Reciprocal Frames, Nexorades and Lamellae: An investigation into mutually mutually supporting structural forms Calder Danz, MS.dc candidate, Design Machine Group, University of Washington Prof. Brian R. Johnson, Advising 12/08/2014
Abstract: Abstract: This paper presents an overview of the current body of literature available on reciprocal frame structures, the history of the RF as an architectural typology, and a discussion of the links and disconnects between research and practice in this area. This information is synthesized by organizing the different morphologies observed into a set of heirarchical lineages or ‘phylogenies’ based on their historical context and the technological requirements for their rationalization processes.
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1.0 Introduction
Reciprocal frames comprise a family of structural systems characterized by the interdependent relations of their constituent parts. The term “reciprocal frame” was coined by the English designer and builder Graham Brown in order to describe a structural paradigm that had, until that time, been without a name. Reciprocal frame building types have a long, though somewhat obscure history, having been developed seemingly in parallel by different cultures in response to the constraints of available materials, but for the most part abandoned following the introduction of modern structural typologies and increased availability of building materials through trade and the development of better transportation technologies. As our global resources are subjected to greater pressure, there has been renewed interest in architectural forms that are highly exible to available materials. Reciprocal frame systems are efcient in their use of small pieces of material to span large volumes. This has bene cial implications for construction in that it makes availabel a material set that is otherwise unsuitable for architectural applications. Hardwoods and lower-quality softwoods that cannot be used in other framing schemes are ideally suited to RF morphology (Thonnisen & Werenfels 2011). This is perhaps one reason that we have seen a surge in the popularity of reciprocal frames as a research topic over the past two decades. Another probable factor is the development of more powerful computing tools tailored to design work. Even simple RF structures have complex geometries (Larsen 2008), and current work explores multiple-unit RF systems that would have been virtually impossible to model (let alone analyze) with the tools available 20 years ago.
A simple 6-strut reciprocal rame
Pavilion by Shigeru Ban & Cecil Balmond, a 3d grillage o multiple 4-strut units
2.0 Relation to Other Space Structures
The reciprocal frame, as a structural type, is most appropriately placed in the family of the “space structures,” a group comprising structural forms that attempt to enclose a maximum uninterrupted interior volume within the limitations of a given material set and context. This includes domes, masonry vaults, spaceframes, tension structures, grid shells, structural panel shells, and a growing list of other structural sub-families. The RF bears important similarities and differences to other members of this family, and these help to explain, in part, why it presents a useful and promising pattern for architectural design in the 21st century.
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Responsive Unit-intelligence
Active Unit-intelligence (Lamella-type RF Vault)
2.1 Masonry Vaulting
Though generally composed of very different materials, reciprocal frame systems bear some similarity to certain types of masonry vaulting. In a well-designed RF, the parts themselves are imbued with the assembly logic of the entire system (Gelez et al. 2011). Keystone arches, gothic vaults and related masonry construction systems share a similar in-built design intelligence. Gothic vaulting tends toward greater “unit-intelligence” as we progess through the historical record. There is more practical customization in the units of a rib vault than those of a groin vault, for example. The fan vaults of the late English Gothic exhibit a higher level of assembly logic still, although it is difcult to know how the builders arrived at the form of each individual element. Most likely, the assembly was shaped and the joints rough-cut on the ground and then trimmed in situ to acheive narrow tolerances. We know that the characteristic modularity of gothic ceilings relates ditectly to the re-use of formwork from one bay to the next. Regarding structural design, masonry vaults of this type must rely on compression alone to transmit loads to their foundations. Because of this requirement, their interior surfaces must have concave curvature to maintain structural integrity. RF structures can be designed to perform in tension and compression, freeing them from this limitation. Whether composed of stone, brick, or concrete, masonry space structures normally involve building elaborate formwork on scaffolding to support the material until completion. In the case of concrete and brick, this formwork is imbued with the project’s fabrication logic. The material itself is ambivalent to its location in the nished whole. The use of formwork is typical in the construction of most space structures, but its complexity varies inversely with ability of building units to self-arrange into the the completed system.
