quickfur wrote:Also, I've also been thinking about a polytope-based game... where the PC is a polytope that has to acquire various "powers" that correspond with the uniform truncations. I've found this rather neat interpretation of the ringings of the CD diagrams that could make the basis for a system of shape-changing for the PC, though I have yet to think of how it can be turned into an actual, fun gameplay.
Marek14 wrote:[...]My idea is basically that instead of race and class, the character would have shape. With uniform polyhedra and Johnson solids, there's quite a number of possible builds
Now, the equipment items you find are polygons. A tetrahedral character could equip up to 4 items, but he could only use triangles, nothing else. A pentagonal pyramid character could equip 5 triangles and 1 pentagon. Equipment of wrong shapes couldn't be used, but there would be a possibility to "scrap" it somewhere and get equipment of different shape in return.
Since more equip slots would make for more powerful characters, the number of slots would be related to experience, characters with many slots would advance slower.
Each piece of equipment would have some core bonuses or abilities. Triangles are sharp, so they would be "aggressive", having attack bonuses and physical skills. Squares would be defensive, while pentagons would be magical, with magic bonuses and spells. Hexagons, octagons and decagons would be similar to their /2 analogues, but stronger - on the other hand, they would be harder to obtain.
Apart from core bonuses, each piece of equipment could also have bonuses associated with its edges and vertices.[...] On the other hand, an icosahedron would have much easier time since all the triangles would make matching attack bonuses for edges or corners easy.
quickfur wrote:Marek14 wrote:[...]My idea is basically that instead of race and class, the character would have shape. With uniform polyhedra and Johnson solids, there's quite a number of possible builds
Nice idea! My idea is more symmetry-based. Characters have symmetry and truncation as parameters, so in 3D there are 3 symmetries (tetrahedral, cubical, icosahedral), and truncation is basically keyed on the CD diagram, giving you the Archimedean polyhedra. The more truncated you are the more power you get; but to get to those states you need to collect truncation attributes (corresponding with each node in the CD diagram). I arbitrarily label them red, green, and blue, since there are 3 nodes in 3D; each color corresponds with a node in the CD diagram respectively. The thing is, each node can only be ringed or unringed, so if you collect red twice, it cancels out and you get nothing. And to make things more interesting, powerups don't come in single colors; they are random and tend to come in combinations (secondary colors, eg. yellow = red + green), so if you pick up a yellow and already have a red, then you lose the red and only get the green. These bonuses are rare, so you kinda hafta make a judgment call when you find an inconvenient combination whether to take it now or to take the chance you'll find a better upgrade later (you might not).
Now symmetry is a more "intrinsic" parameter; you start out with only tetrahedral symmetry, and you need to reach a certain level before you can "graduate" to cubical symmetry, and after that you need to attain to a higher level before you can transition to icosahedral symmetry. It's sorta a similar idea to ship upgrades in the game powermanga (dunno if you've heard of it, it's an arcade shooter where you collect power bonuses and can choose to upgrade weapons or upgrade your ship; ship upgrades are more expensive but more worthwhile in the long run).
Unfortunately, even though this system is nice in theory, it's quite limited in 3D because of the limited number of uniform polyhedra. However, if you upgrade to 4D, you will have 4 symmetries and 15 truncations in each symmetry (plus a lot more inconvenient color combinations to annoy the player with), so it gets a little more interesting.
Now, the equipment items you find are polygons. A tetrahedral character could equip up to 4 items, but he could only use triangles, nothing else. A pentagonal pyramid character could equip 5 triangles and 1 pentagon. Equipment of wrong shapes couldn't be used, but there would be a possibility to "scrap" it somewhere and get equipment of different shape in return.
I like this idea!Since more equip slots would make for more powerful characters, the number of slots would be related to experience, characters with many slots would advance slower.
I like this idea too. It balances out the power you have.Each piece of equipment would have some core bonuses or abilities. Triangles are sharp, so they would be "aggressive", having attack bonuses and physical skills. Squares would be defensive, while pentagons would be magical, with magic bonuses and spells. Hexagons, octagons and decagons would be similar to their /2 analogues, but stronger - on the other hand, they would be harder to obtain.
