Platonic Solids

[DrJB]

Platonic solids are solids whose faces consist of triangles/squares/pentagons only, with no mixing of shapes. It turns out there are only 5 such solids and they do find applications in many areas of engineering/biology/art. Many (before me) have built lego renditions of such solids, and I thought I'd give this my own 'spin'. I was 'lamenting' in a different thread as to the lack of suitable/colorful connectors and as such, I decided to 'decorate' my solids with some trans-orange and trans-blue parts.

The 5 Platonic Solids are:

1. Tetrahedron - also called triangular pyramid

2. Octahedron - made up of 8 triangular faces

3. Cube - Hexahedron

4. Dodecahedron - 12 pentagons as faces

5. Icosahedron - 20 faces (trangles)

In addition to these, there is the well-known bucky ball (soccer/football), or so-called 'truncated icosahedron'.

In fact, the tetrahedron octahedron and cube are mutuals of one another, as are the dodecahedron and icosahedron. You can get one inscribed inside the other, and the vertices of one meet the centers of the faces of the other.

Lastly, and unlike the beautiful renders in THIS thread ...,

... I opted for actually building the solids. The reason is rather simple: Most of these solids are not possible in LDD as some of the parts get 'bent' a bit.

In the photos below, and in addition to the common connectors, I use the following parts:

 and  

The second part came with many Bionicle sets and you can still find it on bricklink

 

And now, for the photos: All Solids together on the kitchen floor - to show 'relative' sizes

Bucky Ball - Truncated Icosahedron

Two Cubes (with bent edges)

Icosahedron (truncated)

Tetrahedron - this one can be built in LDD

Dodecahedron - My favorite (maybe because of the 20 orange globes)

Edited April 11, 2017 by DrJB

[Leonardo da Bricki]

This is really cool! My favorites are the bucky-ball and the dodecahedron. Because of the 20 orange globes.  

[DrJB]

Thank you. I bought those orange globes a while black. I thought they'll make great street lamps ... first time I actually get to use them. 

Also, here is another rendition of the cube, this time an LDD screen-shot.

 

Edited April 10, 2017 by DrJB

[Splat]

Nice work.  I don't think you need the extra 'decoration' though, it just takes away from the purity of the platonic solids.  Well, that's my opinion anyway.

This is a photo of a dodecahedron that I made recently using Mixel ball joint pieces:

https://www.flickr.com/photos/slfroden/33578702861

Yes, I know, this has 'decoration' on it too.  These were the pieces that were available on the Pick-A-Brick wall at my local Lego Store - they didn't have any plain 1x2 plates, otherwise I would have used those.

[aeh5040]

Wow, those are beautiful!  I especially like the truncated icosahedron - I never thought I would be interested in that piece!

Something I found recently that might be useful to you: the dihedral angle of a tetrahedron, arcos(1/3) = 70.5 deg, appears as an angle of certain integer triangles, which can be made with liftarms.  A 2-3-3 triangle is the smallest.

[DrJB]

Thanks. There are many lego renditions of these solids on the net. I did them in Technic, but other people have done them with system parts (plates and hinges) and others with bionicle/hero parts.

In fact, at all the Brickworld events I've attended, there is always one person displaying many geometrical designs. Last year in Chicago, one person had displayed many 'Fractals-based' designs.

[Didumos69]

As some of you may know I like to build 'in system'. Here's my go at the tetrahedron:

[agrof]

I love this topic too, but this time I quit in time, I must work too... 

[Didumos69]

On 9-4-2017 at 8:47 PM, DrJB said:

In fact, the tetrahedron and cube are mutuals of one another, as are the dodecahedron and icosahedron. You can get one inscribed inside the other, and the vertices of one meet the centers of the faces of the other.

It is in fact the cube and the octahedron that are mutuals of one another. In this respect the tetrahedron is a mutual of itself.

[DrJB]

4 hours ago, Didumos69 said:

It is in fact the cube and the octahedron that are mutuals of one another. In this respect the tetrahedron is a mutual of itself.

