For about 30% greater cost of building materials, the Tensile Flexahedron offers much greater shape variability and pleasing esthetics; both important for making people comfortable in hi-tech looking buildings. The Tensegrity requires 4 or more shapes for its weather skin panels and Climate Panels, while the Flexahedron geometry has only one shape Weather Panel, and can make any shape building envelope. Compared with the Tensegrity Climate Envelope, solar efficiency is 25% less.

The other surprising aspect of this structural system is that, in order to tension the transparent plastic weather skin, the whole steel rod frame of the envelope expands 4% during construction. Although geometrically very complex, the hardware is cheap and simple. Figures 4 and 5 show the panel and hubs. These few parts make, (along with a few cables, masts, and anchors), a complete structural frame for Low-E and Cloud Gel for heating and lighting.


The Tensile Flexahedron combines 3 useful properties: First, the cheap flexible skin materials used in tensile structures are kept taut so they do not flap in the wind and tear. Second, the panels this envelope is made from are all the same shape and size, and thus cheap to produce and install. Third, its uniquely variable shape conforms well to various uses and sites.

The extreme simplicity and the use of exactly the right materials results in a heated building shell with the smallest possible footprint regarding: environmental damage from making and recycling construction materials, and lowest initial and operating costs.

The basic cell of this Flexahedron geometry is a pyramid with a hexagon base. Three of these pyramids, joined together at their bases, are shown in Figure 1. The identical triangular facets of these pyramids are called panels. The edges between these panels (green lines) and the edges of the pyramid bases (red lines) can be made of wires, forming a frame for each pyramid, which is held in shape by being under tension, as shown by the arrows. The tension in these wires keeps the panels' skin taut. If these tensions are unequal, then the 3 pyramids' shape will distort together in response, and no longer be symmetrical. Nonetheless, all the wires will remain in tension, and so will the panel skins.

Figure 1: large free-form envelope

Figure 1 is an example of a Flexahedron Climate Envelope of 25,000 square feet (2,300 square meters). The Figure 2 shows the building blocks used to make the example, and also how it would be made to have 100,000 square feet (9,300 square meters).

A net can be formed by joining together at their bases a large number of these hexagon based pyramids, as in Figure 1. This hexagonal bottom net (red lines) can be held under tension along its perimeter (thick red line), and also at each pyramid top (blue dots). Within certain limits, this net under tension can be distorted to conform to a desired shape, with none of its wires going slack. This net geometry with the ability to change shape while maintaining tension is the invention. The photo is of a faceted surface made from pyramids whose joined hexagon bases form a bottom net. It shows how the initially flat Flexahedron bottom net can take a complex shape by distorting its pyramid cells.

Triangular panels for a tensile Flexahedron, made with a skin of strong and durable clear plastic film, and with a wire embedded along their edges, and which zip together along their edges, can form a variable shape net of hexagon based pyramids which is a building envelope. The perimeter of this bottom net is anchored to the ground (small black dots), while the top of each pyramid is pulled up away from the ground. Masts (large black dots) are used to suspend a top cable net (blue lines), which in turn suspends the pyramids from their tops with top wires. The ridge cables (thick blue lines) are part of the top cable net, but are shown as thicker to help visually read the top view of an example building envelope.

The most basic building form or element of this envelope geometry is the barrel vault, or tunnel segment, 2 sizes of which are shown in Figure 2. This tunnel shape, with a width of less than 100 feet, is chosen to make snow slide off. As illustration Figure 2, the tunnel may have an end cap, or it may bend left or right, or three elements may join in a junction. Each of these elements has one mast, and each mast has one element. These 4 elements may be joined in any way, symmetrically or organically, as shown in the example envelope. The path of the ridge

Figure 2: structural modules

cable, going from mast to mast, determines the basic shape of the envelope. With a second kind of freedom, this basic shape may then be adapted, using the Flexahedron's unique flexibility, to fine tune the envelope's shape, which must be concave. To design an envelope, these 4 elements may be photocopied several times, cut out, and taped together.

The lengths of each of the masts and top cables and the location of their anchors, and of the anchors of the bottom net's perimeter, must all be predetermined from a model of the envelope. This model may be physical; of thread and glue, or virtual; in software. The same software could be used for a stress analysis to determine cable, wire, and mast thicknesses.

The first step in the envelope's erection is to put in place the anchors (black dashes and dots) and the bases for the masts. Then the masts are erected. Next, the precut lengths of cable are placed on the ground to assemble the top cable net (blue lines), which is then raised and suspended between the tops of the masts and the anchors. Cables hang down from the top cable net to later attach to the tops of each pyramid.

