Primary-geodesic least surface shapes for predetermined ground plots and functions

Abstract

A means to create the shape for a structure that holds the most space under the least area for any predetermined polygonal ground plot. The ground plot is cut into a sheet material as an open window, called a plot-frame 2. The plot-frame 2 could also be created on the ground itself. A membrane 1 is placed over the plot-frame 2 and a pressure differential is created on both sides of the membrane 1. Functions that take on polygonal shapes are placed on the membrane 1 that would be entryways; connection planes and other flat polygons are achieved by means of constraints 4. The final static display shape of the membrane 1 shows the shape of the intended structure above ground.

Description

This invention is based on my provisional application No. 60/528158, dated Dec. 9, 2003

TECHNICAL FIELD

This artwork relates to all least surface space enclosing structures over predetermined ground plots with functions placed on the surface such as entryways, connection planes, solar panels and the like. Such artwork is called geodesic domes and space enclosing structures. There is an infinite set of shapes other than the sphere that holds space as least surface shape, that was said could be geodesic.

BACKGROUND ART

Within the history of man the first geodesic-like sphere has been under the paws of the Guardian Lions that have been placed outside the temples and gates of China at least 400 years ago. Like all of the current geodesic structures, it seems to resemble a polyhedral made of triangles that resemble a sphere. However, having a geodesic like sphere is not the same as a space enclosing structure.

Next there is a true geodesic dome created by Dr. Walter Bauersfeld in 1922 in Jena, Germany, used for a planetarium-roof that was the embodiment of his work. This structure was made from a divided icsoahedron cast to a sphere to allow his projection devices to remain in focus. It can be assumed he projected rays to the junctures of the material he used as spherical chords that was his framework for this, from a central point. It could also be assumed he didn't see the dome structure which he created, as the roof of the Carl Zeiss optical works, as a single structure. His concern may have just been to have a proper surface for his projectors to function correctly, and the dome itself was perhaps overlooked as a single work.

Richard Buckminster Fuller's first artwork on this subject in 1954, U.S. Pat. No. 2,682,235 made it singular and useful unto itself. It is what Fuller said a geodesic dome could be after this artwork, which matters most of all. Published within Synergetics in 1975, in section 703.01 he says “Geodesic domes can be either symmetrically spherical, like a billiard ball, or asymmetrically spherical, like pears, caterpillars, or elephants.” He said also in section 703.03 “All geodesic domes are tensegrity structures whether or not the tension-compression differentiations are visible to the observer”. “Tensegrity” is a short form of the term “tensional integrity”. Within section 702.01 he also said; “We have a mathematical phenomenon known as a geodesic. A geodesic is the most economical relationship between any two events”. If Fuller is to be given credit for the geodesic dome, the statements should be combined. To me they mean that it is the shape, and not the polyhedral surface that represents what is a geodesic. I call such shapes “primary geodesic surfaces” and the polyhedral representation of them as a “secondary geodesic” no matter the pattern of polygons. He removed the commonly assumed mathematical rules that govern the current geodesic domes and the polyhedral surface. There are no mathematical rules that could be applied to the surface of an elephant Fuller never explained how to create such geodesic surfaces, only that they would be geodesic domes. This current artwork covers the creation of them.

Still looking at Fuller and the bulk of the remaining artwork, most show the polyhedral representation of the sphere as a structure that is called a geodesic dome. Hannula with his U.S. Pat. No. 3,955,329 in 1974 titled as “Hollow Structure” gives a curved set of lines to become a geodesic-like on the surface of the sphere. Also Herrmann U.S. Pat. No. 6,295,785 with a pattern based on the octahedron and not the icsoahedron of Fullers first work U.S. Pat. No. 2,682,235. Within Herrmann's work comes the voice of Yacoe U.S. Pat. No. 4,679,361 saying a geodesic is a representation of a sphere. Leonard Spunt U.S. Pat. No. 3,959,937, “Modular dome structure” in 1976, shows a sphere dome done in circles that is also a polygon much like the triangle, depending on your schooling in geometry. Spunt shows us that the circles could be from cones, all with the tips concentric to the sphere and the axis of each a ray from that center. Circles would come from the intersection of the cones and the sphere. All the commonalities of each of the many geodesic and space-enclosing artworks show a polyhedral representation of the surface of the sphere as a shell of polygons. This leads us to think that there could be more geodesic patents than anyone might care to guess at, that could be copied to a sphere. Also some that look very geodesic seem to just be called “structures”, making the line between the two types very blurred. For all the effort placed into finding common sets of points between the sphere and some solids, rays and cones cast from the center of the sphere, none are able to explain the elephant or caterpillar. Maybe it's time to stop finding all the mathematical sets of points common to the sphere and whatever method or solid, used to divide a sphere into a geodesic. Fuller told us that the sphere is a geodesic if one can see a polygonal pattern or not. Perhaps defining the geodesic dome as a polyhedral representation of just a surface is incorrect. Maybe it should be looked at as least surface relationships that nature can produce as geodesic structures, and what Fuller said they can be. This current artwork shows us how to do just that. It is also able to reproduce with this current artwork, many past geodesic polyhedral surfaces with just 3 parts.