2.2 Tensile Structures
Tensile structures are generally manifested as masted shells, tents, or some similar variant, consisting of a high tensile skin or cable network suspended over the volume of the enclosure. In order to enclose space when built on a level site, there must be an element of the structure in compression (the mast). These structures can be evaluated as an extreme variant on a post-and-beam system, in which each structural element plays an exaggerated role. Tensile shells and reciprocal frames trace a common lineage to post-and-beam structures, though their unique strengths arise from different adaptations on the theme. Instead of specializing the material properties and roles of members, as is
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Exagerated System
Distributed System
done in designing a tensile structure, a reciprocal frame generalizes the role of its constituent parts, such that similar parts perform in both tension and compression. This is a strength and a limitation of the RF typology, as similar units are more or less suited to structural demands of their location and orientation in the structure.
2.3 Gridshells
Gridshells comprise a structural typology that is most closely allied with reciprocal frames in the interdependency of components. Indeed there is some overlap between the two. When a uniform multiunit reciprocal frame system is designed with maximum engagement length between units, its geometry apptoximates a grid. However, there is an essential distinction between the two in the way that a designer thinks of the fundamental units of each structural type and their means of construction. A gridshell’s base units are typically idealized as continuous members that span from one edge of the structure to the other, connecting to all crossing members. The nished structure derives its strength from the ex in the members and this connection network over a (usually) double-curved surface. In construction, this means that the structure is often laid out at and lifted or otherwise deformed into its intended form. This differs from the reciprocal frame in both respects. The units of an RF are idealized as discreet beams that terminate at some length parameter on neighboring units, and in turn provide termination points for other neighboring units. In practice, these structures are assembled in their nal form, following a sequenced assembly plan (Gelez et al. 2011).
Continuous Members
Reciprocally Supporting Members 4
3.0 History
The concept of the reciprocal frame can be traced to prehistoric building types, though the archaeological record and observations of surviving traditions. Similar systems appear in archetypes such as the tepees built by the native peoples of North American plains, the prehistoric “Hogan Dwellings” (Larsen 2008) and the yurts and gers of the northeast Asian steppes. These simple structures may not exactly follow the pattern of a reciprocal frame, but they have in common the design approach of enclosing space with many, relatively small pieces of structural material, arranged in an interdependent system. This efciency in design appears to have been driven by factors such as the demands of a nomadic lifestyle and scarcity of available local building material, conditions often correlated to one another.
epee Frame
Hogan Dwelling
Yurt (closer to gridshell typology)
In the historical record, the earliest known description of a reciprocal frame structure comes from Song dynasty China. A depiction of the “Rainbow Bridge” by Zhang Zeduan dates the concept to the twelfth century or earlier (di Carlo 2008). Bridge designs of this type also appear in the sketchbooks of Leonardo da Vinci (1452-1519c.e.), suggesting that the idea may have migrated from Asia to Europe during the renaissance era. Da Vinci expanded the concept seen in his bridge sketches to depict multi-unit grillage patterns which appear to provide the basis for the concept in western research. Designs by architects Villard de Honnecourt (ca. 1250c.e.) and Sebastiano Serlio (1537c.e.) describe planar systems with a similar type of interdependent structure composed of rectangular-prole beams attached with mortise and tenon or bridle joints (Baverel 2000).
Reciprocal structure o ‘Rainbow Bridge’
Planar system design by Sebastiano Serlio, similar to Da Vinci’s bridge and grillage sketches by Vilard de sketches Honnecourt. 5
The “spiral” roof framing techniques used by contemporary Japanese and English architects is attributed to the framing innovations developed by the Buddhist monk Chogen for rebuilding temple roofs in the late twelfth or early thirteenth century. This type of RF is identical in principle to the type now associated with the term. It is characterized by sloping members arranged in a “fan” around a central polygon and resting upon each other in a closed circuit. This also describes the type of RF structure patented by English designer Graham Brown in 1987 (Larsen 2008).