Now this idea is truly genius. Maybe I'llstealborrow this idea of having shapes corresponding to function. So an icosahedron would be an all-out attacker (all triangles), but a dodecahedron would be an insane magic caster (all pentagons)? That's too cool! And you also have the special case of the rhombicuboctahedron, where both the axial faces and quadrant faces are squares (corresponding to both the cube's faces and edges), so you have an unusually high degree of defense slots plus a good number of attack slots. I vote to name the rhombicuboctahedron the paladohedron (for paladins - who tend to have good armor and defensive skills).Apart from core bonuses, each piece of equipment could also have bonuses associated with its edges and vertices.[...] On the other hand, an icosahedron would have much easier time since all the triangles would make matching attack bonuses for edges or corners easy.
But an icosahedron will have no slots for defensive equipment, so I guess that somewhat balances out, maybe?
Marek14 wrote:[...]
I think this might be quite appropriate for an old-school Gradius-style shooter with those little rockets that were always collecting power-ups How about if you collected the nodes and the symmetries corresponded to different levels instead?
[...]
There would be also some quests Truncated icosahedron would run quests for local soccer team, for example.
Of course, with this division, I'd basically need to have full range of Johnson solids available, since otherwise the pentagon-using characters would always have 12 of them, which is not much variety. But with johnson solid, you might have pentagonal pyramid, quick-growing character that is a good fighter, but also a basic mage. And if you wanted to keep with this triangle/pentagon build, you could also try for gyroelongated pentagonal pyramid, metabidiminished icosahedron, tridiminished icosahedron or augmented tridiminished icosahedron.
As for icosahedron, yes, it wouldn't get any defensive slots. That's not an absolute problem since there would be triangles with defensive properties (or magical properties), they would just be much less common. The main problem of icosahedron would be that he could use only a relatively small portion of items he'd find.
Quests could be divided by presence/absence of some symmetry. There could even be a parallel mirror world that you could only explore if your character is chiral, like snub cube
As a roguelike, the graphics could be basic - but one thing that could be there would be having various dungeons have different internal geometry. One could have normal square net, another could be hexagonal, another could have (3,3,3,4,4) lattice structure...
THE BOOK OF ORIGINS, CHAPTER 1
In the beginning was the Origin, and the Origin gave birth to
the Point, and the Point was singular and lonely. And the
Origin saw that it was not good for the Point to be alone; and
so it stretched forth the Point and it became a Line. Then it
blessed the Line and ordained that it go forth to replenish the
Plane, and so the Polygons were born, and the Origin saw that
it was good.
THE BOOK OF ORIGINS, CHAPTER 2
Now, the Polygons lived in the Plane, and as they grew and
multiplied, they saw that the Plane was no longer sufficient to
contain them. Therefore, the Polygons conferred amongst
themselves, and said to each other, "Come, and let us build
ourselves upon each other, and become the Polyhedra which will
reach even unto Space." And so they came together, and
joined each one unto his neighbour, and thus the Polyhedra
were born.
THE BOOK OF ORIGINS, CHAPTER 3
Now, the Origin came and looked upon its creation, and saw
that the Polygons were creating the Polyhedra, and the Origin
said, "Now they have come together to be of one mind and one
accord, and whatever they wish will not be withheld from them.
Therefore, I shall come down to them, and confound their
angles, so that they shall only form five Polyhedra, and I
shall set this as their boundary and their limit."
THE BOOK OF ORIGINS, CHAPTER 4
And so, the Origin came down to the Plane where the Polygons
were creating the Polyhedra, and turned every Polygon's angle
upon itself, so that the Larger Polygons, those that were of
more than ten vertices, could no longer form themselves into
Polyhedra. Instead, many assemblies of the Polyhedra collapsed
onto the Plane, and thus became the Great Tesselations. Others
disintegrated altogether. Of the remaining Polyhedra, the five
greatest ones, the heroes of old, became known as the Platonic
Solids; and there were thirteen chiefs under them. The thirteen
chiefs, who were known as the Archimedean Solids, ruled over
the Prisms and the Antiprisms, who remain perpetually at war to
this day.
Marek14 wrote:Well, by "internal geometry", I mean that the dungeon could, for example, have rows of square and triangular spaces.
Not like another idea of mine, a RPG that would play inside of an icositetrachoron surface, like a 3D version of Phantasy Star III
As for transformations, something could be probably done with gluing shapes together Two pentagonal pyramids and a pentagonal antiprism could join to form an icosahedron etc.
Keiji wrote:Welcome back, Marek! I hope you'll stick around
The Tiger page is rather bare, if you want to describe your favorite shape, I'll PM you an invite code for the wiki.
