War Gaming

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.

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.

Here is the answer with almost all the details:

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

I'm hoping this question wasn't too difficult. Was planning on it getting some activity in the forms by now :/

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.

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 ;)

Don't forget to include the face angles & dihedral angle like I did.

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

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.

Wow, you are definitely a competitive gamer if you play advanced squad leader. Games like that are way before my time but I have had a chance to draft a team & play a couple of skirmish matches of advanced squad leader and even though it felt like a lot to remember, I had a great time ;) Even tho it's not a tile-based game it's defiantly one of the better grid-based games with intense combat. I just wish they simplified the rules to make the game accessible for new players, that rule book is a boor :/

You have earned my respect as a gamer, thank you for your input & great pick for a conversation starter ;)

Welcome back USPG!

I'm afraid I didn't understand your math puzzle (as usual).

Quote: Wizard

Welcome back USPG!

I'm afraid I didn't understand your math puzzle (as usual).

Not a problem & thank you for letting me know. I will attempt to rephrase the question in a different context. This is good for me to know because there probably are others that don't understand the question either. The key point in the question is your looking for optimal functionality. Now there are many different aspects of 3d shape functionality, physics can define 3d shape functionality by stress or load tolerances, topology can define 3d shapes by how they are affected through continuous change. Game design is a science that uses a wide range of mathematics to create optimal game conditions, it has its own way of defining 3d shape functionality. My previous comments on this form have been trying to explain this science & I think this question can help people see the math behind game design. I will try and go into more details as to what the question is asking just in case there is still more confusion. I really do believe that multiple people should be able to get the answer right if they just gave it some deep thinking.

1st let's start with listing the criteria to determine game design functionality & why it's criteria, to begin with. This list will be in priority order from highest to lowest.

*The face(s) of the polyhedron must be a size so that an inspired circle can fit which has a diameter size between 1.15 & 1.25 inches. {This is because the standard miniature's base is a 1-inch circle in diameter. The game designer should always make compatible accessories whenever possible + the game designer wants to license this intellectual property to the people who chose to make the industry standard size}{Also the chosen shape must be a polyhedron because grid-based wargaming is the industry standard and grids require straight lines}.

*The shape must have more than 2 face angles (don't count faces below 90°) when the object is oriented flat on 1 on its faces, relative to gravity. {The entire point of using a 3d tile is to create as much of a 3d environment as possible. & to do that you need a shape that provides more than 2 "unique" face angles. It's very important to realize that faces at the bottom half of the object with not be counted since other tiles will be connected to the bottom half of most tiles + miniatures can't hang upside down.}

*Shape must have a volume < 10inches³. {This is just because it's not practical to ship a 50 tile set if the tiles are 20 inches³. A polygon with 120 sides might seem ideal because of all the unique face angles it provides but if every face has an inscribed circle of 1.15 inches that 120 sided shape because extremely impractical.}

*Shape should provide the maximum number of moveable spaces. {The more spaces a shape can provide for miniatures to travel on the more complex game strategies can flourish. Again a perfect example of this is how the hexagon tile replaced the square tile because of the more complex strategies that it provided as a grid shape. But just remember that many polyhedra come with multiple different faces but because we are attempting to develop a new 3d grid-based game, all grid-based games have consistent movement space design as to always represent equal distance in movement. So if your polyhedron has multiple different faces you need to specify which will be the space that miniatures can occupy and move to. Lastly, it's important to reimagine movement since most are used to game movement on a 2d plane. When someone asks you to move a piece 1 space, how do you define that? Surely we would define the nearest space if any edge of that space touched an edge of the space that the piece occupies. But what if just the vertices touched? In chess, the king can move 1 diagonal space which I would argue is a 1 space move instead of a 2 space move. Movement by vertices touching alone should be counted in this analysis.

*The shape should tile itself. {This does not mean it needs to fill in 100% of empty space when tiling, but the shape must not have any uneven angles which would cause it to not repeat it's orientation in a stackable continuous pattern.}

I'll post an example & details tomorrow.

Let's start with a good example of a bad shape to choose for war game functionality.

Let's imagine we choose a cube for our 3d game tile. Now let's use dice since the faces are numbered so that there is no confusion about the analysis of the cube being a 3d game tile.

Now let's compare the dice (cube) to the list of criteria that the question has established for functionality. But 1st let's establish the face numbers of the cube so that orientation can be accounted for. Let's make face #1 the cube's top which is facing up towards the sky. So #6 is now the bottom of the cube facing the ground. Let's make #2 on the front, facing forward away from your perspective which makes #5 on the back, facing towards you. Lastly, let's make #3 on your left side & #4 on your right side. These will be the default orientation position for the cube.

Criteria:
1. The face(s) of the polyhedron must be a size so that an inspired circle can fit which has a diameter size between 1.15 & 1.25 inches.

2. The shape must have more than 2 face angles when the object is oriented flat on 1 on its faces, relative to gravity.

3. Shape must have a volume < 10inches³.

4. Shape should provide the maximum number of moveable spaces.

5. The shape should tile itself.

Cube's criteria analysis:

1. & 3. Criteria 1 only makes sense when combined with criteria 3. I separated these criteria because if the shape chosen already has a difficult time creating an inscribed circled that fills up most of its face area then the shape should immediately be abandoned. However, even if the shapes faces can create inscribed circles with a large enough surface area ratio, the shape might still not meet it's criteria because the amount of faces on the shape extends the volume size of the shape > 10 inches³.