Compund RF roo by Kazuhiro Ishii
Simple RF roo
John Wallis, in his Opera Mathematica (1699) was the rst to describe the geometries of planar and inclined systems. Through physical and mathematical models, he also attempted explain load path resolution for planar RF structures (Baverel 2000). The work of Olivier Baverel is perhaps most responsible for the current popularity of the RF typology. His PhD thesis was an important step in dening these structures in a modern context. Here, the ge ometry of 3-dimensional RF grids is explored using analytic geometry and later generative processes using genetic algorithms. Extensive structural analysis was also performed to test the effect of varying the assemblies’ parameters. Baverel coined the term “nexorade” to describe these systems. Derived from the latin, ‘nexor’: link, “nexorade” apparently plays on architectural terminology to place the RF in a family of repeating-element assemblies (arcade, colonnade, nexorade). The term is elegant, but competes with Graham Brown’s “reciprocal frame,” creating a somewhat confusing and redundant jargon in the research regarding these structures. [graphics: baverel photo and diagrams]
Design or a quadrilateral ‘nexorade’ and diagrams describing analysis o orrces and moments (Baverel 2000)
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Patented framing systems, such as the lamella roof invented by Friedrich Zollinger (1920’s) further confuse the vocabulary. Certain types of lamella vaults operate on the same principles as an RF or nexorade system. A 2010 publication (Tamke et al. 2010) documents the design of a free-form “lamella” structure using an agent-based form-nding algorithm. The project produces a novel approach to rationalizing reciprocal frames, but uses a different terminology. 4.0 Morphology
The anatomy of a reciprocal frame assembly can be described as consisting of one or more ‘fan’ units, each composed of a number of beams. Parameters specic to this structure include the number of beams per fan, axial eccentricity, and engagement length in the assembly (Baverel 2000). Multiple ‘fan’ reciprocal frame assemblies offer unique opportunities for decorative patterning in the structural system of a building. They also present unique challenges in the generation of form. In any system of more than two RF units, there will be at least one member whose ends are both attached along the mid-span of adjoining members. In non-planar, assemblies, the arrangement of connections makes it impossible for the axis of one member to intersect with those to which it must connect. This can be ameliorated by modifying the members individually, or by applying numerical goal-seeking techniques to the assembly as a whole. After some study, it appears that the domain of reciprocal frame structures can be broken down into ‘families’ based on the desired macro-geometry and the appropriate means of rationalization. One interpretation of the RF taxonomy is presented on page 8 of this document, looking at the eld through a geometric and historical frame. The families may look very differ ent if parsed using different parameters (e.g. structural or material.)
4.1 Macro: Surface Geometries
Depending on intended form, aesthetic, and the degree of direct control desired in the design process, different approaches are more or less appropriate to the task at hand. In the case of designing reciprocal frames, families of geometric forms respond more favorably to certain families of strategies. As is the case with other problems in architectural form generation and rationalization, these strategies can be divided into two groups: top-down and bottom-up. There is some correlation between these two strategy groupings and the relative complexity of the system, with some notable exceptions.
4.1.1 Top-Down Approaches
Depending on the complexity of the form desired, the rationalization strategies available will be different. When designing RF geometries for radial vaults, simple geometric constructions allow for analytic solutions with a relatively low degree of complexity in the resulting assembly (e.g. Zollinger lamella and “rainbow bridge” patterns.) Domes are most easily rationalized by supplanting the input surface with a polyhedron that approximates a sphere, and applying transformations to produce the RF geometry (Senechal et al. 2011). Computational tools have been developed that allow any surface geometry (within limits) to be rationalized as a reciprocal frame structure (Parigi 2014, Song et al. 2013). These appear to be built on the geometric relationships proposed by Baverel, improving the applicability of the theory with new optimization strategies and interfaces that allow end-users greater control.
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4.1.2 Bottom-Up Approaches
The basic parameters of the individual RF fan can be manipulated directly (with a physical model) to ‘sculpt’ the nished form of the assembly. This is one viable way to design irregular forms without a numerical approach (Larsen 2008). However, the propagation of change through a system can be difcult to predict, making this a cumbersome method for more complex systems. A digital equivalent of this approach is found in constraint-based solid modeling packages (e.g. SolidWorks, Catia.) These provide a more precise denition of parameters, but can be equally difcult to control when propa gating change in large assemblies. For simple systems, analytic methods provide a great degree of control for the expert user. Other inventive solutions have employed agent-based goal-seeking and similar algorithms to design self-resolving RF systems. However, it is unclear whether such systems would allow the degree of user control necessary in a conventional architectural design context.
Constructive parameters o an RF unit or ‘nexorade,’ as defined by Baverel (2000).
