What is 3D shape of hexes? Hex dipyramid or Hex cylinder?

[Hari Seldon]

The problem is there is no such thing as a 3D object with all identical faces, where each face has 6 sides or more. If such a shape existed distances in Erfworld would be the same in all directions (if a unit standed at the center of a hex it would be an equal distance to go any direction at any angle to get to the edge of the hex. The dodecahedron gets it up to 5 but that isn't far enough. Yes I know dodecahedrons (as well as impossible "hexahedrons" since the more perfectly symmetrical faces you get the closer you get to a sphere so obviously the 3D stacking gaps would still hypothetically be there) won't stack in 3D without gaps, but if the dodecahedrons (or impossible "hexahedrons") were stacked as if they were cubes and the empty spaces ignored, and units in Erfworld "teleported" between the gaps to the next hex ... the space the Erfworld units could occupy would still have the same distance traveled (for all practical purposes) going in any direction.

To make 3d hexes in Erfworld the simplest way you could just do what you do when you turn a circle into a cylinder, just add height to the 2d shape. [ The best I can do is say that the hexes turned 3D must either be those cylinder-types with only one face top and bottom OR be hexagonal dipyramids (but that would make hexes above and below each other be 1/2 shifted horizontally from each other, if you drew a vertical line through the middle of the ground hex it would intersect with the boundary of the hex above it then intersect with the middle of the hex above that. The benefit, though, is that hexagonal dipyramids DO stack perfectly in 3D with no gaps.

[moose o death]

they are hexes

erfworld is flat

like a table

[Drifter]

they are hexes

erfworld is flat

like a table

heh heh heh! Eloquently put, Moose.

It would work best for the narrative for Erf to be flat. I wouldn't want to add timezones to the what is already a confusing turn/daylight issue...

Besides, Hari, a flat map gives the author (and us readers) a limitless variety of sides, terrain, rulers, adventures, plot lines, etc.

[Hari Seldon]

they are hexes

erfworld is flat

like a table

Error does not compute.

(I added bolding)
"The dwagon J-hooked into a dive that took it swooping over the rooftops, directly toward them. Its wings were stiff and swept back as it covered most of the intervening distance in a single glide. As it neared their tower, it pumped its wings hard, three times. It took the tower like there was an invisible skate ramp up the side." - Summer Updates – 032

How would you calculate 3D aerial movement, like that? If there are only 2D hexes that extend all of the way up and down then movement up and down in the air must be free because it's all in the same hex. Obviously movement up isn't free, or else Bats would dive-bomb like no tomorrow (not so much that they'd hurt themselves but enough to hit harder) since that would give impressive bonus damage to their attacks.

Though, as I mention in my Falling without move and Duty physically enforced for Summoned thread, movement down appears to give you fractional positive move per hex because there is Potential Energy in Erfworld.

[DevilDan]

While I don't take it as an article of faith, it wouldn't surprise me if Erf were flat.

As to flight, don't forget that we are dealing with magical flight; as such, the capabilities and motions of a flying unit are established by arbitrary parameters rather than by a necessarily self-consistent physics model.

[Hari Seldon]

I agree that it wouldn't surprise me if Erf's surface is flat instead of a sphere, but where does that require the airspace above Erfworld to be flat? True the Lookamancer Table was 2D here and wasn't 2.5D - which would have helped make the flying groups easier to see - (though the Table could display in 3D and they were talking about flights of dwagons ... though why this wasn't shown again in 3D or 2.5D to help see where the dwagons actually are (instead it was 2D in the other link) doesn't make sense unless it was some sort of "strategic overview" and not "tactical overivew" ...) which would have helped to distinguish the flying units from the ground units so I don't think the Lookamancer Table was the end all and be all of what Erfworld's hexes look like. Especially since Hex cylinder would make more sense than the cubes shown ... it could just have been an approximation ... but these people LIVE in a hex world so I can't think of a reason why they should approximate with cubes instead of Hex cylinders.

