I live!

Discussion of shapes with curves and holes in various dimensions.

I live!

Postby Marek14 » Fri Nov 11, 2011 11:28 am

Random browsing brought me to this page... wow, is it really five years since I was here? And you're still using the word "tiger"! :) Might be my most permanent legacy...

I read some of my old posts, and I could still understand them, so that's something, at least.

I apologize if this is the wrong place, but I was most active in this part of forum, I think, so I came to say hello :)
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Re: I live!

Postby Secret » Fri Nov 11, 2011 11:46 am

hi
nice to meet you

Here's your legacy
Image
(Wendy have checked it is accurate)

Current status of the higher dimension forum (From my perspective):
Quite active, with quickfur doing 4D renders as usual, mrrl and shark also have their own rendering program to render 4D objects. Keiji, Wendy and some others discussing 4D maths etc.
There are also some newbies asking questions which allow us to investigate 4D things in depth
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Re: I live!

Postby Marek14 » Fri Nov 11, 2011 6:45 pm

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" :)

I wandered here while I was trying to find whether anybody has tried to extend Johnson solids into 4 dimensions, or whether there is even a ballpark estimate of how many there would be. I got THERE from wondering about a roguelike game where a character would be a polyhedron and he would find polygonal items which he could equip on his faces.

I have little time for math these days, my life is more or less consumed by my translator work. I had a math crisis of sorts when I found that lots of my cool ideas were already done, and to find something truly new, I'd have to specialize more than I'm capable of... Ironically, tiger might really be something I found first :)
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Re: I live!

Postby quickfur » Sat Nov 12, 2011 4:55 pm

Welcome back!!! Hope you'll stay this time. :) The forums have been kinda in a lull... recently there has been more activity, but it hasn't been like the good ole days for a long time now.

And speaking of roguelikes... I've been developing this ASCII-based 4D maze game on and off over the years, and have been experimenting with a version with gravity (so you have 3D floors and can climb up 1 block at a time, and you fall into pits, etc.). The major hangup with this version is that the maze diggers often produced unreachable areas; recently I've started to implement a reachability analysis algorithm as a first crack at solving this problem. I have some ideas on how to ensure levels are solvable, but haven't gotten around to implementing it yet. I'm hoping that if all goes well, I might actually start adding more rpg-like elements to the game (to make it qualify as "roguelike", haha), add items and NPCs, etc..
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Re: I live!

Postby quickfur » Sat Nov 12, 2011 4:58 pm

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.
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Re: I live!

Postby Marek14 » Sat Nov 12, 2011 6:12 pm

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.


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. These would be synergies and to activate them, you'd have to position the equipment correctly. Equipment can be moved between faces, and arbitrarily rotated or flipped. For example, if you put two polygons together so their shared edge will have attack bonus on both sides, that bonus will become active - but if one side will have attack bonus and the other defense bonus, you will get neither. For vertex bonuses, you will need to have at least three bonuses of the same type to activate them.

This means that for example cuboctahedral character will have hard time matching edge bonuses since every edge is formed by a triangle (which tends to get attack bonuses) and a square (which tends to get defensive bonuses). You'd need to find a triangle or square with rarer combinations to match the edges. 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.
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Re: I live!

Postby quickfur » Sat Nov 12, 2011 7:17 pm

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'll steal borrow 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?
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Re: I live!

Postby Marek14 » Sat Nov 12, 2011 7:39 pm

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.


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?

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'll steal borrow 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?


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...
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Re: I live!

Postby quickfur » Sat Nov 12, 2011 8:43 pm

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?

Hey, I like that idea! It would be a flying polyhedron shooter. :) You start off as a flying tetrahedron shooting triangles at other flying polyhedra (who shoot back at you), and each time you shoot down a polyhedron you have a chance of getting either a node powerup or a symmetry powerup (rare). Collecting the same node twice nullifies it, so you have to watch what you collect. Furthermore, you can shoot polygons that you have (so a tetrahedron can only shoot triangles), and polygons can only damage polyhedra that contain that polygon (so shooting triangles at a cube doesn't cause any damage).