4.2 Micro: Pattern and Tesselation
Among the many benets of RF (or nexorade) systems is their aesthetic quality of geometric pattern ing. Depending on the strategy adopted for generating the RF geometry, there may be a limited set of patterning options available. In top-down approaches that employ pattern re-mapping, virtually any planar pattern of straight lines can form the basis of an RF system. However, certain polygonal tesselations are optimal because of their repeating properties and tendency to distribute loads more evenly (Song et al. 2013). The most commonly studied patterns are the quadrilateral and the tri-hex forms, but many hybrid forms also exist. In the case that patterning is applied to the surface geometry by means of a mapping function, the tesselation can be generated in 2D and transferred to 3D space. In the case that a mesh or polyhedron is used as a framework for approximating desired geometry, the resulting RF will follow the topology of the framework. In these cases, manipulation of the mesh may offer some control in patterning. For example, geodesic subdivisions of the cube and icosahedron approximate a spherical surface to a greater extent with each level of subdivision, yet the tesselation patterns of the two are distinctly different. Furthermore, any polyhedral mapping can be altered to represent the dual of the original solid by changing the engagement length parameter.
Aesthetic variations in an icosohedon RF transitioning to its dual.
Repeating mappable tesselations as defined by Song et al. (2013)
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Scaffolding tubes and swivel con- Pin-jointed moment truss units nectors: 3r1t-DOF joint[12] (1r-DOF joint) [6]
imber Joinery: 0-DOF joint [Kazuhiro Ishii][9]
4.0 Connection Strategies
The materials used in the construction of RF structures are varied. Joinery methods for constructing reciprocal frames differ widely depending on the material chosen. Following da Vinci’s sketches, many physical models are built with struts of circular cross-section. These are easy to connect and adjust, and provide insight into the parametric nature of the system. Built architectural examples are most often built with squared timbers, making joinery at complex angles a technically demanding task, or with identical units, sometimes consisting of multiple sub-units (e.g. a planar truss.) Depending on the material and joinery system, the design may be pre-dened to a greater or lesser degree. Timber-frame joints, for example, require a high level of geometric certainty, where hinges and scaffold clamps may allow the structure to be dynamically edited during construction. However, systems with many degrees of freedom, such as that designed by Baverel for his experiments, are also highly constrained by their dening parameters during design. For example, as seen in the case of a geo desic RF, eccentricity (strut diameter) is dynamically linked to engagement length and surface curvature (Senechal et al 2011). In structural design, it is desirable that these parameters be independent of one another so that members can be sized appropriately. In constrast to the bypass-strut model, systems with highly customized joinery correlate with a more malleable design process, as constraint parameters such as ‘eccentricity’ are absorbed by the joint (e.g. timber framing) and are able to respond to changes in the system without driving other parameters. The choice of strut/beam material and joinery strategy is dependent on the intended function of the structure. Deployable and temporary structures, such as the Rice University Pavilion or the archaeological shelter erected at Bibracte (Gelez et al. 2011), demand a smaller unit with an easily decoupled mechanical joint. More permnent structures, such as the works of Kazuhiro Ishii and Graham Brown, may use joinery methods motivated by material aesthetics, traditional technologies, or even spiritual representation (Larsen 2008).
5.0 Structural analysis
Structural analysis was not a central focus of this research. However, initial experiments were carried out using Karamba, a nite element modeling plug-in for the Rhino/Grasshopper environment. It is also worth mentioning the intensive research and discussion that has clustered around this structural typology in the civil and structural engineering disciplines. Analytical methods can be used to show
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that Simple RF systems are subject to innite load paths (Nelson & Kotulka 2007) and can be dened as statically determinate or indeterminate. (Gelez et al. 2011). Finite element analysis has been used to show that RF structures can match traditional structures in strength and rigidity (Garavaglia 2013) while being less sensitive to settling. The greatest potential of these structures appears to be in their application as deployable architecture, producing spaceframes with relatively simple assembly compared to conventional multi-layer systems.
Initial test setup using Karamba: Beams are defined in two halves, split at the engagement parameter.All connections are defined as fixed joints with 0 degrees o reedom.
6.0 Experiments
The nal models developed during this project are most closely related to the Zollinger lamella-type RF system, in that they create beam elements with rectangular cross-sections and curvature on the top edge. There are two benets to this geometry in practice. First, the beams are deeper at the mid-point, where connections are necessarily out-of-axis. Second, the curvature on beam tops more accurately approximates the input surface, making it easier to plan an elegant sheathing or paneling strategy. While their units may require more cutomization in systems with variable curvature, lamellatype reciprocal frames are more exible to input surface geometry and structural constraints because of their unit cross section parameters and joinery strategy. 6.1 Tools
The test cases for this project were created using Rhino 5 as a modeling and rendering environment, and Grasshopper as a parametric design engine. The community of Grasshopper users has been interested in reciprocal frames for several years, and has produced an array of scripts for creating the geometry. Some of these examples served as inspiration for this project.