The cube passes criteria 1 & 3 because the cube can have an inscribed circle of 1.15 inches diameter & have a volume of 1.52 inches³

2. This criteria is probably very confusing to understand & it doesn't help that I'm language challenge but I think this example will help everyone understand the criteria. When judging these criteria we must have the cube oriented in it's default position. Other shapes must be judged with 1 of its faces laying flat against the ground surface since there are many ways to orient 3d objects. The reason for requiring the shape to be on lying on a face is because, in order for this shape to be used as a game tile, the tile itself must lay flat on a table. It defeats the point of using a tile-based board if a ground platform is required. For the cube example, imagine that instead of the cube laying flat, there was a triangle pyramid base that you oriented the cube sideways so that the cube lied flat on its vertices with 3 of its faces touching the baseboard. Yes, you could do that but again it defeats the purpose of having a free-forming board game since now you have to have your entire game board fit inside a predetermined ground surface size.

That being said let's look at these criteria from the cubes default position. What we see is only 2 different types of face angles with respect to gravity. The 1st face angle is the one we are all familiar with and that is face #1 which is 90° from gravity's direction. The 2nd face angle is #2, 3, 4 & 5 which are at 0° degrees from cavities force. Notice how we don't include any faces that are facing in any direction towards the ground.

The cube fails to pass this criterion since it only has 2 distinct face angles at are usable. Again the goal is to create a rich 3d environment and the cube only has 2 face angles to provide 3d gameplay.

4. Movement can be defined by the total number of possible faces that touch vertices to the center face minus 1. So for the cube, its movement value is 3 which is extremely weak for a 3d shape. We start at the center face #1 (or any face closest to the center) & count all the faces which connect by vertices (#2, 3, 4, & 5) then minus 1 (we minus 1 because, in order for a player to move a miniature into the center face #1, the miniature must 1st have moved from a vertex connected space or from a tile face that connects to a vertices space of #1. Therefore the effective movement must always subtract 1 when doing this calculation.

The cube definitely fails at providing a large amount of movement in comparison to most other polyhedrons. It's important to note that all 2d times are going to have faces with 90° angles with respect to gravity, so it's only the tiles that are different than 90° that truly bring out the 3d properties.

5. Lastly the shape must tile itself so that all angles in a tile formation are uniform. A cube does this perfectly in that it tiles itself without any wasted space. A dodecahedron looks like it could tile itself but the angles at which it tiles itself are not uniform and eventually add up to a slight angle in which it can not continue to connect its faces. Now that I think about it, I probably should have put this as criteria #1 since if the shape doesn't meet this requirement it's better off being excluded from possible answers.

So in analyzing the cube for a 3d game tile we can include the cube is not the ideal candidate and definitely does not express maximum functionality as a game tile. The cube is such a poor choice as a game tile that the only benefit to its use over its 2d square counterpart is that the cube allows miniatures to be placed on a 0° angle from gravity (faces #2, 3, 4, 5). Everything else in terms of functionality is exactly identical. The cube provided the same number of movement spaces, with only 2 extra points of contact (squares 4 edges to a cubes 6 faces).

Btw a cube's "true" movement value is 11, but I don't want to get into that right now. I think the criteria I established is good enough but the real movement value is MUCH more difficult to calculate :(

The simplest way to explain it is that every 3d tile has movement that is localized (meaning movement that is still connected to a single tile) & globalized (meaning movement starts and ends on different tiles). All 2d tiles have globalized movement but 3d tiles have 2 different distinct movement types. I have asked only for the localized movement value minus 1. The formula for the true movement value of a 3d tile is

M = L + G - 1

A cube would express its movement formula as 11 = 4 + 8 -1

P.S. 8 is the highest possible globalized movements value but in game design we prefer localized movement because not every tile can be in a board position that allows for maximum globalized movement.

Here let's do another example so everyone is clear what shapes don't meet the criteria. Let's use the dodecahedron shape!
____________________________________________________________

Criteria:
1. The face(s) of the polyhedron must be a size so that an inspired circle can fit which has a diameter size between 1.15 & 1.25 inches.

2. The shape must have more than 2 face angles when the object is oriented flat on 1 on its faces, relative to gravity.

3. Shape must have a volume < 10inches³.

4. Shape should provenvironmentximum number of moveable spaces.

5. The shape should tile itself.
__________________________________________________________

1 & 3. Shape has a length of .85 inches with faces that have inscribed circles with a radius of .585 inches which give the total volume of 4.706 inches³

So the dodecahedron these criteria.

2. There are only 2 viable face angles, 1 standard 90° & 5 18° angle faces.

This is a fail because 2 is just kit enough to simulate a 3d environment.

4. Localized movement is a 4 (5-1) which is acceptable, but I think we an do better ;)

5. It doesn't tile with itself so a definite failure.

Quote: Wizard

Welcome back USPG!

I'm afraid I didn't understand your math puzzle (as usual).

Does any of the new info help in understanding the question? I went all out trying to explain everything, (it's difficult for me). Should I do some more examples of bad 3d game tile shapes?