Of all the works remaining on space enclosing structures there are three people that need to be noted. The first is Helmut Bergman U.S. Pat. No. 4,258,513 showing that a structure can have rectangles providing a function on the surface for solar collectors by providing a place to mount them. Also from Helmut Bergman there is U.S. Pat. No. 4,364,207 that can allow for a changing ground plot titled “Extended Space Enclosing Structure”; it shows a somewhat variable ground plot. His works are creative because they show that such structures can be close to the geodesic sphere and have the surface changed to also include rectangles, for some required function such as for solar-panels, doors or clustering. His creations are also based on the icsoahedron as are many others, but one has to look closely. However creative this falls short of showing a least surface over any ground plot. The extended structure is limited to an elongated circle only. However it is the idea of polygons other than triangles as an intended function that makes my current artwork more valuable. I have to extend a full measure of credit to Bergman for showing us this. I hope to have fully exploited his teaching with this present artwork.

The last two remaining artworks worth mentioning starts with David B South U.S. Pat. No. 4,155,967 in 1979 and U.S. Pat. No. 4,324,074 in 1982. He inflates an inelastic membrane that is formed to be spherical-like; because of the manner in which it was made and used. His system relies on the pneumatic pressure within it to become stiff. Later in 1999 with U.S. Pat. No. 5,918,438 he told us this again, when he found the need to place a net over the membrane to help it retain that shape. Because this is a sphere based structure, I consider it a primary geodesic, because there can be seen no “tension-compression differentiations” as noted by Fuller above. Unfortunately, with the work of David South, the shape of the structure is predetermined at the time of manufacture of that membrane. It only becomes rigid under pressure and is unable to produce a least surface area for the space it binds between it and the ground when inflated. It is unable to conform to any ground plot. Making a membrane in that manner for a different ground plot, as a least surface above that ground, would require a guess at best. He has not been able to teach us the means to find the other shapes Fuller talked about. This current artwork has no such short comings.

Along side and before David South there is the work that can be seen on the internet at www.binisystems.com/binisystems.html. Here Dante N Bini shows us a system much like that used by David South, but the membrane has the ability to be elastic and can be stopped at any level of inflation. However as the video shows within that web page, connections to the planes are made by cutting away the surface of one dome to come into contact with the next. The perimeter of the second dome has to encroach into the perimeter of the first. This tends to limit when and where such a system can be employed, and requires virgin ground. This system would be unable to maintain the least surface with the connection plane in place as a flat polygon. Such would be the case if the dome had to connect to a flat surface from an existing structure. Even if this lesson from Dante Bini is most close to this current artwork, it can't show us how to implement the polygonal functions that Helmut Bergman has given us. He did not explain how to produce the other asymmetrically spherical shapes Fuller told us about. This current artwork takes care of the need for virgin ground and cutting away sections to connect the dome to other structures, and any encroachments into that structure.

To account for the geodesic dome as to what Fuller said it can be it may be time to let go of the idea of projecting points, lines and circles to the sphere. The sphere is only a mathematical real world model of a least surface shape for the volume it holds, and the best at that. It may be that the sphere has been used almost exclusively until now, because its math is relatively easy to work, the points easy to produce. This current artwork removes the math of the surface and allows us to use a simpler means to create even more complex geodesic surfaces with functions on its surface.