6.2 Strategies
The rationalization approaches explored in this project fall into two groups. Both are top-down processes. The rst set uses geodesic polyhedra as reference geometry to approximate domed sur faces. The second employs u,v coordinate mapping to transfer a 2D pattern to a 3D surface. Both are considered top-down processes. In order to gain some understanding of reciprocal frame, and the relationship between patterning and geometry, initial experiments looked at geodesic polyhedra as a base for constructing the system. Later tests adopted the lamella-type construction logic as a means of rationalizing varied curvature and doubly-curved surfaces. The two surface types chosen were a catenary-prole barrel vault and a torus patch.
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6.3 Patterning
‘Test 0’ establishes a simple method for constructing the basic pattern of the reciprocal frame. In this case of these experiments, the basic unit is a 4-member fan. Lines are dened by midpoints on a bounding rectangle and tangents on a central circle of variable radius. The base unit is propagated by mapping to a rectangular grid, which can be re-sized to change the proportions and density of patterning.
Generating base pattern or rationalizing surace as RF system 6.4 Testing the Viability of U,V Mapping
Starting with this pattern structure, the script rst generates a vaulted form by dening a catenary cross-section and sweeping it along a straight base curve. The parametric catenary curve is valuable in designing a shell form, as it represents a minimal load path, and can be easily varied in height relative to plan dimensions. However, as height increases, the radius at the apex of the curve decreases. This parameter can be used to test the limit of the rationalization strategy relative to surface curvature. Step 2 provides a lattice of geodesic curves coincident with the catenary surface. These curves are used to dene the top proles of the RF beams. This curvature has two purposes in a RF vault. As the endpoints of each beam’s axis lie on a curved surface, all other points on that axis are bound to lie some distance away from the surface. Curvature in the top of the beam accounts for divergence from the surface between the beam end and the points along its length where other beam ends
Mapped pattern
Vectors defining beam ends
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intersect it. As in the Zollinger roof system, the top curvature allows sheathing material to be wrapped evenly over the frame. Lines are placed at the endpoints of each top curve, normal to the curvature of the vault surface. These act as place-holders to approximate the beam ends. Connecting these lines produces a bottom edge and closes the loop of the beam outline. Using unique vectors to dene the beam ends creates closed loops, but the majority are non-planar and therefore cannot be used to dene planar surfaces. The problem is remedied by nding the best-t plane for the vertices of each beam outline and projecting the geometry to that plane. Another approach denes a single vector for each pair of end lines. An advantage of using a best-t plane is its ambivalence to changing curvature in the surface.
Non-planar loops
Best-fit planar loops
The planar surfaces developed in the previous steps approximate the beam geometry, but without accounting for the presence of neighboring beams. The next process is a trimming operation involving several steps. Rough surfaces are exruded and intersected with trimming planes to create a new set of beam geometry that is responsive to neighboring elements in length and angle. The data structure established through this process will also help to locate joinery detailing later on. The following page shows examples of the variation generated by this denition.
Capture10Detail.PNG
Capture11Detail.PNG 13
Successful variations on a lamella-type RF catenary vault
beamDepth = 0.5 beamThickness = 0.1 catenaryLength = 1.25*base catenaryGravity = x0, y0.25, z1.0
beamDepth = 0.5 beamThickness = 0.1 catenaryLength = 1.50*base catenaryGravity = x0, y0.25, z1.0
beamDepth = 0.5 beamThickness = 0.1 catenaryLength = 2.0*base catenaryGravity = x0, y0.25, z1.0
beamDepth = 0.5 beamThickness = 0.1 catenaryLength = 2.0*base catenaryGravity = x0, y0, z1.0
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Failures occur when beam length is out o proportion with surace curvature. Te struts created are unrealistic and prone to breaking the trimming operations in the definition. Tis limit can be quantified as the ratio o the lengths o geodesic cuves between beam endpoints and straght lines between the same sets o points.