The panels are zipped together on the ground to form the pyramids, whose bases are zipped to each other. Zipping them together forms the bottom net (red lines) of steel rods from the pyramid bases, as well as forming the net made of rods (green lines) from the pyramid edges. As the pyramids are formed, their tops are fastened to the cables hanging from the top cable net. The top view of the example envelope shows all 3 nets at this stage, with the joined pyramid bases flat on the ground.

Next, the top cables hanging from the top cable net are used to haul the pyramids up into place below the top cable net (blue lines), as shown in the cross section of the example. In the top view of the example, the effect of lifting up the pyramids is to move the bottom net perimeter (thick red lines) about one third of the way in towards the ridge cables (thick blue lines). Finally, the bottom net perimeter is fastened to the perimeter anchors (small black dots).

Alternately, the cable nets may be erected, and then the panels installed by zipping them together from a cherry picker. This method allows the panels to be handled more delicately.

The building envelope shown in Figure 1 is made from 3 tunnel segments, 4 bends, 2 junctions, and 4 end caps. Or, looked at another way, it is made from 68 of 6-sided pyramids and 12 of 5-sided pyramids, for a total of 468 panels. The panels are 10-20 feet high, so the floor area of the envelope, which is 60% of the total panel area, is 11,000-44,000 square feet (1,000-4,100 square meters), and the free span of the tunnel sections is 30-60 feet(9-18 meters).


The complete envelope web consists of 3 nets: the top cable triangle net (blue lines), the pyramid edge diamond net (green lines), and the pyramid base hexagon net (red lines); and a fourth set of top wires that connect the top of each pyramid to the top cable net (short blue lines in the cross section of the example, and blue dots in the top view). So the complete web is made of 4 subwebs, with all of their different shape facets meshing. Any reasonable spider would turn green with envy. Like a network of roots seeking water, this array of nets collects from the panel edges the wind and snow forces applied to all of the panel surfaces, then carries them to, and drains them into, the mast tops, and the anchors for the top cable net (black dashes), and the anchors for the bottom net perimeter (black dots), as shown in Figure 1.

The top cables hanging from the top cable net and pulling up the pyramid tops (short blue lines in the cross section) are there to allow the pyramids' steel rod frames to move and flex a bit in order to distribute their tensions more evenly. (For the same reason, these top cables may have springs.) For a pyramid's edge rods to share the tension from their top cables as evenly as possible, the top cable should be on the center line of the distorted pyramid. The top cables act like the air pressure inside an inflated building's envelope. Top cables are not necessary for the pyramids on the perimeter of the bottom net, since their tops could not move even if they had top cables.

Although not used in the present example, pyramids with 4, 5, or 7 sided bases may be substituted in the bottom net of hexagon base pyramids, as shown for the rigid Flexahedron in Section 13, Figures 5 and 6. This is done in order to determine the envelope's shape; or where sharp curves; or other than six-fold symmetry are desired. But all of these different pyramids are made by zipping together a different number of the same triangular panels. When many of the pyramids are not six sided, the triangular grid of the top cable net must be modified accordingly.

Further, the use of the previously described 4 building elements and the consequently regular triangular grid of the top cable net may be discarded for a more organic, free form envelope shape, as shown for the rigid Flexahedron in Section 26, Figures 2 and 3. In this case, using a model, the envelope's floor area is first covered with a bottom net of hexagon pyramids. Then, where desired, 4, 5, and 7 sided pyramids may be substituted in. Utilizing the Flexahedron's flexibility of shape, the bottom net is then pulled up by its pyramid tops into its desired shape: one that fits the envelope's use and site, and that also sheds snow well. Next, the placement of masts is determined by trial and error, looking for the minimum number of masts to support all of the pyramid tops through a top cable net. The position of the masts then determines the top cable net's geometry, and the position of its anchors.

To make larger span envelopes from the same size panels, the triangles in the top cable net's grid can each be divided into 4 equal triangles (the first step in making a fractal), thus suspending 4 times as many pyramids, thus covering 4 times the floor area. This is illustrated as expanded geometry tunnel segments. Of course, the other 3 building elements can be expanded similarly.

Unlike the tensile Flexahedron made from wires, the triangular panel frames discussed in Section 26 are rigid, and have compressive strength. They are joined at their edges with hinges. Removing the restraint of having to keep under tension every single rod forming a pyramid base or edge gives this rigid panel Flexahedron net much more variability of shape than the tensile Flexahedron, whose bottom net can form only concave shapes. Their rigidity allows these panels to push each other into position, unlike thin rods. The steeper the pyramids, the more flexible a net they form. However, for a building's envelope, the smoother and more simple snow shedding shapes of a tensile Flexahedron with shallow pyramids are usually fine, as shown in the example. For large span envelopes, the much greater structural efficiency and consequent lower cost of tensile envelopes is best.

Figure 3: Weather Panel Edge Assembly

Figure 4: Weather Panel Skin Hub Assembly


Figure 5: Structural Frame Hubs