line increment as shown in the figure on the left below. The single line increment can join with other line increments to produce networks of line increments termed foundation sutures. The foundation sutures are the fundamental networks of line increments that are used to generate polyhedra. There are two fundamental foundation sutures, which have been termed the canine foundation suture (composed of two suture line increments) and bovine foundation sutures (composed of three suture line units), both of which are shown on the left below. All line increments are conserved in length throughout the Suture Mechanics System. The drawings on the right show the point and edge contacts of the canine and bovine foundation sutures, the significance of which is described below.
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For simplicity in illustrating the generation of polyhedra using the Suture Mechanics System, the following examples of the form generating process focuses on the bovine foundation suture. The same processes can also be used for the canine foundation suture.
Primary Point Junction and Secondary Edge Junctions - When the foundation sutures interact, they conjoin at point to point contacts or edge to edge contacts. As an example, when the bovine foundation suture is conjoined point to point, it is referred to as the primary bovine suture. When conjoined edge to edge, it is referred to as the secondary bovine suture.
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Replication of the Foundation Sutures - The replication of a foundation suture is simply the duplication of the suture 2, 3, 4 ,5 or 6 times in a linear string. This replication operation is shown below with an example of three replicates of the primary and secondary bovine sutures.
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Wrapping of the Replicated Foundation Sutures to Create Equatorial Rings - Initially, the replicated foundation sutures are constructed (replicated and conjoined in a linear string) on a flat horizontal plane. In order to construct a three-dimensional form, the replicated suture must undergo a wrapping operation. The flat suture lattice is wrapped in a spherical form to produce an equatorial ring. Examples of this wrapping operation are shown below using 2 and 6 replicates of the primary and secondary bovine sutures.
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Foundation Suture Caps - Caps are line increments that act as capping components on the poles (top and bottom) of the equatorial ring. When the capping operation is complete, a closed polyhedron is formed. The capping units in the Suture Mechanic System, shown in drawing 1 below, are classified into three categories: circumferential caps (Cap A and B), radial caps (Caps C and D) and compound caps (Cap E). In addition to showing examples of the three types of caps, drawings 2 and 3 below also show an equatorial ring (of a three-replicate primary-bovine suture) with the circumferential cap (Cap A) and a radial cap (Cap C) floating above and below the equatorial ring, followed by the resulting polyhedron when the caps are added to the top and bottom of the equatorial ring.
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Caps as Appendages to the Foundation Suture - Preferably and as an alternative to representing the caps as separate components from the foundation sutures, the line increments that make-up the caps can be attached (or appendaged) as individual line increments to the suture. This concept is illustrated below. Drawing 1 is a suture lattice (a three-replicate secondary-bovine suture) without the caps appendaged to it. Drawing 2 shows the same suture lattice with a circumferential cap (Cap A) appendaged to it. When the suture lattice is wrapped, the polyhedron on the right is formed. Drawing 3 shows the same suture lattice with a circumferential cap (Cap B) appendaged to it and the resulting polyhedron. Likewise, drawing 4 shows the same suture lattice with a circumferential cap (Cap C) appendaged to it and the resulting polyhedron. By changing the type of cap, new polyhedra are formed with the same suture lattice. In fact, five different polyhedra can be formed from the same suture lattice by simply changing the type of cap.
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Generation of Polyhedral Families - In the Suture Mechanics System, polyhedra are generated in families. By replicating a suture 2, 3, 4, 5 and 6 times, a family of polyhedra are generated using the same foundation suture. An example of the family of the Twist Octahedron is shown below. The suture lattices are shown across the top of the drawing with the corresponding polyhedra below. Remember, the suture lattices are constructed in a flat plain and then wrapped in a spherical form to produce the polyhedra.
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Additional Important Concepts of Suture Mechanics - Additional concepts including Polyhedral Hemispheres, Glide/Reflection Transformation of Polyhedra, and Twisted and Untwisted Polyhedral Hemispheres are described in the section on Other Features of the Suture Mechanics System.