[...]
There would be also some quests :) Truncated icosahedron would run quests for local soccer team, for example.

lol!! I like that. And the platonic solids would have to run quests for the local RPG club as die?

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.

Hmm, you're right. Maybe I'll need to take a look at the Johnson solids myself sometime. The problem is that it's not easy to come up with a system for transitioning between them, since there's no nice symmetry that we can exploit to make a self-consistent system.

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 :)

I've tried to make a game before where the PC is a polyhedron, but doesn't know this. His movements are restricted by doors that have different shapes; the PC can't pass those doors if he's the wrong shape. Eventually, though, he will find a mirror and learn what he is. After that he has to find shape-changing machines (truncator, rectifier, dualizer, etc.) to change his shape so that he can get through those doors.

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 problem with internal geometry is that it's non-obvious. I mean, I had been playing Hunt the Wumpus for years before I had the slightest idea that it was in the geometry of a dodecahedron.
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Re: I live!

Postby Marek14 » Sat Nov 12, 2011 9:20 pm

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. :D
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Re: I live!

Postby Keiji » Sat Nov 12, 2011 9:38 pm

Welcome back, Marek! I hope you'll stick around :D

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.
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Re: I live!

Postby quickfur » Sat Nov 12, 2011 10:55 pm

The problem with freedom of assembly is that you might end up with highly self-intersecting shapes that are very hard for the player to see where the facets are, and also very hard to handle programmatically when you want to do collision detection and stuff.

As for "ascension rooms", I've thought of that before too. I like your idea of gathering other characters (perhaps some are NPCs) for assembling into higher polytopes. Another idea is to go into a hall of mirrors and get constructed via the CD diagram method. :)

As a humorous aside, I once started an Interactive Fiction piece where the PC was polyhedral, and wrote up some kooky origin story for them, which I think you'll find amusing:

Code: Select all
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.

Code: Select all
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.

Code: Select all
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."

Code: Select all
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.

It goes on, but you get the idea. :) Unfortunately that IF never got off the ground... I implemented a couple o' complicated puzzles but just could not manage to get it to the point where it was actually fun to play. So I've given up on it.
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Re: I live!

Postby quickfur » Sat Nov 12, 2011 11:19 pm

Marek14 wrote:Well, by "internal geometry", I mean that the dungeon could, for example, have rows of square and triangular spaces.

Ohhh, I see. Well, that would be cool. :) How would it tile, though, if it was, say, pentagons?

Not like another idea of mine, a RPG that would play inside of an icositetrachoron surface, like a 3D version of Phantasy Star III :)

I started writing a 4D story once, based on the idea of a team of explorers who found an ancient buried temple with strange objects that were apparently as solid as rock but spontaneously shape-shifted, and were unusually dense. Turns out they are actual 4D objects trapped in the 3D world (well, they are the cross-sections of 4D objects). Later, the team finds a projection room where they could see shadowy shape-shifting forms - which are actually projections of polychora. As they puzzled out this strange discovery, they accidentally activate something and found themselves inside a strange spacecraft (or what looked like a strange spacecraft) made of 5 pyramidal rooms interconnected in an impossible way (i.e., the surface of a 5-cell). And they could not get out. Eventually, they found a way to activate something... and found themselves in another strange spacecraft with pyramidal rooms. Except, this spacecraft is larger, and appears to have 16 rooms instead of just 5. Bwahahaha...

I outlined the story up to the 24-cell, and then started giving up because it was getting too boring: it was just room after room of identical octahedra, and I was running out of ideas for external plot elements to drive the story onwards. And just thinking of how I would write the part about the 600-cell gave me such a headache that I scrapped the idea immediately. :)

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. :D

And the lack of convex shapes could be hand-waved away as the non-convexity causing an instability which causes the constituent parts to break apart again. :)

The interconnections between shapes would be quite complicated, though. You'd have to precalculate them beforehand so that the program can deal with them symbolically. Otherwise, if you model them by floating point coordinates, you run into the risk of floating-point errors causing strange glitches later in the game.
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Re: I live!

