Dodecahedron


Dodecahedron
#1
After seeing this in LEGO House: https://photos.app.goo.gl/bAAXaEebvrTMJfZZ6... I feeled urged to create a LDraw model!

I was too lazy to create a flexed axle template for LDCad so I used 12L rigid tubing instead, but it does the job!


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.mpd   dodecaedron.mpd (Size: 179.91 KB / Downloads: 4)
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RE: Dodecaedron
#2
(2019-06-04, 18:03)Philippe Hurbain Wrote: After seeing this in LEGO House: https://photos.app.goo.gl/bAAXaEebvrTMJfZZ6... I feeled urged to create a LDraw model!

I was too lazy to create a flexed axle template for LDCad so I used 12L rigid tubing instead, but it does the job!

Very cool, how did you get the positions of those axle joiners?
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RE: Dodecaedron
#3
awesome.
I also wonder like Roland how you found the positions of the connectors in advance
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RE: Dodecahedron
#4
(2019-06-04, 22:29)Steffen Wrote: awesome.
I also wonder like Roland how you found the positions of the connectors in advance
Problem is not so much to get connector positions (dodecahedron vertices coordinates are known: https://en.wikipedia.org/wiki/Regular_do...oordinates), but to have them in the right orientation.
Here is the method I used:
- Get the distance between connectors. I used a visual method for that, placing two connectors with roughly the right orientation, then changing the distance until I got the desired length (96mm = 12L) for the flex part I put between them. Turns out the distance is 252 LDU.
- Calculate the radius r of the circumscribed sphere[url=https://en.wikipedia.org/wiki/Circumscribed_sphere][/url] (https://en.wikipedia.org/wiki/Regular_do...Dimensions): 353.117LDU.
- Because of symmetry, a connector must be tangent to this sphere. So I placed a connector at y=-353.117, above the middle of top face.
- Then calculate the angle I must turn that connector around origin to place it on one top pentagon vertex. We get alpha = arcsin(r1/r)=37.377° (r1 is radius of circumscribed circle of pentagon with side a = a/2/sin(36°) )
   

After that, it's only a matter of duplication, rotation by 72° around world Y axis and rotation by 120° around of Y local axis of already placed connectors...
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RE: Dodecahedron
#5
(2019-06-05, 7:01)Philippe Hurbain Wrote: Problem is not so much to get connector positions (dodecahedron vertices coordinates are known: https://en.wikipedia.org/wiki/Regular_do...oordinates), but to have them in the right orientation.
Here is the method I used:
- Get the distance between connectors. I used a visual method for that, placing two connectors with roughly the right orientation, then changing the distance until I got the desired length (96mm = 12L) for the flex part I put between them. Turns out the distance is 252 LDU.
- Calculate the radius r of the circumscribed sphere[url=https://en.wikipedia.org/wiki/Circumscribed_sphere][/url] (https://en.wikipedia.org/wiki/Regular_do...Dimensions): 353.117LDU.
- Because of symmetry, a connector must be tangent to this sphere. So I placed a connector at y=-353.117, above the middle of top face.
- Then calculate the angle I must turn that connector around origin to place it on one top pentagon vertex. We get alpha = arcsin(r1/r)=37.377° (r1 is radius of circumscribed circle of pentagon with side a = a/2/sin(36°) )


After that, it's only a matter of duplication, rotation by 72° around world Y axis and rotation by 120° around of Y local axis of already placed connectors...

Don't we all just LOVE maths?  Big Grin
Jaco van der Molen
lpub.binarybricks.nl
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RE: Dodecahedron
#7
I love this.

Lego should make a set with the 5 platonic solids
https://en.wikipedia.org/wiki/Platonic_solid
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RE: Dodecahedron
#8
Witchcraft. I love it.

If I remember correct, Isn't there a way to inscribe all the Platonic solids into one another?
There is a cube inside the dodecahedron, and there's a octahedron inside the cube, and so on...
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RE: Dodecahedron
#9
(2019-06-06, 21:24)Magnus Forsberg Wrote: Witchcraft. I love it.

If I remember correct, Isn't there a way to inscribe all the Platonic solids into one another?
There is a cube inside the dodecahedron, and there's a octahedron inside the cube, and so on...

Yah. Stellate the corner point of the faces out to a center point and then connect those points. 
Pyramid -> Pyramid
Cube -> Octohedron
Dodecahedron -> Icosahedron.
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RE: Dodecahedron
#10
(2019-06-06, 21:41)Orion Pobursky Wrote: Yah. Stellate the corner point of the faces out to a center point and then connect those points.
I think Magnus rather meant something like this:
https://www.youtube.com/watch?v=gwxQfujwWrw
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RE: Dodecahedron
#11
Oh, yes.

Beautiful visualisation. I have to make something like that one day.
Thank you for finding it.
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RE: Dodecahedron
#12
yes.
beautiful.
love it also.
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Truncated icosahedron
#6
Using a similar method (a bit complicated because all edges are not equivalent) I also made a truncated icosahedron (soccer balls, geodesic domes, fullerenes, etc...). Enjoy!
   


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.mpd   truncatedicosahedron.mpd (Size: 323.6 KB / Downloads: 1)
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