Polyhedral Hemispheres - One of the unique features of the Suture Mechanics System is the ability to separate polyhedra into two hemispheres (north [top] and south [bottom]). The separation occurs along the foundation suture equatorial ring. The foundation suture forms the pathway or dividing line (gap) between the polyhedral hemispheres. An example of this hemisphere separation is illustrated below, along with a plan view of the sutures for this polyhedron.
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Glide/Reflection Transformation of Polyhedra - When switching between the primary and secondary modes of a foundation suture, a glide/reflection transformation takes place. An example of this is shown below with the shift between the primary bovine suture to the secondary bovine suture. The sutures are represented as the white spaces.
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The drawings below further illustrate this shift with the suture of the twist octahedron transitioning to the rhombic cube.
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Twisted and Untwisted Polyhedral Hemispheres - The twist dynamics between the polyhedral hemispheres is initiated by the glide/reflection symmetry operation. When the sutures are wrapped into rings, the glide/reflection operation is transformed into a twisting and untwisting operation on its axis of symmetry. The polyhedral hemispheres are able to twist and untwist in relation to each other and transform from one polyhedron to a totally different polyhedron. Based on the twist/untwist operation, all polyhedra are referred to as Primary or Secondary. Primary means the hemispheres are twisted in relation to one another and secondary means the hemispheres are untwisted in relation to one another. The drawing below illustrates this twisting operation which transforms the cuboctahedron to a hendecahedron.
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Based on the mechanics involved in generating polyhedra using the Suture Mechanics System, it is possible to analyze polyhedra by breaking them down into their fundamental sutures.
One of the unique features of the Suture Mechanics System is the ability to separate polyhedra into two hemispheres (north [top] and south [bottom]). The separation occurs along the foundation suture equatorial ring. The foundation suture forms the pathway or dividing line between the polyhedral hemispheres. Separation of a polyhedron into its hemispheres allows the foundation suture to be recognized as well as the capping components. This results in breaking the form down into its fundamental construction components (suture and cap components). The polyhedra can then be categorized to determine which family of forms it belongs to or if it is a hybrid form. An example of the analysis of the hendecahedron and twist octahedron are illustrated below.
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This technique has application to all types of regular and hybrid polyhedra as well as other polyhedral forms such as polyhedral tubes. Additional examples of analyses of polyhedra are included in the section on Hybrid Polyhedra and Analysis of the Buckyball.
For a sculptor, the Suture Mechanics System provides a physical and conceptual vehicle for studying natural systems and for creating forms of art. Some examples of wood and metal sculptures and drawings that were created based on the ideas from the Suture Mechanics System are presented below. The sculptures are based on the author's Twisted Loop System that is described in Part II of this site (Introduction to Twisted Loops).
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In the process of generating polyhedra using the Suture Mechanics System a new self all-space filling polyhedron was discovered. This new polyhedron, termed the "Twist Octahedron" by the author, is equivalent to a pair of trigonal prisms twisted together. The drawing below shows the Twist Octahedron and it's self all space-filling configuration. It is a self all space-filling form when packed in interlocking alternating orientations. Prior to the Twist Octahedron, eight self all-space filling forms were known (shown below). The Greeks discovered five of the eight and the remaining three were discovered by Lord Kelvin, Buckminster Fuller, and Keith Critchlow. The Twist Octahedron is the ninth self all-space filling polyhedron.
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Hybrid Polyhedra
By mixing the equatorial ring and cap components among themselves in the Suture Mechanics form generating process, it is possible to generate a great number of hybrid polyhedra. The first drawing below illustrates a hybrid polyhedra with mixed circumferential and radial caps on a canine suture. The second drawing shows a hybrid tetrakiadecahedron (of R.E. Williams), which is generated by mixing the canine and bovine sutures. An infinite number of polyhedra can be generated with these hybrid mixtures. This process shows the utility and flexibility of the Suture Mechanics System for generating both new (hybrid) polyhedra and analyzing known polyhedral structures.