BRIEF SUMMERY

This current artwork was created to confirm an equation I found for a geodesic dome with a square foot print to the ground. The shape that the membrane 1 displayed, matched the equation well. It did show a square geodesic dome can have an area only 3.0-4.5 percent above the area of a sphere that holds twice the volume that is displaced by the membrane 1. In other words twice the area of the displaced membrane 1, is only 3.0-4.5 percent above the area of a sphere that can hold twice the space displaced by the membrane 1. Also I didn't know that Fuller had said they could be anything but spherical at that time. After it came to my attention what Fuller said geodesic domes could be, it was still many weeks later that I realized what I had on my desk. I also have to credit a math book that was resting on the surface of the membrane 1, for showing me the membrane 1 is self-correcting.

The system works because of the nature of the membrane 1 and the pressure across its two surfaces. The tension in the membrane 1 caused by making it larger in area by that force, will act to return it to a least area. In turn the pressure that caused the change in area wants to expand in all directions. The net result is that the membrane 1 shows a static display of the two opposing forces in balance. The membrane 1 shows the least area for the amount of space it has displaced. If the pattern cut into the plot-frame 2 happens to be the similar polygonal shape of some predetermined ground plot; the membrane 1 shows all least surface areas for any amount of space that could be above that ground-plot, regardless of target height or the amount of space within the final structure. If the amount of space that is bound by the membrane 1 and the membrane 1 itself become very large, it also becomes very spherical. Because of this action when made massive in size, it shows clearly that the membrane 1 is spherically packing the pressure agent. That tells us the membrane 1 is always seeking a least area. Much as a small drop of water free from outside forces would. The reduction in area is a natural event for such systems.

With the complexity of the math I used to create this shape, the membrane 1 shows the more correct result. It shows surfaces for equations that may never be found. The membrane 1 allows us to map and collect data for the shape it shows, so it can be reproduced in scale, or as the final size of the structure. All solutions that the membrane 1 shows are least surface shapes. I would call such shapes “primary-geodesic” shapes. Expanding the membrane 1 from the open window of the plot-frame 2 shows one side of the asymmetrically spherical shapes that Fuller said are geodesic. The second half would be the mirror reflection of the membrane 1 past the edge of the window cut into the plot-frame 2. The primary shape maybe represented as a polyhedral, and than becomes what I call a secondary geodesic. The math of the surface is no longer required. The collection of data points that make up the surface is all that is necessary, or as already noted, the surface itself.

It has to be the nature of that surface that the membrane 1 shows, which bonds any amount of space, how that surface is shaped, and the functions each surface has imposed on itself; that makes something geodesic. Just as it is the surface of the pear, elephant, and sphere, that allows them to be geodesic. All self seeking least areas are therefore geodesic. The mathematical rules used to explain the geodesic sphere will not work for all the other asymmetrically spherical cases. As nature shows us the most well-fed elephant is the most spherical, that too is a least surface seeking system.

Most of all the past artwork on geodesic structures can be reproduced with this current one by making use of an appropriate plot-frame 2 and a single constraint 4. The constraint 4 might look something like a sea-urchin, with its tips in all the right places. With the membrane 1 over this and the correct pressure differential across its face, even the polygonal surfaces that comprise the polyhedral geodesic dome can be recreated. By adding/removing, or deflecting the end point of any arm on the constraint, any number of geodesic surfaces can be produced with one plot-frame 2. Considering that there can be an infinite number of constraints and plot-frames 2, in any combination; the practical use of this invention comes clear.

Looking at the membrane 1 and its reflected surface and area past the plot-frame 2 there are some things worth noting. All secondary-geodesic surfaces that are polyhedral representations of the primary-geodesic surface presented by the membrane 1 have a higher surface/volume ratio than the primary shape. All non hemispherical shapes produced by the membrane 1 have a higher surface/volume ratio than the sphere that holds the same volume as twice the one displaced by the membrane 1. As I have found with a square window within the plot-frame 2 the amount that its surface area to volume relationship is higher than the sphere, becomes unimportant when the functionally functionalism of the structure is considered. Each plot-frame has a null or dip in the surface area to volume relationship as the amount of space displaced by the membrane moves from zero to infinity. Each different or geometric none-similar polygon, cut into the plot-frame 2, has a different null number. Only a circle cut into a plot-frame 2 and a good quality membrane 1 will produce a null value of zero. This happens only when the height of the displaced membrane 1 from the plot-frame 2 is equal to the radius of the circle cut into the plot-frame 2. That is because one would have to compare the S/V of the membrane to the S/V of the sphere as noted above. This is true for all none-constrained membranes 1. There can be no negative null numbers. All constrained membranes 1 should have an S/V ratio above the null for that plot-frame 2.