Also I thought Parson was figuring out the self-consistent physics model of flight (or thought he was) when he said:

He put the handkerchief to his bleeding elbow. "Well, yes," he admitted, "yes it was." He grinned. The faces looming over him were suddenly hilarious. "But you gotta admit, you gotta admit! I learned something today!"

- Erfworld Summer Update - 032

[DevilDan]

The cubes always seemed to me like a representation of a single unit or some icon or symbol.

Presumably the colors and symbols of the pure 2D map are presumably more meaningful to the commanders reviewing the map than to us.

[moose o death]

they are hexes

erfworld is flat

like a table

How would you calculate 3D aerial movement, like that?

the miniature was modelled to be above the ground. the narrator embellished the details.

i know it's against the erfworld is not a game mantra, but your trying too hard. their world is as flat as ours isn't, ours isn't quite spherical and theirs isn't completely 2d.

[Guurzak]

"He wondered if there was an altitude limit to the airspace zone, which led him to wonder if different kinds of flying units had different max altitudes. If so, would they be able to use the highest-flying units for spy plane missions?"

The world is flat. Hexes are hexagonal prisms. Whether the prisms are infinitely tall is unknown.

[DevilDan]

We cannot assume any physical phenomena on Erf. That magic stuff gets on just everything. You should see an average magician's dry cleaning bill.

[Welf von Ehrwald]

Or it's not flat, but a globe like ours. Or how else do you explain the existence of a sun and a moon? For me, it's easier to imagine a globe than a game board that floats trough space. Okay, I can imagine that, I read discworld, but still.
And on topic: my bet is on hex cylinder, with the limit just above the highest clouds.

[Hari Seldon]

the miniature was modelled to be above the ground. the narrator embellished the details.

i know it's against the erfworld is not a game mantra, but your trying too hard. their world is as flat as ours isn't, ours isn't quite spherical and theirs isn't completely 2d.p

Agh ... people ... fine do you want the gliding Dwagon Banana to get a Special Bonus for Gliding which Simulates Potential Energy Without Acutally ... I thought Parson decided there was Potential Energy when he said that he learned something from flying Banana ... that the "Potential Energy" game mechanic trumps the "Heavy units can't fly Dwagons" game mechanic.

The thing with tabletop hex games is you don't want to sit there calculating potential energy and crap if a Heavy unit attemts to fly your Dwagon and has a controlled descent (so you simply say he Can't Attempt It in the first place). Attack of Opportunity rules are bad enough already in D&D.

Erfworld is different. It's not so simplified, though it is different from Earth.



And people remember there's also a dungeon zone so there would have to be (whatever you want to deal with airspace zone) in that area, too. How can we calculate ground units' move in dungeons? We can't use the old 2D system ... what if the unit goes up a slanted ladder (ex: to scale a castle wall)? I'm telling you guys Erfworld is almost required to have 3D hexes stacked in 3 dimensions.

And I never said Erfworld had to be a sphere. Doesn't matter one way or the other.

[DevilDan]

I guess we took literally your question about the shape of hexes. Surely hexagonal prisms tiling a sphere would have a larger "top" hexagon?

[Hari Seldon]

Yes I suppose so, but that's only if the hexes are enormously tall (like in there is ONE hex and it reaches up into the sky and into the ground ... if Erfworld did happen to be a Sphere then at the center of the Sphere move would be ... rather expensive ...).

If there are hexagonal prisms or hexagonal dipyramids stacked in 3D space that are roughly 1 kilometer in diameter, then if Erfworld is the same size as Earth ...
Radius of Erfworld average ground level (at the equator since Erfworld is actually a bit egg-shaped so geographic pole-to-pole circumference is actually bigger): 6,378 km
Radius of Erfworld +1 3D hex(whatever) from ground level: 6,379 km

Circumference of a Circle: 2*pi*r
2*pi*6,378 km = 40,074.1559 km
2*pi*6,379 km = 40,080.4391 km

40,074.1559 km - 40,080.4391 km = 6.2832 km difference in circumference

There are 40,074 hexes (each imperceptibly wider in he horizontal direction to make up for 0.1559 of a hex left over ... 0.1559 km left over / 40,074 hexes (each a kilometer wide) = 3.89030294 × 10^-6 km that each hex is lengthened in horizontal direction)