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Polyhedral Tubes
In addition to generating traditional and hybrid polyhedra, the equatorial rings can be stacked, before capping components are added, to form tubes with variable lengths and diameters. The tube length depends on the number of equatorial rings that are stacked and the tube diameter depends on the number of foundation suture units in the equatorial ring. The diagram below shows a six unit secondary bovine tube with the equatorial ring (or wave ring), polyhedral source, the polyhedral hemispheres and the shape of the suture gap between the hemispheres.
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A sixty-atom cluster of carbon, called the "Buckminsterfullerene" or Buckyball, was discovered rather recently. Analysis of the Buckyball using the Suture Mechanics System has revealed another foundation suture unit composed of four line increments. It is similar to the Bovine suture but it has four line increments instead of three. The four line increment suture meanders through the halves of the polygon pattern of the Buckyball. When the four unit suture is replicated and stacked edge-to-edge, it becomes a tube of hexagons.
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Angular Polyhedra and Twisted Loop Forms of Polyhedra
The Twisted Loop concept of Suture Mechanics was inspired by thinking about how the angular polyhedra could be represented as angular and curvilinear loops. The fundamental mechanics of the polyhedra in Part I are contained in a process of polyhedral transformation that involves alternate and opposite point and edge junctions of the structural components used to generate polyhedra. The fundamental mechanics of Part II (functioning as twisted loop complements of the angular polyhedra of Part I) are contained in twists performed upon a neutral closed loop.
The loop in suture mechanics is a circular ring or torus. It has no surface area and its circumferences can be infinite in scale. All of the potential twists, no matter how complex, can be performed on a single loop. Initial neutral conditions become dynamic with the advent of the twist. Most important in the twist mechanics of the loop system, what ever the potential and complexity of the forces generating the twist, is the fact that the loop is always capable of returning to its initial configuration. This dynamic condition permeates the entire twisted loop system.
The twisted loop anatomy and its complementary association with polyhedra may be difficult to visualize at first. To show this relationship, the suture mechanics twisted loop system of Part II begins with a twisted loop exhibited by a familiar object – the suture loop that meanders between the material parts in the cover of a tennis ball (see figure below). The material of a tennis ball is composed of two parts, but the suture loop that separates the two parts is one continuous twisted loop without parts. When viewed form different angles, there are three hemispherical views of the tennis ball suture loop, as shown in Figure A below. The three views change dramatically if the tennis ball is transformed into a transparent sphere so we can visualize the suture loop circumnavigating the sphere, as shown in Figure B below. If the three spherical views are confusing, just concentrate on Figure A and compare it with Figure B. The middle spherical view in Figure B, more clearly shows how the loop twists to form the sphere. This example emphasizes the importance of the orientation of the viewers’ perspective on recognition of the twists in the loop. The next Section describes the nomenclature, mecahnics, and concepts of the twisted loop system.
FIGURE OF TENNIS BALL LOOPS
Twisting in nature can be observed in a river meander, a DNA double helix structure, mid-ocean ridges, and a magnolia leaf to name a few. Classical scholars were mostly interested in how these systems were formed. Suture Mechanics focuses more on how these systems behave in relation to one another instead of how they are formed individually. The differences in classical polyhedra and the foundation suture generated polyhedra and twisted loop configurations are like the differences in the behavior between solids and liquids. In the classical idea, polyhedra resist change and are actually thought of a solids. In the suture mechanics idea, polyhedra are more fluid, they can twist and untwist and the hemispheres can separate and rejoin to form secondary configurations as presented in Part I of this site. The twisted loop is even more fluid because it has an oscillatory, inside out and outside in capability as described in the following Section on the Fundamental Twist Mechanics.
Part II
Funamental Twist Mechanics
The fundamental mechanics of twists that can be performed on a loop are illustrated in the figure below. The loop upon which the twists are performed is without junctions or intersections – it is a continuous toridal loop. The figures below shows two kinds of twists “single twists” (Figure A and B) and “double twists”(Figure C and D)and two kinds of loop structures “exostructures” (Figure C) and “endostructures” (Figure D). A single loop twists to the left or the right one twist at a time. The single twist loop, when replicated, generates a spiral tube. Double loop twists are different, they twist right and left simultaneously to generate spherical forms (Figures C and D). Exostructures are formed with twists on the outside of the loop (Figures A and C) and endostructures are formed with twists on the inside of the loop (Figures B and D).