You're absolutely correct. My typing and thinking were not in-sync that day :)

The tetrahedron has 6 vertices, and these correspond to the faces' centers of a cube.

Vice versa, the cube has 8 vertices, and those fit inside an octahedron, meeting the centers of the triangular faces.

One additional fact, the tetrahedron and cube can be 'filled' with smaller replicas of themselves without any gap. This is obvious or the cube, but a bit more of a mental challenge for the tetrahedron. This is not true however for the dodeca/icosa-hedrons. For the octahedron, I'm not sure ... 

Edited April 11, 2017 by DrJB

[Didumos69]

8 minutes ago, DrJB said:

You're absolutely correct. My typing and thinking were not in-sync that day :)

The tetrahedron has 6 vertices, and these correspond to the faces' centers of a cube.

Vice versa, the cube has 8 vertices, and those fit inside an octahedron, meeting the centers of the triangular faces.

There is a relation between the tetrahedron and the cube though: You can build a tetrahedron from six diagonals of a cube, which is where my tetrahedron above is based on:

[DrJB]

Neat. I wonder if one can build two 'intersecting' tetrahedrons (with lego) i.e., one as you showed, the other using the 'other' diagonals... Will try tonight at home. 

Edited April 11, 2017 by DrJB

[Didumos69]

16 minutes ago, DrJB said:

the tetrahedron and cube can be 'filled' with smaller replicas of themselves without any gap. This is obvious or the cube, but a bit more of a mental challenge for the tetrahedron.

I would be curious to know how. You can put smaller tetrahedons - with half-length edges - in the corners of the containing one, but the remaining space is an octahedron.

[aeh5040]

3 hours ago, Didumos69 said:

It is in fact the cube and the octahedron that are mutuals of one another. In this respect the tetrahedron is a mutual of itself.

That's correct, except that the word is "dual". http://en.wikipedia.org/wiki/Dual_polyhedron

2 hours ago, DrJB said:

Neat. I wonder if one can build two 'intersecting' tetrahedrons (with lego) i.e., one as you showed, the other using the 'other' diagonals... Will try tonight at home. 

Of course one can!  Just use #2 connectors for the two intersecting edges, since they intersect at right angles.

2 hours ago, Didumos69 said:

I would be curious to know how. You can put smaller tetrahedons - with half-length edges - in the corners of the containing one, but the remaining space is an octahedron.

Didumos is right.  You can see that it's impossible to tile a regular tetrahedron with regular tetrahedra by the fact that the dihedral angle, arcos(-1/3)=109.47 deg, is not a multiple of 360 deg.  On the other hand, tetrahedra and octahedra together will even tile space.

I used essentially the same construction (but a slightly more compact version) for the vertices of the tetrahedra in the compound of five tetrahedra.  The five tetrahedra are not connected but weave through each other, and their vertices lie on the vertices of a regular dodecahedron.  It was extremely difficult to assemble this (one of the hardest things I've ever made, in fact).

 

Edited April 11, 2017 by aeh5040

[aeh5040]

In addition to the Platonic solids there are 13 Archimedean solids: http://en.wikipedia.org/wiki/Archimedean_solid.

Can someone make a snub cube?

[Didumos69]

9 hours ago, aeh5040 said:

You can see that it's impossible to tile a regular tetrahedron with regular tetrahedra by the fact that the dihedral angle, arcos(-1/3)=109.47 deg, is not a multiple of 360 deg.  On the other hand, tetrahedra and octahedra together will even tile space.

I was already thinking that if the tetrahedron could be filled completely with smaller ones, all with same size, it would mean that one should be able to fill space with tetrahedrons alone, as multiple instances of the tetrahedron filled with smaller ones would fill a bigger one and so on, and I was quite sure that would not be possible. Filling space with tetrahedrons and octahedrons would be a nice starting point for a 3D pattern with 2 different molecules in the chemistry lab .

Edited April 12, 2017 by Didumos69

[DrJB]

@ Didumos69 and aeh5040 ... You're both right. I was playing with 'magformer' shapes few days ago and thought I could do few tetrahedrons in a larger one. I was wrong.