3.89030294 × 10^-6 km = 3.89030294 × 10^-3 m = 3.89030294 mm

So out of an entire kilometer-wide 3D hex, the bottom is 3.89030294 millimeters longer than a kilometer to account for extra decimals in the circumference of Erfworld not breaking down into a neat rational number

As for the top of the 3D hex, that would be:

0.4391 km left over / 40,080 hexes (each a kilometer wide) = 1.09555888 × 10-5 km = 1.09555888 x 10^-2 m = 1.09555888 cm

So the top of each hex is 1.1 cm wider at the top to account for the rounding

6.2832 km difference in circumference / 40,080 hexes = 1.56766467 x 10^-4 km to account for different circumfrences = 1.56766467 x 10^-1 m = 0.15677 m difference at top of ground hex vs bottom to account for different circumferences

So bottom of ground 3D hex = 1 km + 3.89 mm rounding fix = 1.00000139 km
Top of ground 3D hex = 1 km + 1.096 cm rounding fix + 0.15677 m circumference difference fix = 1.00016773 km

1.00016773 km - 1.00000139 km = 16.63 centimeters

A kilometer's a little big for a hex but a decimeter sounds just about right (100 meters is a little big for a Gate to a City to be 1 Hex big for example), but regardless of the hex size you choose the Circumference difference remains proportionally the same magnitude. So if the hex size was 10 meters (a decimeter) then we'd just change a few 10^Xs and end up with 1.66 millimeters as the difference between the bottom and top of the ground hex that is a decimeter wide as our answer. That's hard to spot, 1.66 millimeters difference in 10 meters (about 30 feet)!



Alternatively, you could have hexes be say pi*whatever in diameter and that would get rid of a lot of the decimals. Let's assume that's the case (and that it's pi decimeters, which is about 90 feet ... a little big but about 9 feet (pi meters) is too small for a hex, period)):

(6378 km radius of Erfworld / pi * 10^-2 km/hex diameter) * 2 * pi = 1,275,600 hexes at the surface of Erfworld
(6379 km radius of Erfworld / pi * 10^-2 km/hex diameter) * 2 * pi = 1,275,800 hexes at +1 hex above the surface of Erfworld

1,275,800 hexes - 1,275,600 hexes = 200 hex difference between circumferences

200 hex difference / 1,275,800 hexes = about 1.56764383 x 10^-4 hex stretch per hex to account for difference
= 1.56764383 x 10^-4 * pi decimeters
= 1.56764383 x 10^-3 * pi meters
= 1.56764383 * pi millimeters
= about 4.92489834 millimeters
= about 4.925 millimeters

Note: this does not take into account Earth's non-perfect-sphere egg shape as that makes its pole-to-pole circumference a slight ellipsoid instead of a circle (though the equatorial one might still be a perfect circle more or less), but ... Jesus gimme a break. This does account for 3D hexes being polygonal solids, though, so long as the 3D hexes aren't flat on the bottom and instead wrap around the surface of Erfworld's spherical shape so that the bottom of a hex is a little convex and the top of a hex is a little concave. If Erfworld was just a sphere floating in space and not the center of its universe, perhaps we shouldn't wrap the hexes and then the stretching the top to make it bigger would GO AWAY because the hexes stack without regard to fitting around a central point ... unfortunately then how would hexes be oriented in such a way on Erfworld, with the hexes always being hexagons ... ex: hexagonal bipyramids stacked oriented with pyramidal base being perpendicular to Erfworld's geographical equator ... the equatorial hexes would be a vertical cross-section of the hexagonal bipyramid and be very strangely shaped with two points and a middle while the polar hexes would be very close to the familiar hexagaon except when the horizontal-ish cross-section happened to be not quite a perfect hexagon perfectly in the middle of the figure (which is almost all of the time).