Postby Marek14 » Sun Nov 13, 2011 12:09 am

Keiji wrote:Welcome back, Marek! I hope you'll stick around :D

The Tiger page is rather bare, if you want to describe your favorite shape, I'll PM you an invite code for the wiki. :)


Not sure if I'd call it a "favorite" shape - I don't play favorites in higher dimensions. I consider it my shape, period :D

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.


Yes, but I specifically didn't want this. The convex polytopes mean that there's upper limit -- unlimited extension would get cumbersome.

Another idea I was thinking about was to have a game like Carcasonne played on uniform planar tilings or on hyperbolic plane :)

As for the icositetrachoron idea, it was meant with each cell to be not a room, but a complete hollow world. They can be even spherical - as long as each has eight "access points" where you can switch to a different world. The icositetrachoron can be naturally decomposed into three sets of 8 octahedra such that every pair connected by face has different colors. In my idea, this was represented by having a sun in the center of each world, and the suns differed by color - there were worlds with red suns, which were cold and frozen, temperate worlds with yellow suns, and hot worlds ruled by blue suns.

My idea of races relations is not too developed yet, but basically the tetrahedra are the original race, which developed two main lines of octahedral and icosahedral symmetry. In fantasy terms, octahedral symmetry is like humans, tetrahedral is like dwarves (small and simple), while icosahedral symmetry is like elves (complex and strong in magic).

Then there are dihedral and pyramidal symmetries of prisms, antiprisms and various Johnson solids - the three main cultures consider them inferior, but they have their own surprises. Various diminished and gyrate versions of rhombicosidodecahedron are "imperfect elves", born into a world that considers them abominations. Triangular orthobicupola, elongated square gyrobicupola and pentagonal orthobirotunda are artificial homunculi, being similar to octahedral/icosahedral symmetry heroes, but subtly wrong. The strange Johnson solids from the end of the line like triangular hebesphenorotunda are otherworldly creatures with their own agenda. Heptagonal and enneagonal prisms/antiprisms are mysterious beings whose logic and morality goes completely besides the rest of the world. Their unique polygons give them strange and weird powers.

If you want, I can probably make some unique histories and powers for all the solids :)
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Re: I live!

Postby Marek14 » Sun Nov 13, 2011 8:16 am

BTW, when I was lying in bed late at night, I thought of a perfect name for this game :)

The idea of character as geometric shape was already explored by Abbott in his famous "Flatland". So, if we use polyhedra instead of polygons, the game could have a beautiful (and oxymoronic) name:

FLATLAND 3D!
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Re: I live!

Postby wendy » Sun Nov 13, 2011 9:36 am

Welcome back!

I've been wrangling electromagnetics, gravimagnetics, and a few other things, mostly for three dimensions. It's in connection with a monograph on dimensional analysis.

given that magnetics is a result of moving electric charges, then gravimagnetics is the result of moving mass. Apart from the fact that they share the same equations, there is no further connection. If you use the equity that F = GMm/R² = cQq/4 pi r², you can find a term M/Q, about 270,000,000 lbs/vb. It comes out in metric too, but SI is too assymetric to make it work properly (apart from being generally obsolete in the more fashionable parts of the galaxy).

Still, i've been looking at dipoles etc. A normal dipole, with say x Verbers at each end of a foot, gives x verber-feet, and the magnetic moment, produced by way of a current around an area, both make different dimensional versions of the same thing: a ring covering a definite (vector) content. However, we measure the linear dipole in terms of its surround (ie along the line), while the ring-dipole in terms of its arounds (vertical to the plane of the dipole).

Just have to figure out how to make the two connect: ie how to make an 'around-measure' on the dipole, or a 'surround-measure' on the curl function.
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Re: I live!

Postby Marek14 » Sun Nov 13, 2011 2:53 pm

Another weird idea - I was thinking of fair dice.