The nomenclature for the twisted loops in the figure below are as follows: Figure A is a single exostructural twist, Figure B is a single endostructural twist, Figure C is a double exostructural twist and Figure D is a double endostructural twist.
INSERT FIGURE
An infinite number of single and double exostructural and endostructural twists can be performed on a loop. The forces that drive the loop to twist are the control variables and the loop itself is the topological invariant. The loop has this property of invariance because it is always capable of returning to its initial condition. Consequently, the twisted loop does not qualify as a topological knot because it cannot pass through itself without cutting or splicing. It is an unknot. The tetraspherical loop presented below and cuboctahedral loop presented in the next section, further illustrate the mechanics of the loop system.
Tetraspherical Loop Example
The tetrasherical loop shown in the figure below illustrates the fundamental left/right, single/double, and exo/endo twist mechanics with greater clarity. The spacial coordinates are the same as those for the regular tetrahedron. Examine the changes in the loop closely as it twists over and under itself. Remember, the single twists turn left or right one twist at a time, the double twists turn right and left simultaneously, the exostructure twists are turned outside the loop, and the endostructure twists are turned inside the loop.
INSERT FIGURE
Cuboctahedral Loop
The cuboctohedron is shown in firgure A below. The hemispheres of the cuboctahedron twist in relation ot each other on its three-fold axis. Every spherical polyhedron with twisted axes can be mapped into a twisted loop configuration.
The two twisted loop forms of the cuboctahedron, the exoloop and endoloop as shown in Figure B below, are examples of more complex loop configurations. The loop is formed with three levels of twists: three twists int eh front, six around the middle, and three in the rear. The dots in the exoloop diagram mark the exact location of the twelve verticies of the cuboctahedron. The loop twists around the twelve vertex coordinates in such a way that loops become a space filling meander in the form of a sphere. Although it may be difficult o see, the twists in the exoloop and endoloop can be transformed into each other on the same configuration loop.
The spheres that are circumnavigated by the exolop and endoloop of Figure B, are depicted in the close packed spheres in Figure C. Figure C shows twelve spheres that are close-packed around a thirtennth sphere (N). The three levels of the loop twists are coordinated with the three levels of spheres: three spheres around the top pole, six spheres around the middle, and three spheres around the bottom pole. The polar spheres of Figure C and the twists representing the spheres at the front and rear loops in Figure B are twisted in relation to each other on their three-fold axis.
INSERT FIGURE
Loops with and without Intersections
Part II
Angular Loops
Hybrid Loops
Twisted Loop Spheres
Hypercube Loops
The twisted loop of Suture Mechanics can be used to analyze more complex structures such as the hypercube, shown in the figure below. The hypercube can be transformed into a pair of twisted loops that fit inside one another – one loop as a nucleus and one loop as an outer shell. Exostructural (outside) twists form the shell (figure 1 a and b) and the endostructural (inside) twists form the nucleus (Figure 1 c and d). The two loops are conjoined point-to-point as shown in the exo/endo configuration of Figure 1 at the bottom left.
Taking this isea further, the endo/exo configuration of the hypercube is shown in Figure 2 (on the bottom right). In this endo/exo configuration, the outer shell loop and the nucleus loop change places from that of the exo/endo configuration in Figure 1. The former shell becomes the nucleus and the former nucleus becomes the shell and they are conjoined edge-to-edge instead of point-to-point. Actually, it is not necessary for the loops to change places because in Suture Mechanics, the exoloop and endoloop can metamorphose into each other without changing places. This is possible because the initial conditions of the loops are conserved without junction sites. The only junction sites are located between the loops where the shell and nucleus loops meet.
Twisted Loop Table
Add Text
Loops
Loops