Now, I like the idea of extending this beyond Plato's shapes, but that would require investing in many more connectors than I have. The real difficulty here is that many of the shapes require angles that are 'not' available within the standard lego connectors and as such, any attempt of virtual modeling might not work.

I have quite a few of the triangular shaped connectors, and will get to exploring more shapes during my evenings. One good starting point though would be the magformers set from Amazon ... and the sample shapes below. One word of caution though. The magnets work fine for small structures. Large structures tend to be too heavy for the magnets and collapse easily.

      

I just realized that all above shapes require 4 edges per vertex ... Not sure this is doable in Lego. Might need to restrict to three edges/faces per vertex.

20 minutes ago, Didumos69 said:

 Filling space with tetrahedrons and octahedrons would be a nice starting point for a 3D pattern with 2 different molecules in the chemistry lab .

Yes, very good point. The only 'connection/vertex' that is easy to do with any shape is the one below. It requires two thin wheels, 3 connectors ... and quickly gets real busy.  Also, and because of gaps in the parts, the resulting construction is 'not' solid.

 

Edited April 11, 2017 by DrJB

[aeh5040]

17 minutes ago, DrJB said:

I just realized that all above shapes require 4 edges per vertex ... Not sure this is doable in Lego. Might need to restrict to three edges/faces per vertex.

It depends in part what you are willing to count as a vertex.  Your "Icosahedron (truncated)" below can already be viewed as a distorted small rhombicosidodecahedron (which is the solid pictured above).

On 4/9/2017 at 11:47 AM, DrJB said:

Icosahedron (truncated)

However, the snub cube (and snub dodecahedron) are genuinely different.  That's why I particularly mentioned that as a challenge...

[ColletArrow]

Impressive but more and more mind-boggling the further I read, especially late at night...

[DrJB]

6 hours ago, aeh5040 said:

Of course one can!  Just use #2 connectors for the two intersecting edges, since they intersect at right angles.

Here it is ... Two 'intersecting' Tetrahedra.

 

and an LXF file for those willing to experiment further. 

tetrahedron.lxf

 

4 hours ago, aeh5040 said:

It depends in part what you are willing to count as a vertex.  Your "Icosahedron (truncated)" below can already be viewed as a distorted small rhombicosidodecahedron (which is the solid pictured above).

...

However, the snub cube (and snub dodecahedron) are genuinely different.  That's why I particularly mentioned that as a challenge...

You're right ... this calls for a clear mind, not for a late night activity. :)

Edited April 12, 2017 by DrJB

[Didumos69]

2 hours ago, DrJB said:

Here it is ... Two 'intersecting' Tetrahedra

Nice! The same should work for your real life tetrahedron.

[aeh5040]

Here is a quick rhomicuboctahedron:

[aeh5040]

If I've got it right, this beauty from the "Chemistry" thread is in fact an instance of the tetrahedron-octahedron tiling of space.  Replace the 2L axles with something much longer (e.g. 32L) to see the tetrahedra and octahedra...

[DrJB]

Very nice ... You mean like this?

This is a periodic structure, and might as well reside in the Chemistry thread.

I do see the octahedra, not sure about the tetrahedra though... unless I need to do 3D tiling (above figure has only 1D tiling).

Edited April 12, 2017 by DrJB

[Didumos69]

9 hours ago, aeh5040 said:

If I've got it right, this beauty from the "Chemistry" thread is in fact an instance of the tetrahedron-octahedron tiling of space.  Replace the 2L axles with something much longer (e.g. 32L) to see the tetrahedra and octahedra...

Thanks, I appreciate your kind reference. In fact it was the (space filling) truncated octahedron honeycomb that served as a primitive for this.

Only the square faces are rotated 45 degrees. The gaps between the primitives have the same shape as the primitive itself.

But you're right. When you untruncate the primitives, they become octahedra and the gaps between the primitives become tetrahedra.

However to obtain this, as opposed to the attempt by @DrJB, you still need to align the octahedra as balls would be aligned in a (square) pyramid of balls:

 

Edited April 12, 2017 by Didumos69