Usually, fair dice are taken to be regular and Catalan polyhedra, but I think that this is actually too restrictive. Consider this:

If I make an uniform cuboctahedral die, it stands to reason that it would land on square faces more often than on the triangular ones. Not only are the square faces closer to the centre of mass, but there is less of them, so each square face definitely has a higher chance to show up than any given triangular face.

But if we move the truncation point and shrink the squares until they become points and the shape becomes an octahedron, triangular faces will be obviously the only ones that would show up.

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?
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Re: I live!

Postby quickfur » Sun Nov 13, 2011 4:21 pm

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. :)
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Re: I live!

Postby Marek14 » Sun Nov 13, 2011 5:01 pm

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. :)


The thing is that I'm not quite sure how many parameters snub dodecahedron has. With great rhombicuboctahedron, for example, you can change three lengths, and if we keep one of them fixed, it gives us two independent variables. Wanting to equalize three kinds of faces gives us two equations, so since we have two equations and two variables, a solution will most likely exist.
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Re: I live!

Postby quickfur » Sun Nov 13, 2011 6:50 pm

The snub dodecahedron can be constructed as an alternation of the great rhombicosidodecahedron. The latter has 3 parameters, just like the great rhombicuboctahedron, so I would assume that the snub dodecahedron has 3 parameters as well. One of the parameters can be eliminated if you fix the radius, say, or fix the centroids of the pentagonal faces (since they must be vertices of an icosahedron).

But now that I think of it, I think you might be right about the probabilities... we don't really care about the complexity of the underlying physics; what we care is that when we toss the dice, it should land on any of its faces with equal probability.

As to how to derive equations for them, one consideration is the relative stability of the polyhedron when resting on faces of different types, since this will change the probability that it will/won't keep rolling when sitting on this face. A pentagonal face is rounder, so I'd expect it tends to resist rolling motion more than a triangular face, although the triangular face also has some directions in which it resists rolling more than others. I'm not sure if area adequately captures "equivalent" behaviour here.

Perhaps one way to find out how to equalize the faces is by running a physics simulation to sample a large number of initial velocities & rotations and based on that calculate an approximate probability for the different types of faces. Assuming that the probabilities shift in a linear way, we can probably adapt Newton's method or some such divide-and-conquer technique to arrive at parameters that give the most even probabilities. At least, this will give us some idea of where to look, even if it only gives crude solutions (that people will probably nitpick, esp. if they feel the dice is somehow not 100% fair).
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Re: I live!

Postby quickfur » Sun Nov 13, 2011 6:54 pm

Actually, I just realized that the snub dodecahedron will have non-regular faces if we vary the 3 parameters, which will probably shift the probabilities in the wrong way (the triangular faces will split into two different types, which can't possibly have even probabilities). But the uniform solution clearly also has uneven probabilities, because of the pentagonal faces. So I'm not sure if a fully even solution exists at all!
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Re: I live!

Postby Mrrl » Thu Nov 17, 2011 3:44 am

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" :)


One of the viewers (it's 3D section-based, not renderer) is here: http://astr73.narod.ru/4DView/4DviewV2.zip
Model file for tiger (tetragonal model with 96000 cells): http://astr73.narod.ru/4DView/Examples/tiger.zip
Instructions are in this post: viewtopic.php?p=16328#p16328

I've not checked formulae from Wiki page, just coded them. So I'm not sure that it is the object that it should be. But its cross-sections are very impressive :)
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Re: I live!

Postby Marek14 » Thu Nov 17, 2011 9:56 am

Is there an archive of the renders?

I already have some suggestions for the render... one, to put the controls in a text file and pack it with the program.

Two, to add a function (perhaps LEFT+RIGHT click) that will reset the view to the original state.
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Re: I live!

Postby Keiji » Thu Nov 17, 2011 4:54 pm

Thanks to your viewer and model, I've made an animation of cross-sections of the tiger rotating with whatever plane {shift+left button+horizontal mouse axis} is:

Image
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Re: I live!