6.5 Failure Cases
During the development of these experiments, several cases were observed in which the denition failed to produce a buildable geometry. The limits of the system have proved difcult to diagnose, as they lie in the relationships between parameters, not all of which are direclty controlled in the deni tion. One failure case occurs when beams meet at extreme angles. This manifests as beam ends trimming improperly or beam proportions exaggerating to unfeasible dimensions. This failure mode is a function of beam length, pattern aspect ratio and surface curvature. Pattern aspect ratio is a function of pattern grid extents and surface dimensions. Curvature (the radius of the osculating circle) of a catenary is a function of length and gravity. Beam length is a function of the pattern grid extents and the u,v aspect ratio of the target surface. Effectively, all controllable parmeters are capable of breaking the denition depending on the values of the others. These parameters could be linked in a way that keeps the denition from breaking, but this would make it more difcult to customize as well. Fail ure cases so far have been found only at extreme instances. In general, the denition performs well in producing vaults of realistic proportion.
An exaggerated orm: Tis surace geometry causes the definition to break initially due to the relationship between apex curvature and beam length. Te problem can be fixed by increasing pattern density, thereby decreasing beam length. Tis remedy is scale-specific, as it eventually breaks down when pattern density becomes uneasible to build.
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6.6 Testing Double Curvature and Grid Warp
The second test case for mapping is a toroid surface patch. This is used to test the generality of the the rationalization scheme developed in the previous case. The toroid patch was chosen for multiple reasons. First, working with a regular form means that the difculties encountered can be attributed to an iso lated set of new factors not present in the previous test case (i.e. grid warp, double curvature.) Second, the toroid surface is among the more complex solid forms that can be tiled with planar quadrilateral surfaces (Woodbury 2010). The fact that this geometry can be panelled efciently makes it a valuable test case from an architectural perspective, and suggests that results may be more easily generalized to other families of forms tileable with planar quads.
A toroid patch surace and its parent solid. Te architectural value o this orm is well documented. Extracting patch suraces at different parameters allows the designer to generate a wide variety o interesting orms that are tame and efficient in regard to rationalization.
This test was successful in producing a method for controlling warp during the re-mapping of the RF pattern to the target surface. As the u,v grid approaches the inner and outer radii of the torus, the cells of the grid warp in aspect ratio. By applying an inverse warping function to the cells of the pattern grid prior to mapping, the proportions of the RF units can be regulated. This is a valuable step in rationalizing the RF system as it allows for greater control over the angle at which dependent beams meet.
Mapped pattern resulting RF wirerame
Pattern and wirerame afer applying warp
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A lamella-type RF structure mapped to a toroid patch surace
7.0 Future Work
The scripts developed in this study address many of the geometric challenges posed by complex lamella-type RF systems. Continued development will look at generalizing the script to arbitrary surface geometries and address joinery details for fabrication using CNC equipment. The application of other tesselation patterns and optimization strategies presents avenues for future studies as well. Preliminary structural analysis has been explored using Karamba’s nite element modeling tools for Grasshopper. Subsequent experimentation will apply these tools to the structural types described in these studies.
8.0 Conclusions
Reciprocal frames, as a family of structural forms, compose a complex mosaic. Various manifestations throughout history suggest that the system’s material efciency and unique aesthetic of inter dependence resonate with designers across cultural and temporal domains. The complexity of these systems has historically constrained their applicability in architecture to small-scale and relatively simple implementations. New developments in digital modeling, fabrication, and communication tools are allowing a community to grow around shared interest in this still-obscure typology. This facilitates the development of new techniques for managing the complexity of these structures, and reinforces a consistent vocabulary for their description. From a pedagogical perspective, the reciprocal frame is an exciting teaching medium. In physical explorations, elaborate structures can be created from inexpensive, lightweight material. In the realm of design computing and structures education, the RF presents a range of challenges that push the boundaries of conventional methods and encourage deeper understanding of tools and materials.
