3V icoahedron geodesic sphere

[Drewpan]

Hi All,

There are two other forum posts sort of related to this post, but they are not quite what I need, so here goes.

I am designing a Sculpture that incorporates a geodesic sphere. Geodesic spheres are created by projecting either a tetra-hedron, octa-hedron or iscosa-hedron onto a sphere. These are the only regular polygon shapes that are constructed of regular triangles only. The more triangles involved in the projection, the more spherical the ultimate figure. 3V refers to the original triangle shapes being split into three triangles on each side. There are tools in AutoCAD that will do this, but the problem is that they end up not being quite accurate enough and give many different strut lengths. A 3V sphere should be able to be constructed of only 3 different strut lengths that consists of 12 pentagonal shapes and 80 hexagonal shapes. In the Sculpture this will consist of the 92 machined connectors and the 270 A,B and C struts. The great thing about the design is that it can be scaled naturally to ANY size by multiplying the Strut ratios by a Constant. Simply - if you want a bigger sphere then multiply the strut ratios by the same constant, as long as the struts still fit the connectors then it will work.

The connectors are fairly straight forward to design and are identical, the problem is drawing the finished dome accurately so that it can also be scaled. I know all of the angles, but manually calculating angles for each strut will be a nightmare.

Example.

For a 2m diameter sphere, the struts will be, A = 348.6 mm (60 required), B = 403.5mm (90 required) and C = 412.4 mm (120 required)

The connectors will be based on a Pentagon made of A and B struts, and Hexagons made of B and C struts. The Pentagon angles will be 54 degrees around the edges, 72 degrees in the middle with a rise from the edge to the centre of 10.135 degrees. The Hexagon will be 60 degrees around the edge and at the centre and a rise of 11.992 degrees. Assembly will look like this:

This graphic is from the Desert Domes website: http://www.desertdomes.com

I don't know how they generated this graphic, probably with a computer calculating and building the image. What I want is to do the same thing in AutoCAD without having to manually work it out. Then being able to extract the co-ordinates of each connector and list them.

My thoughts run along the lines of this:

Drawing the Hexagonal and Pentagonal shapes and locking them into a Block shouldn't be too hard, but I still need to work out the angle of the plane to paste each Block (92 of them). The angles will be constant and the edge of the Hex or Pent will be a mirror plane for the adjacent shape. The thing that bollocks it all up is the 12 Pents will be in regular positions but scattered around.

There must be some kind of script that can calculate this (all be it unwritten so far) but I cannot get my head around how to write it myself. Simply splitting the faces of an icosahedon into three triangles each side them projecting it onto a sphere will give struts of irregular length. There must be a way of fixing the length and angle then recursively (?) building the sphere.

Ideally the script would take the strut lengths as parameters so that a sphere of any size could be generated.

Any help would be appreciated but I am thinking this might be a Guru who likes a Challenge solution somehow.

Cheers

Andrew

[Drewpan]

BTW this is part of my Assignment on Managing Self in an Engineering Environment. It is not about how to draw in AutoCAD, it is about what I would do if I was working as an Engineer and had this problem - I would ask for help. 😎

Andrew

[tramber]

My first suggestion would be that you build it on Excel.

Building entities with a simple script is so easy as long as you know well how to talk to Autocad.

x,y,z

x,y<angz

dist<angxy,z

dist<angxy,angz

Spécifying spherical coordinates is not a problem at the command line or in a script.

Have you ever sent a script to the command line ?

If you don't want to use external means, you can ask for some help in the lisp forum....

[chris_lee_3]

I am not understanding the problem. If you know the strut length, then you are done. Color them the same as your diagram and connect them. The sphere will come together organically.

[tramber]

http://tutoriels.abcad.fr/programmation/script/xls-dwg-acad.php

It is in French but it explains how to build a script with Excel.

CONCATENATE(...) must be the formula in English, in Excel 😉

[Drewpan]

The problem is that I can manually calculate all of the polar coordinates but I don't want to do that 270 times.

Each node will be at 60 degrees or 54 degrees and 11.9 or 10.1 degrees and the strut length long. I can manually input the polar coordinates 270 times or I can work out a script for it. That is the problem. It will take me much time to manually calculate it and if I make a mistake I may not be able to identify it until it is far to late and will have to start again.

It is one of those problems that is not as simple as it looks.

Andrew

[leeminardi · Accepted solution]

For a geodesic v3 dome that has a radius of 1 the line lengths are as follows:

A = 0.3486

B = 0.4036

C = 0.4124

A good source on the math of geodesics can be found here:

Here's one way that with this information you can construct a geodesic model in AutoCAD.

Start with a circle of radius 1.0 center at 0,0,0 on the XZ plane (set an appropriate UCS for this construction where the UCS Y axis is coincident with the world Z axis). Create a line at an angle of 63.4356° as noted below and two circles of radius 0.3486 (the length of A).

Create the yellow (A) and cyan (B) lines as shown below.

Return to the world UCS and use array polar to make a total of five copies of the 3 lines.

u

With UCS ZA set the z axis in the direction of the radial (magenta) line and then select the 3 lines as shown.  We want to use these three line (two A's and a B) to complete one spherical triangle of the geodesic sphere.

Use arraypolar with item = 2 and angle = 72° to construct the base of the spherical triangle.

Construct the 3 additional cyan edges as snapping to the ends of the 3 existing cyan (B) lines.

We now need the center point for the cyan (B) hexagon projected to the sphere's surface.  Construct a line from the 0,0,0 to the mid point between two opposing vertices of the hexagon.  Then use lengthen to make it 1.0000 long resulting in the white line below.

Create the red (C) lines from the end of the white radial line to the vertices of the cyan hexagon.  Verify that their length is 0.4124.

From here you can use arraypolar about different radial lines to complete the geodesic dome remembering that each cell of the geometry will be repeated 5 times about pentagon center (angle 72°).

[chris_lee_3]

Cool, tried to do a tetrahedron during lunch and realized I don't remember squat about 3d.   

[Drewpan · Accepted solution]

Hi All,

Thanks for your replies.

This is one of those "simple" problems that is quite difficult underneath.

Between the two forums I have sort of worked out the problem but I am still getting my head around it.

Cheers

Andrew