Postby Marek14 » Thu Nov 17, 2011 6:54 pm

I'm interested in the other toratopes, rotopes, etc. in this program. And maybe graphotopes, you remember those? :) I remember I was explaining those to a psychologist when we went to pursue my Asperger's diagnosis...

First of all, there was this one note I put here years ago, about tiger's relation to tetratorus. Since both can be formed by taking a torus and rotating it around a plane, I speculated that you could continuously morph one into other by moving that plane and taking a snapshot of the resulting shape.

As for graphotopes, those were kind of a supergroup of the various combinations of "square" and "circular" shapes. Basically, you had n nodes connected into a graph by edges. This defined an n-dimensional figure where the cross-section by any coordinate plane is a circle if the two nodes corresponding to the plane's axis are connected by an edge, and square if they are not. The cubinder, spherinder, duocylinder and so on are special cases where the graph is made of disjoint components that are complete graphs. The simplest one that doesn't work that way is 3D object called "crind" - I called it a "dome", and its implicit equation looks like this, I think:

(max(x,y))^2 + z^2 = 1
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Re: I live!

Postby Keiji » Thu Nov 17, 2011 7:26 pm

Yes, I remember the graphotopes :)

I thought they were pretty interesting, although it didn't look like there was much you could do with them, as opposed to the toratopes, tapertopes and uniform polytopes.
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Re: I live!

Postby quickfur » Thu Nov 17, 2011 8:01 pm

What are the equations of the tiger again? I've been thinking about approximating curved 4D objects with polytopes with my renderer, and thought that since I have a much improved facet enumeration algorithm now, I should revisit this subject. :) I've been stuck on polytopes too long due to technical challenges in rendering curved objects. But at least I can do approximations. :)

Also, I remember inventing a notation for objects that included the duocylinder and the crind. I believe keiji calls it "tapertopes". Basically, you start with a point, denoted by a dot (.), then apply any series of the following "tapering" operations: '|' (extrusion), 'A' (pyramid), 'X' (bipyramid), 'O' (rss taper). The O operator basically tapers the polytope using sqrt(r^2) so .O is a line segment, .OO is a circle, .OOO is a sphere, etc.. The n-cubes are obviously .|, .||, .|||. etc., and the n-simplices are .A, .AA, .AAA, ..., and the n-crosses are .X, .XX, .XXX, and so on. The crind would be .||O.

Since you always start with a point, the convention is to drop the initial dot from the notation. Cartesian product is also available, by concatenating shapes (without dropping the dot). So duocylinder is OO.OO and 3,3-duoprism is AA.AA. As you can see, the number of operators = number of dimensions.

This notation covers quite a large variety of shapes, especially crind-like shapes, although it doesn't include toroidal shapes or some of the more symmetrical polytopes.
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Re: I live!

Postby Marek14 » Thu Nov 17, 2011 8:24 pm

The tiger equations (parametric) should be:

x = A cos a + C cos a cos c
y = A sin a + C sin a cos c
z = B cos b + C cos b sin c
w = B sin b + C sin b sin c

I think.
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Re: I live!

Postby Keiji » Thu Nov 17, 2011 8:25 pm

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? :(

quickfur wrote:Also, I remember inventing a notation for objects that included the duocylinder and the crind. I believe keiji calls it "tapertopes".


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 :D
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Re: I live!

Postby quickfur » Thu Nov 17, 2011 9:04 pm

Keiji wrote:
quickfur wrote:What are the equations of the tiger again?


They're on the wiki, where they always have been: Tiger

Thanks!

This right after I added a prominent wiki link to the header bar? Really? :(

For some reason, the header bar broke on my browser. I saw it once and then in subsequent pages it was just text superimposed on the title banner in with unreadable contrast (missing CSS maybe?). I'm not sure if it's a problem with the code, or with my browser. I suspect it's my browser.

[...]
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 :D

I haven't kept up with the wiki for who knows how long. I'm out of date! I feel so last century! :P

I was just going to invent another shape notation system, but I guess now i better be catching up on SSC2 before I try anything else, otherwise I'd just be reinventing the duocylindrical wheel. :P
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