As for the polytopic roguelikes... That's a really nice idea!
I've gotten a few thoughts of my own for it, on reading your posts...
You could start as a 2D being in a 2D world, not only finding edges to equip and fighting enemies but also finding friendly polygons to team up with forming a party. Each party member could specialize in different things, you could have an RPS mechanic where triangles beat pentagons beat squares beat triangles.
When you have enough party members of the right shape, you can take your party to a special place, let's call it the ascension room, where the world is expanded into the next dimension and your party members pull together to form a new shape one dimension higher. So, for the first ascension room, you could have just 4 triangles and form a tetrahedron, or 6 squares, for a cube. Then you proceed as a single 3D being and go to find 3D allies until you can form a 4D polytope, and so on
Reminds me of Spore in a way (mainly the cell phase to creature phase transition).
To make things even more interesting, you could include concave shapes, such as gluing two dodecahedrons by a face (and then removing the two glued pentagons). That would give you an infinite number of possible polytopes within each dimension.
Marek14 wrote:Another weird idea - I was thinking of fair dice.
[...]
So, I think that if we truncate an octahedron, there must come a point where it will become a fair 14-sided die - where each square face will have exactly the same probability as each hexagonal face. I think that it might be the point where the area of square and hexagonal faces are the same, but I'm not sure how to prove it - I'm not even sure how you would define a probability of landing on a given face. Perhaps through a solid angle from the center of mass?
I tried similar exercise for square pyramid, and I found that a square pyramid of base 1 and height sqrt(7)/2 should be fair if the theory of face areas is correct. (The lateral sides would be isosceles triangles of base 1 and height 2).
Similarly, there should be a fair 32-sided die in shape of sufficiently truncated icosahedron, and maybe a 30- and 62- sided dice with carefully chosen parameters for great rhombicuboctahedron and great rhombicosidodecahedron. What do you think?
quickfur wrote:Marek14 wrote:Another weird idea - I was thinking of fair dice.
[...]
So, I think that if we truncate an octahedron, there must come a point where it will become a fair 14-sided die - where each square face will have exactly the same probability as each hexagonal face. I think that it might be the point where the area of square and hexagonal faces are the same, but I'm not sure how to prove it - I'm not even sure how you would define a probability of landing on a given face. Perhaps through a solid angle from the center of mass?
The problem with this is that it presumes that fairness can be measured exactly as a probability. But this may not necessarily be the case, because the different geometry around the faces of different polygonal degree alters the physics of the dice landing and stabilising on a surface in a way that may involve more than a single parameter. It may end up being a matrix value, then there is no way of equalizing the behaviour of different faces.I tried similar exercise for square pyramid, and I found that a square pyramid of base 1 and height sqrt(7)/2 should be fair if the theory of face areas is correct. (The lateral sides would be isosceles triangles of base 1 and height 2).
Similarly, there should be a fair 32-sided die in shape of sufficiently truncated icosahedron, and maybe a 30- and 62- sided dice with carefully chosen parameters for great rhombicuboctahedron and great rhombicosidodecahedron. What do you think?
A snub dodecahedron "equalized" in this way would give you 92 faces.
But again, this presumes that fairness can be equated with face area (or solid angle); but in terms of practical use, it would have to be equality of the physics of the dice landing on a particular face. Now if there are only two kinds of faces, then it may be possible, since tweaking the sizes of each type of face would in theory shift the two probabilities around. But if you have 3 types of faces, it may turn out that the probabilities shift around in a way that they will never be equal.
But I could be wrong.
Marek14 wrote:Oh, really? I'd like to see that rendering program I wonder if someone already published on the shape known as tiger... I hope the name will become official, as it certainly possess the "fearful symmetry"
quickfur wrote:What are the equations of the tiger again?
quickfur wrote:Also, I remember inventing a notation for objects that included the duocylinder and the crind. I believe keiji calls it "tapertopes".
Keiji wrote:quickfur wrote:What are the equations of the tiger again?
They're on the wiki, where they always have been: Tiger
This right after I added a prominent wiki link to the header bar? Really?
[...]
The tapertopes do not include the RSS operator, and the bracketopes (which do include the RSS operator) do not include pyramidal tapering. Use SSC2 if you want to represent a mix. The notation you describe is linear and linear notations went out of fashion years ago
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