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References
1. Parigi, Dario, Kirkegaard, Poul Henning. Design and Fabrication of Free-Form Reciprocal Structures, Nexus Network Journal Vol. 16 pp. 69-78. (Turin: Kim Williams Books, 2014) 2. Garavaglia, E. (2013) Collapse behaviour in reciprocal frame structures. Structural Engineering and Mechanics, 46, ,533-547,,-. 3. P. Song et al. Reciprocal Frame Structures Made Easy, ACM Trans. Graph., vol.32, no. 4, July 2013, New York, NY 4. B. Senechal, C. Douthe, O. Bavarel. 2011. Analytic Investigations on Elementary Nexorades. 5. S. Gelez, S. Aubry, B. Vaudeville. 2011. Nexorade or Reciprocal Frame System Applied to the De sign and Construction of a 850 m^2 Archaeological Shelter. 6. Tamke, Martin; Riiber, Jacob; Jungjohann, Hauke. Generated Lamella, ACADIA 10; ISBN 978-14507-3471-4] New York 21-24 October, 2010), pp. 340-347 7. Olga Popovich Larsen, Reciprocal Frame Architecture. (London: Architectural Press, 2008) 8. Di Carlo, B.,The Wooden Roofs of Leonardo and New Structural Research. Nexus Network Journal, 10, 27-38. (2008) 9. Udo Thonnisen, Nik Werenfels Reciprocal Frames: Teaching Experiences. International Journal of Space Structures Vol. 26 No. 4 (2011) 10. Olivier Baverel, Nexorades: A Family of Interwoven Space Structures, PhD Thesis, University of Surrey, December 2000 11. Robert Woodbury, Elememts of Parametric Design (London: Routledge, 2010) 12. Erik Nelson & Brandon Kotulka. “Innite Load Path?” (Structure Magazine: Oct. 2007)
Image References
1. Model of a 6-strut RF: http://www.reciproboo.org/#/hexayurt-roof-options/4558840976 2. Pavilion at Rice University: http://www.ricegallery.org/new/exhibition/bambooroof.html 3. Diagram of a fan vault: http://www.lookingatbuildings.org.uk/styles/medieval/roofs-and-vaults/stonevaulting/fan-vaults.html 4. Lamella framing: https://www.ickr.com/photos/13274317@N00/2699170812/ 5. Tension shell: http://www.autoorb.com/singularitiesof-architecture-textile-structures-/3.bp.blogspot. com*-JEgXFctNcNg*UPZxAqe2ptI*AAAAAAAAAIg*y9WGG4Kg1Ro*s640*tensile-structure.jpg/
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6. RF vault model: http://gramaziokohler.arch.ethz.ch/web/e/lehre/166.html 7. Gridshell during assembly: http://wvcarch64.wordpress.com/2009/03/10/gridshell/ 8. RF grillage model: http://www.coroot.com/m_marcarelli/Reciprocal-Frame 9. Tepee frame diagram: http://lifestyle.howstuffworks.com/crafts/seasonal/summer/camping-activi ties-for-kids1.htm 10. Hogan Dwelling illustration: https://www.google.com/search?q=hogan+dwelling&tb m=isch&tbo=u&source=univ&sa=X&ei=GkeGVLW-OsnwoASkxYD4BA&ved=0CC wQsAQ&biw=1680&bih=882#tbm=isch&q=hogan+dwelling+&facrc=_&imgdii=_&img rc=OnE5Y4QSA7mFuM%253A%3B8-NjXTL15PdHvM%3Bhttp%253A%252F%252Fwww. merriam-webster.com%252Fart%252Fdict%252Fhogan.gif%3Bhttp%253A%252F%252Fiipr. com%252Fdeck%252Fcell_practice%252F196313%3B301%3B225 11. Yurt Diagram: https://quadralectics.wordpress.com/4-representation/4-1-form/4-1-3-design-in-citybuilding/4-1-3-1-the-circularradial-model/ 12. Rainbow Bridge: http://www2.kenyon.edu/Depts/Religion/Fac/Adler/Reln270/links270.htm 13. Da Vinci Sketches: http://www.hiroshi-murata.com/the-da-vinci-grid 14. Serlio Grillage: (Larsen 2008) 15. Iishi Roof: http://www.ibiblio.org/japanwood/phpBB2/viewtopic.php?t=789&sid=134d5ecc14cbc83 10e13d167fa4583ee 16. Stick RF roof: http://tanglewoodproject.co.uk/ 17. Baverel thesis diagrams: (Baverel 2000) 18. Geodesic nexorade models: https://groups.google.com/forum/#!topic/geodesichelp/5fTgkbtWxqw 19. Tesselation Mappings (Song et al. 2014) 20. Swiveling scaffold connectors: (Baverel 2000) 21. Pin-jointed truss units (Gelez et al. 2011) 22. Timber joints: (Larsen 2008)
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