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USpapergames
USpapergames
Joined: Jun 23, 2020
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November 30th, 2020 at 10:27:41 PM permalink
Hi & Welcome Back Everyone!

Got a question here & it's related to my newest invention, which I'm re-entering the toy industry! I'm curious how many other mathematicians can come up with the same solutions I have.

Question: Find the optimal polyhedron for a wargaming tile. What are it's face diagonal lengths?

*Notes*

*The tile must provide the maximum number of different spaces for movement.
*The tile must also provide the maximum amount of unique angled spaces.
*Movement through the faces of the tile will be by connected vertices & not edges.
*The volume size of the tile must be < 10 inches³ with no dimensions exceeding 4 inches.
*The inscribed circle diameter on each face must be between 1.15-1.25 inches.
*Different shaped faces are allowed but all movement spaces must be symmetrical.
*The shape should tile itself without leaving any empty space (there's a word for these shapes but I'd rather not help that much). Only 1 shape should be used to tile 3d vector space.
Last edited by: USpapergames on Nov 30, 2020
Math is the only true form of knowledge
USpapergames
USpapergames
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November 30th, 2020 at 10:41:33 PM permalink
To come up with the solution I need to teach you guys a little game design knowledge so that way you will understand how to apply your math skills! For the longest time, I have been a fan of wargaming & it's bothered me that the genre has primarily had 2d game environments. There are a lot of old men that spend hundreds of dollars on these game boards with miniatures (sometimes filling up complete rooms of space) & yet many of these expensive game boards are nothing more than square tiles that provide only 3 different directions of movement. Many of today's wargame tiles claim to be 3d but are real 2d interlocking tiles with 3d artwork.

It's important to understand how the industry has evolved over the years to see why tiles are the standard. War games were invented as real war simulators that generals would use to strategize while currently in a war. Many of these game simulation boards were 2d maps (chess & it's early variations were some of the first war game simulations) any many became 3d to simulate the actual battlefield with units moving from the shorts paths between distances around the earth's curvatures (way before the math branch of geodesics was discovered). Now many of these war game simulations had grids but the generals wanted to simulate ever more realistic games for increased accuracy & eventually removed all grid lines to simulate free movement. This solution however was dramatically inferior to grid-based game boards in that movement was now based on measurements!

I've played with wargamers who haven't realized the advantages of using tile as a form of movement over measurement, so unless you're a game designer this is probably something you haven't thought about. As silly as this system sounds you still have franchises like Star Wars who came out with a war game not even 10 years ago which uses measurement movement because what I'm sharing with you still isn't well known in the industry:

1. Dispute over Distance - You're going to need a referee to confirm distances. Your opponent will argue to the death that your not within range to attack them (especially if the measurement is barely within distance). It's not even the distance that you can argue about but also where to start measuring your miniature (War game simulations used to have referees, seriously).

2. Exposing your Strategy - Every time you want to check to see if your minion is within the distance you need to check with a measurement which gives away your thought process (Again war game simulations did this by having players leave the room on their opponent's turn. Because of this hidden movement was eventually added to the complexity).

3. Turn Time - Turns take considerably longer due to every player needing to check their measurements before making a move to verify no superior opportunity cost is lost (Yet again it was common for war games to take days to complete).

Many of these hobbyists care more about the aesthetics than the gameplay but much more often the gameplay is most important, which is why grid based boards have become the industry standard. There is a need for grid systems to overcome the previous problems but static grid boards just can't provide a unique game experience compared to tile based boards. So tile is the solution because it allows players to make unique grid maps using uniform board pieces that can be arranged in various formats. Over the years hexagon tiles have replaced square tiles & have become the standard in wargaming because there are 5 different directions a character can move throughout the board instead of a square's 3 (since tactical game movement doesn't involve back and forth movement).

*As for the inscribed circle measurement, many of today's miniatures come on 1-inch diameter bases so in order for gamers to use the miniatures they love on these new tiles there needs to be an inscribed circle slightly bigger than the base size so that my patented connectors will lock on to the miniature base to the tile so that miniatures stay connected to the title form various angles of gravity.
Last edited by: USpapergames on Nov 30, 2020
Math is the only true form of knowledge
USpapergames
USpapergames
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November 30th, 2020 at 11:08:10 PM permalink
Here is the answer with almost all the details:

The best Parallelohedron to chose for a 3d board game is a Rhombic Dodecahedron. I'll let the form discuss this before I give an explanation because I believe it will be obvious once people think about it & discuss it.