Constraints only hold the membrane 1 to points/lines/planes that the membrane 1 may or may not reach on its own. These constraints may hold the membrane 1 to a needed door frame size on the edge of a ground-plot. The remaining free area of the membrane 1 will still show the least area for its displaced space no matter the amount. The only problems that can occur come from the failure to make precise parts and the ability of the membrane 1 to maintain an even surface tension.

A good test for the quality of the membrane 1 is when it is expanded from a circular plot-frame 2 and how close it comes to a hemisphere when the height is equal to the radius of the circle. Because the shape of the sphere is known so well, the surface of the membrane 1 would be mapped and compared to that of a true hemisphere.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an expanded view of unit with a square within the plot-frame 2;

FIG. 2 is an assembled view of FIG. 1;

FIG. 3 shows a static display of FIG. 2;

FIG. 4 the final shape of membrane 1 and the space inside;

FIG. 5 the space of FIG. 4 with a set of nodes for polygon framework connections;

FIG. 6 one of any geodesic framework over the shape;

FIG. 7 a realization of Fullers Elephant with this system;

FIG. 8 shows one basic system of FIG. 1 unit with constraints for doors;

FIG. 9 gives shapes of the membrane 1 found in FIG. 8;

FIG. 10 is a more complex plot-frame 2 and simple wire constraint 4;

FIG. 11 is the primary geodesic surface shown in FIG. 10;

FIG. 12 gives one of many cluster systems that can be made with this artwork.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 shows an expanded view of a plot-frame 2, with a square as the open window that the membrane 1 will be pushed from. The shape cut into the plot-frame 2 can be any closed shape. The plot-frame 2 is best made from a stiff, thin sheet material. It can also be that the ground serves as both the plot-frame 2 and the base 3. In that case the plot-frame 2 would be a closed polygon, or cross section of any object upon the ground.

The square in the plot-frame 2 of FIG. 1 could be a square predetermined ground-plot of any size. The base 3 allows us to seal the system and place a pressure on one side of the membrane 1. The base 3 can be made from about anything that will hold up to the forces that will act on it. The amount of space between the membrane 1 and the base 3 form a containment vessel. The pressure agent within that containment vessel and used to push on the membrane 1 can be any suitable and finely divided substance. Some such elements could be but are not limited to; air, oil, none set plaster-like substance or water. The pressure can also come from heat, chemicals or the removal of ambient air. It only matters that a pressure difference can be created and maintained. If an entry-port is required for some pressure agent, it is best located within the base 3 and within the open area of the plot-frame 2. The membrane 1 is said to be one, that is able to become easily expanded, and will display an even surface tension when its surface is distorted. Most times a rubber-like compound will work best, but the membrane 1 is not limited to this. A hot plastic in an almost liquid state, may also be employed, and allowed to cool and set hard.

FIG. 2 shows the assembled view of the unit. It shows a starting zero displaced volume across the open window of the plot-frame 2. The unit is assembled by any means that will hold the membrane 1 tightly to plot-frame 2 and to the containment vessel, that the pressure is found within. Any means to bond this is usable, such as glue, screws, clamps and locking clips or a combination thereof. What ever the means to combine and seal the unit is unimportant, so that has been omitted for clarity of view. At this stage the unit is assembled and ready. It can be that the plot-frame 2 is also mounted to the side of a tank. The volume of space in that tank is disregarded, because the only concern is with the amount of space that the membrane 1 displaces across the plot-frame 2. However if the plot-frame 2 was mounted to a tank large enough, the pressure differential across the surfaces of the membrane 1 could come from a reduction of pressure within that tank. In that case the higher pressure would come from the ambient air. The membrane 1 will show the same result no matter the direction of its displacement, as long as it is free to do so.

