I've been doing a bit of research about the Platonic solids for a forthcoming Masonic lecture ... and discovered something that I did not know.
A Platonic solid is a regular, convex polyhedron that can be constructed from congruent, regular polygons. Furthermore, Platonic solids can be tessellated with other solids of the same type and size with no void space between them.
There are five solids that meet the criteria shown above and are Platonic solids. These are:
These are, of course, the shapes of the D4, D6, D8, D12, and D20 dice ... and before I began my research, I had not realised that they were known as Platonic solids.
A Platonic solid is a regular, convex polyhedron that can be constructed from congruent, regular polygons. Furthermore, Platonic solids can be tessellated with other solids of the same type and size with no void space between them.
There are five solids that meet the criteria shown above and are Platonic solids. These are:
- The tetrahedron
- The cube
- The octahedron
- The dodecahedron
- The icosahedron
These are, of course, the shapes of the D4, D6, D8, D12, and D20 dice ... and before I began my research, I had not realised that they were known as Platonic solids.
... and a subsequent lecture about the Archimedean solids? :-)
ReplyDeleteBTW, the statement that "Platonic solids can be tessellated with other solids of the same type and size with no void space between them" seems strange to me, and I'm not sure this is true. Do you have a source?
Phil Dutre,
DeleteThe Platonic solids feature in the ritual of Holy Royal Arch Freemasonry, hence the research. It is from there and an online mathematics website that I gleaned the information about the shapes tessellating in such a manner that there are no voids between them.
All the best,
Bob
If you have enough dice, it's testable at least.
DeleteKaptain Kobold,
DeleteI hadn't thought to try that!
All the best,
Bob
I knew them as Platonic solids, and once, long ago, even had to make some out of light cardboard once, as a classroom resource. But I didn't know that they could form 3D tessellations, though. It is certainly true of hexahedrons (cubes), and whether or not it is so of the other Platonic solids should be easy enough to prove, demonstrate or refute!
ReplyDeleteHaving a set of 'Shogun', I have 6 dodecahedral dice. From these I arranged a kind of tetrahedral structure that seemed to fit together very well. Indications are, then, that dodecahedrons are tessellatable in 3D.
DeleteNot conclusive, but fairly persuasive.
Indications are, though, that octahedrons are not tessallatable in 3D. This could happen only if the internal face angle was 2π/3 also. But that is not possible - in fact the internal face angles are 2.arccos(0.577), and angle rather narrower than 2π/3 (2.arccos(0.5)).
Archduke Piccolo,
DeleteI suspect that finding an answer as to whether or not a particular Platonic solid will tessellate with another of the same type and size is going to take some time ... and a big box of dice!
All the best,
Bob
Archduke Piccolo,
DeleteIt sounds as if Phil Dutre is way ahead of me, and has shown that they don't. Oh well, I must have misunderstood the sources that I used ... but it does make a interesting side note for my lecture.
All the best,
Bob
Had a few of these in my hands tonight
ReplyDeleteAlong with the cheeky newcomer the d10
Do the rules makers of Under a Lilly Banner in The League of Augsberg have any Masonic connections as they love moving between the Platonic Solids to simulate wargaming factors (instead of adding +1 like other rules the keep the same result but give you more chances to get it by "going up a solid")
Ex need a "5 to hit" but I am advantaged, instead of using a basic d8 I get to use a d10 instead
GZG were using that mechanic in their sci-fi ground combat rules over 20 years ago - modifiers changed the dice you rolled rather than the number.
DeleteChanging die type instead of adding a modifier to the number rolled is one of my favourite mechanics. See http://wargaming-mechanics.blogspot.com/2017/02/opposed-die-rolls.html for some analysis.
DeleteGeordie and Exiled FoG,
DeleteI don't think that there is any connection between the writers of UNDER THE LILY BANNER and the Masons; I suspect that the use of the Platonic solid dice is just a mechanism that meets a particular need.
All the best,
Bob
Kaptain Kobold,
DeleteI must admit to having seen something similar used in a game at a COW many years ago. I don't think that it is that a unique mechanism used by game designers.
All the best,
Bob
Phil Dutre,
DeleteI've tried using different Platonic solid dice in a similar manner in the past, and found that in the heat of a solo game I kept picking up and using the wrong dice, so it has become a mechanism that I tend to avoid if at all possible.
All the best,
Bob
That's a fair point. That's why I have invested in dice al of the same colour, e.g. all my D4's are blue, D8's are yellow etc. Then the colour is an additional clue for picking up the right die.
DeletePhil Dutre,
DeleteThat is what I should have done ... but I bought mine in sets that were all the same colour. I will give some thought to buying some sets of multi-coloured dice when I next visit a wargame show.
All the best,
Bob
I ordered mine online, in exactly the colours and quantities I wanted for each dice type. E.g. from https://em4miniatures.com/
DeletePhil Dutre,
DeleteThanks for the link. I've usually bought my dice in pre-packed sets from games shops so buying them online will be a new experience for me.
All the best,
Bob
As I expected, the platonic solids do not tesselate 3D space. The reason is that the dihedral angles (angles between 2 adjoining faces) are not divisions of 360 degrees, except for the trivial example of the cube. See e.g. https://www.youtube.com/watch?v=ARicbryMMMs&t=207s for a nice explanation.
ReplyDeleteHowever, some combinations of different types of platonic solids can fill the space, e.g. tetrahedra and octahedra, see e.g. https://www.princeton.edu/news/2011/06/27/princeton-researchers-solve-problem-filling-space-without-cubes
See also http://mathworld.wolfram.com/Space-FillingPolyhedron.html for some mathematical formulations.
Phil Dutre,
DeleteHaving read your references, I can see how I may have misunderstood the ways in which the Platonic solids tessellate. I had understood it to mean that they could only tessellate with solids of the same type, but that is now patently wrong.
In my defence, all I can say is that when I studied A Level Mathematics (back in the late 1960s), tessellation of Platonic solids wasn't part of the curriculum.
All the best,
Bob
That's why we have blogs ... to help each other out with our own favourite obscure corners of knowledge :-)
DeletePhil Dutre,
DeleteVery true ... and thanks for your very helpful input.
All the best,
Bob
I feel I hear the footsteps of the ghost of Mr Kepler walking with us ;)
ReplyDeleteGeordie an Exile FoG,
DeleteMethinks you are referring to his Platonic solid model of the universe ... which might just be one step too far for my audience of (probably elderly) Freemasons!
All the best,
Bob
;)
DeleteGeordie an Exiled FoG,
DeleteI mentioned it in passing in my lecture ... and several of the attendees had heard of it!
All the best,
Bob
Kepler and the Platonic solids was also nicely visualized in the 80s documentary series Cosmos. See here, around minute 32: https://www.youtube.com/watch?v=pDYMF1RGthQ
DeletePhil Dutre,
DeleteThanks for the link. I'll try to watch it later tonight.
All the best,
Bob
It is nice when the audience surprises you ;)
ReplyDeleteGeordie an Exiled FoG,
DeleteI learned a lot as a result of this blog entry.
It's one of the real benefits of the Internet.
All the best,
Bob