Transparent, Cube Transformation, Face Rotation, Tile Rotation, Tile Formation

The measurements of the shape are:
Face Small Diagonal - 1.5 inches
Face Large Diagonal - 1.5√2 inches
Side or Edge - 0.75√3 inches
Face Incircle Diameter - .5√6 inches
Surface Area - 13.5√2 inches²
Volume - 6.75 inches³
Maximum Dimensions (laying flat) Length = 3 inches | Width & Height = 1.5√2 inches
Last edited by: USpapergames on Dec 1, 2020
Math is the only true form of knowledge
USpapergames
USpapergames
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December 1st, 2020 at 12:18:24 AM permalink
Here are some short educational reference videos to help get you guys on the right topic! 🤞

https://youtu.be/-seIA9tukDs

https://youtu.be/eHgLWPjQ_M0

https://youtu.be/bTW422CspTk
Last edited by: USpapergames on Dec 1, 2020
Math is the only true form of knowledge
USpapergames
USpapergames
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December 1st, 2020 at 10:23:52 AM permalink
I'm hoping this question wasn't too difficult. Was planning on it getting some activity in the forms by now :/
Math is the only true form of knowledge
gordonm888
gordonm888
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December 1st, 2020 at 12:44:45 PM permalink
I find your posts in this thread to be very interesting. In the past I have looked at 3d geometric shapes, but i haven't studied them closely.

When I'm an ignorant slug on the topic of a thread, my policy is to "read but don't post." I suspect that others are that way as well; and that may be why you're not getting any responses.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
USpapergames
USpapergames
Joined: Jun 23, 2020
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December 1st, 2020 at 1:09:26 PM permalink
Quote: gordonm888

I find your posts in this thread to be very interesting. In the past I have looked at 3d geometric shapes, but i haven't studied them closely.

When I'm an ignorant slug on the topic of a thread, my policy is to "read but don't post." I suspect that others are that way as well; and that may be why you're not getting any responses.



Thank you so much for your input. This question is definitely tailored for a specialist in geometry, which are there many since geometry is such a large branch of mathematics. Tho I think my skills in 3d vector space are adequate for high-level analysis, I by no means consider myself an expert in the field. But would love to find one & pick their brain for a while tho ;)
Math is the only true form of knowledge
USpapergames
USpapergames
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December 1st, 2020 at 3:13:53 PM permalink
Don't forget to include the face angles & dihedral angle like I did.
Small Face Angle = cos⁻¹ (⅓)°
Large Face Angle = 180° - cos⁻¹ (⅓)°
Dihedral Angle = 120°
Last edited by: USpapergames on Dec 1, 2020
Math is the only true form of knowledge
USpapergames
USpapergames
Joined: Jun 23, 2020
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December 2nd, 2020 at 5:44:00 PM permalink
Anyone here play war games? Favorite games? Rules? Combat mechanics? Themes?
Math is the only true form of knowledge
ThatDonGuy
ThatDonGuy 
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December 2nd, 2020 at 6:10:25 PM permalink
Quote: USpapergames

Anyone here play war games? Favorite games? Rules? Combat mechanics? Themes?


Don't get me started. I am not so much into miniatures as board wargames (GMT, the old Avalon Hill, that sort of thing).

If you are familiar with the Squad Leader/ASL series, I was both a "purple box owner" (the first edition of the game had a purple, black, and white box, before the four-color orange box was released) and a "Towson boy" (ASL was released at Origins 1985 at what is now called Towson University near Baltimore (back then, it was Towson State), and while the first module (Beyond Valor) was available, the rulebook was not; I bought a copy of the module (in June), took it home, and put it in a closet for six months until the rulebook that I had pre-ordered arrived in early December). I have also, at one time or another, owned most of the old SPI "monster games" - if anybody remembers the game "The Campaign for North Africa" from an episode of The Big Bang Theory, I owned it when it first came out in the early 1980s. BTW, it's not that complicated (Advanced Squad Leader is much more complicated than that game); it's just that it requires a lot of bookkeeping and paperwork. For example, you have to keep detailed records of all of your supplies, including having to include extra water for the Italian troops as somebody came up with the brilliant idea of providing them, in the middle of a desert where water is hard to come by, with unboiled pasta.

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