In FIG. 3 the membrane 1 is displaced via some pressure increase between it and the base 3. The surface of the membrane 1 is larger. The amount of space that the membrane 1 moved past its rest state within the open window in the plot-frame 2, and the area upon the membrane 1 represent the size and shape of the intended final structure. It is showing the least surface for that amount of space passed from that plot-frame 2, regardless of the volume of that space. It is at this stage that the surface of the membrane 1 is mapped to within the three dimensions of space. The more detail in this mapping and analysis of the surface of the membrane 1, the better any reproduction will be. The final target structure will be similar to the shape of the membrane 1, be that larger, smaller or of equal size. Also the membrane it self might become the inner or outer shell of the structure it self; and then be removed and reused, or kept in place.

FIG. 4 shows the membrane 1 by itself or the shape of the space between the membrane 1 and the base 3 if plaster was used as a pressure agent and allowed to harden. In such a manner as that, the shape can be removed, copied and measured in more detail. It can also provide a scale copy, showing the shape and volume of the structure. This would be of use if a model was required.

FIG. 5 shows some locations over FIG. 4 that could be nodes which would be used to divide the surface with a set of chords producing a polyhedral representation of the surface in FIG. 4. The amount of nodes and the distances between them is all relative to the wishes of the person that is to produce the final structure. One such wish could be to have all the chords about equal. A different intention might be to have the least waste from an unlimited pile of lumber made up of 8 foot pieces. The larger the number of nodes the more the final structure will resemble the surface of the membrane 1. When the surface of FIG. 4 was mapped in to three-dimensional spaces, so were the nodes. It is than up to the rules of analytical geometry to show us the distances and angles, or any other information that will aid us to be able to produce the polyhedral surface. The polyhedral surface for FIG. 5's nodes can be seen in FIG. 6.

At this stage all the details have been covered to produce the elephant or any other object Fuller talked about. The only change is to the window cut into the plot-frame 2. In FIG. 7A, there is a plot-frame 2 with the shape of the side view, or horizontally lit shadow of an elephant upon a wall cut into it. The base 3 is omitted or assumed to be under the plot-frame 2 for clarity of view. FIG. 7B shows what the membrane 1 would be like when reflected past the open window cut into the plot-frame 2 in FIG. 7A. For the view of 7B to be clear the base is omitted. What can be seen in FIG. 7B is that any increases or decreases with the pressure differential on the sides of the membrane 1 will show an elephant that could be better or lesser fed, much as nature would in the real world. To represent the caterpillar, I would cut the top view of one as seen on the ground into the plot-frame 2. The result would be a tunnel like shape on the membrane 1. This approach would even work if the caterpillar was bent, as if it was in the middle of making a turn. For the pear one would be cut from stem to navel, a copy that cross-section would be cut into the plot-frame 2 as an open window. If the membrane 1 is expanded to a proper size even the indentation at the location of the navel is reproduced, if it was somewhat hidden within the pear to start with. The membrane 1 would not show perfect detail. Nature imposed more functions than the membrane 1 would be able to show. But the overall representation would be close and appear “child like”. In the case of the elephant the membrane would produce one with only two legs, but nature required the real one to be able to walk so it has four. The topic and use of constraints 4 is next. However constraints 4 could be added to the bottom of the legs on the plot-frame 2 to give it flat pads. The more proper constraints 4 added the more close it would become in appearance to the real thing. One should even be able to reproduce some of the features that are in the head and joints of the elephant if care is taken in the formation and placement of them.

FIG. 8 shows the same plot-frame 2 and base 3 as FIG. 2 did with some constraints 4a and 4b applied to the base 3 and within the plot-frame 2. In FIG. 8 the membrane 1 is removed for clarity. Constraint 4a could be made from a bent rod, or from sheet or block material as with 4b. It could also happen that each constraint 4A and 4B are just two poles ending at the proper locations, that of the top outside corners of the ones shown in the FIG. 8. Constraints are made to hold the membrane 1 to any point/line/plane required, for the function intended. As just noted there can be more than one that will satisfy the same condition and impose the same function on the membrane 1. Constraints 4 can be made in any fashion that they need, as long as they hold the membrane 1 to the required placement, and need not be in place until the membrane 1 is partly of fully inflated; aiding in keeping an even surface tension upon the membrane 1. They can be of any shape and on any area or side of the membrane 1, also could be of any number upon the membrane, but can not cover the whole surface of same. They could be made in sets so they hold the membrane on both sides, act only to confine that point/line/plane of the membrane 1 to a required location for providing some function. A function on the membrane 1 could be for a door, connection plane to allow for a dome to be added to an existing wall that the dome needs to team up with. A function on the membrane 1 acts only on the surface of it, and because of that only on the surface of the structure itself. They can hold a portion of the membrane 1 flat and vertical to mount some object to the shell of the structure that wouldn't work well if the surface of it had a continuous curve. One such object might be a set of cabinets. Constraints need not be in contact with the edge of the plot-frame 2. One such constraint 4 could be a match set used for a set of solar collection panels, and be held above and beneath the membrane 1.

