This tutorial will show you how to create a Temcor-style dome using only a little bit of math.
Most of the info in this tutorial was gleaned from TaffGoch's reverse-engineering of the subdivision method of the old Amundsen-Scott South Pole Station, so a huge thanks to him!
A major advantage of Temcor domes is their low unique strut count - it increases arithmetically with frequency, not unlike Duncan Stuart's regular triacontahedral geodesic grid (Method 3*), but the result looks much more pleasing.
For simplicity, the frequency of the dome we're making is 14, so the chord factors can be cross-checked against TaffGoch's Temcor model.
The Inventor 2016 .ipt is included at the end of the tutorial.
I described Method 4 as Duncan Stuart's regular triacontahedral geodesic grid, but it's not. The method was actually invented by Christopher Kitrick, who, in his 1985 paper, "Geodesic Domes", described its construction. In addition, in his 1990 paper, "A Unified Approach to Class I, II & III Geodesic Domes", he outlines 8 other methods, one of them being Duncan Stuart's Method 3, the other his own "Method 4", and, surprisingly enough, a method analagous to Temcor's, which he calls "Method aa" (Step 7 shows how Temcor modified "Method aa"). In a future instructable, I will describe the construction of the methods outlined in the latter paper.
Step 1: User Parameters
Before we start on constructing the dome, enter the parameters shown:
Phi — The Golden Ratio. Defined as ((1+√5/)2
Circumsphere — This is the circumsphere of a dodecahedron, defined as ((Phi*√3)/2)
PatternAngle — This is the central angle of a dodecahedron. Since the frequency of our dome is 14, we divide this central angle by half the frequency, in this case, 7.
Step 2: Sketching a Golden Rectangle
Start a sketch on the YZ plane, then create a three point rectangle as shown, referring to the image notes for additional information describing the creation of a Golden Rectangle.
Step 3: Creating a Golden² Rectangle
Create a work plane using the X axis and the line highlighted in the first image, then start another sketch on this work plane. Construct a centerpoint rectangle starting from the origin, then dimension the rectangle as shown in the third image.
Step 4: Creating the 2v Triacon Triangle
Now that we have all the geometry we need, form the boundary patch in the second image using whatever method you prefer. I chose to do a 3D sketch, but sketching on another work plane would work just as well.
Step 5: Creating the Intersection Planes
Start another sketch on the first work plane ("Work Plane 1") you created, project the corners of the Golden² Rectangle, then connect these points and the origin to form the central angle of the 2v triacontahedron. Divide it by half the frequency of the dome, as if you were starting a Method 2 breakdown. Place points on the midpoints of the chords.
Exit the sketch, then create a plane using one of the chords and its midpoint, as shown in the second image. Then, create another work plane using "Angle to Plane around Edge". Select Work Plane 1 and one of the construction lines shown in the middle right and lower left image. Accept the default angle of 90 degrees, otherwise the rest of the subdivision wouldn't look right. Repeat the process using the rest of the chords and construction lines to obtain the result in the lower right image.
Step 6: Creating the Intersection Curves and Forming the Subdivision
Start a 3D sketch, then create intersection curves using the work planes you just created and the boundary patch, forming the lines shown in the top image.
Draw lines coincident to the endpoints of the intersection curves as shown in Image 2. Make them all equal to the radius of the dome. Draw the chords joining the lines that lie on the intersection curves. Connect any geometry that looks close enough to form a triangle of the subdivision. Refer to the next 10 images for which chords to mirror across the intersection work planes - they can explain it better than mere words can.
Step 7: Completing the Dome
Create a Thicken/Offset of the bottom rows, omitting the last two rows of triangles. Pattern the new OffsetSrf 6 times, or ((Frequency=14)/2)-1. Hide the OffsetSrf, stitch the patterned surfaces, then mirror the stitched surface with the YZ Plane. Create work planes resting on the vertices of the top triangle, as shown in Image 6. Trim the stitched and mirrored surfaces using these new work planes, then stitch the remaining surfaces together. Pattern this last surface across the Z axis, then stitch these final surfaces together, and you're done!
Step 8: Checking Chords
So, our dome is finished, but let's see if the numbers match with TaffGoch's model:
Going by the reference parameters, it looks like they are a perfect match!
Dividing the chord lengths by 1000, we can clearly see a perfect correspondence with the chord factors of TaffGoch's model, as well as the footprint radius and apex factors.