I have allowed the ones in FIG. 8 to be used for two different sized doors, or connection planes. Each of the constraints in FIG. 8 could be at any location, and the membrane 1 would still show all least surface area solutions. All shapes that the membrane 1 takes on would be primary-geodesic shapes. Any polyhedral representation of the membrane 1 would be a secondary-geodesic.

FIGS. 9a and 9b show the final shape of the membrane 1 that would be produced in FIG. 8. FIG. 9c is one of any secondary-geodesic polyhedral shapes that could be copied to the primary-geodesic surface, much in the same manner that all past geodesic artworks have been copied to the sphere. It was done with the membrane 1 expanded or displaced from the square plot-frame 2 above and herein. Because the constraints in FIG. 8 apply a fixed height to part of the surface of the membrane 1, they need to be to scale for the plot-frame 2 and target height and size of the final structure, and of the intended doors. There may or may not be similar considerations for constraints intended for other functions on different plot-frames 2.

FIG. 10 shows a simple arched rod or bent wire type of constraint 4 and a symmetrically cut plot-frame 2 for the membrane 1 to expand from. One or both halves of the surface of this shape of the membrane 1 as seen in FIG. 11 may be constructed at any one time, as the finances of the person doing the work would allow. This adds a freedom to the engineering, manufacture and finance sides of the structure. As was the case before, there could be constraints 4 placed around the edge of the plot-frame 2 for entryways. The option to add and remove constraints 4 to FIG. 11 is endless. For the first time the primary geodesic surface can take upon it the wishes of the person that will have it in the end. There is no longer a need to confine geodesic surfaces to the sphere and the rigidness of assumed rules of math, which has been used to give the polyhedral surface.

FIG. 12 shows what can be done with repetitive use of just a square and rectangular plot-frame 2 and an expanded membrane 1. This cluster arrangement could also be made as the finances allow. Constraints 4 would be made to match the tunnel pathways that run between the domes, as well as the intended size of the doors. Locations of the tunnels or pathways and the doors are open to the whims of the designer. Each unit in FIG. 12 would have a least area, for the functions imposed on the membrane 1. It could be that the plot of ground that is covered by the cluster could be cut into a single plot-frame 2 and the least surface and primary-geodesic surface could be found for that system as a whole. The membrane 1 expanded from that plot-frame 2 would be more flowing and less blocked than the one seen in FIG. 11. It would appear blocked only where it was constrained for some intended function such as entryways.

The idea of primary-geodesic shapes allows a structure to take advantage of the functions that people require and the cost of maintaining controlled environments. The advantage of having the most volume of space inside a structure could come in handy when the outside environment would tend to “bleed away” the air inside; such as would be found on Mars or the Moon. This system however is most creative when used to cover the foundation or pad left behind from a structure that was removed by wind, fire, or water. Than edge of the pad or foundation would than act as the plot-frame 2, and the base 3, with constraints 4 in place at the required time during the final display of the membrane 1. This would allow one to at least create a solid temporary cover in times of need. There are a number of means to fabricate both the primary and secondary geodesic surfaces found with this system. This artwork is not intended to cover such assembly of the structure as there is already many means available.

Claims

I. I claim a least surface geodesic shaped structure that matches up to any ground-plot and has the required polygonal functions such as connection-planes, entryways, view-ports and other flat or curved predetermined surfaces that could be demanded upon its surface by,

fixing the shape of any plot of ground that the structure is required to cover to an open window within a plot-frame (2);

expanding the membrane (1) from the open window within the plot-frame (2) with a even pressure force between the plot-frame and the base (3);

while creating functions on the membrane (1) as polygonal surfaces whereby, constraints (4) hold the membrane (1) to points/lines/planes by fixing the surface of the membrane (1) to the places that it needs to be for the polygonal shapes that it is required to have, so that the membrane (1) shows the least surface above the plot-frame (2) with the proper shapes for the functions that the structure is required to have.