In conclusion, assuming Erfworld's hexes' diameters are irrational numbers (that can multiply up to pi) so that the irrational number pi doesn't cause Erfworld to have partial hexes left over when it is a perfect sphere (unlike Earth which is egg-shaped) then ... for a pi decimeter diameter 3D hex that is curved around Erfworld's surface like the hexagons are curved around Bubble Shields in the game Halo, we are looking in general for differences in millimeters between the top and bottom of the hex. Whether that is stretching or displacement we don't know.

If the hex isn't curved then the only difference is that the bottom faces of hexes in Erfworld aren't actually on the spherical surface of Erfworld, the bottom face is slightly below it except on its edges ... they touch the surface of Erfworld. This is like how you can draw lots and lots of straight lines inside of a perfect circle to approximate it.

The big question is whether or not 3D hexes (for this example think about hexagonal prisms) should be ADDITIVELY stretched (the stretch that made the top of the hex below bigger being the starting point for the bottom of the hex above ... which is streched more to make the top of the hex above ... ad infinitum) because if they aren't then the 3D hexes as you go up in altitude are displaced (slightly) horizontally from the others below them were to account for the fact that to keep no gaps between the hexes and there are more hexes than there were before. So the choice is:

1. keep the hex where they were horizontally and start getting larger and larger gaps as you go up in altitude, until those gaps can fit extra hexes in them ... this looks ugly trust me ... basically right before you can add in more hexes the units are almost teleporting an entire hex distance between every hex because that's almost how big the gaps are!)
2. Best choice in my opinion: compact the 3D hexes in the new layer to remove the gaps so that the bunch of tiny gaps are converted into just a few larger gaps outside of your compacted hex formation ... voila! That's where the spare hexes go ... the ones that make up the increase in circumference of the sphere they exist in compared to the hexes below. Note: if hexagonal dipyramids were used then the displacement from this would be a problem since the pyramids would be invading the below hexes' personal space, and there would be empty spaces where the pyramids would have gone were they not displaced. Attempting to curve the pyramids to compensate doesn't help because the displacement could cause the pyramid to end up anywhere ... near the edge of the lower hex ... in the middle of the lower hex ... depending on which displaced hex you choose.

In the end ... if Erfworld is spherical then Hexagonal dipyramids would have serious problems with gaps, displacement, or stretching to ridiculously-sized hexes at higher altitudes ... but Hexagonal prisms would do just fine (except that there would be gaps, displacement - but at least the pyramids would not invade the other hexes space - (vertical movement would still be buggy though), or stretching).

These choices are completely unacceptable (if Erfworld is a sphere). I have to conclude that actually, gravity in Erfworld is what fixes this problem. Imagine these cubes were hexagonal prisms. The top and bottom faces of which were oriented to point away from the center of Erfworld. The stretching here is exactly the stretching I was discussing earlier to prevent gaps between hexes of different spherical layers. BOOM ... relativity isn't scienticically proven to be in Erfworld, but with electricity and magnetism probably working the same (Parson's solar calculator watch works) ... well ... Lorentz contraction (a part of Relativity) came directly from the failure of the Michelson–Morley experiment and ... all of that was because electromagnetism works the way it does. If it works the same in Erfworld then someone could be standing still observing an electric field and someone zooming by would see a magnetic field and ... the only way for the two reference frames to be consistent was if time slowed down for the observer in motion, relative to the electric field (Look at the appropriate chapter in The Cartoon Guide to Physics it does a good job of coming up with Relativity without invoking gravity or acceleration in the proof).






But I thought Erfworldians said that time is relative but not in the Earthworld relativity way?

Yes ... yes they did. There can only be two explanations:

1. Erfworld is really actually a flat world with gravity that ONLY points downwards ... downwards like from above the tabletop to below the tabletop, regardless of if that part of Erfworld had a mountain on it. Since gravity doesn't point to the center of an object, the curving space to fix the hexes and also pop out with relativity thing never comes up, since the hexes are truly stacked vertically in a cube-shaped volume above and below Erfworld's surface, without trying to point to the center of a sphere they simply have their top and bottom faces exactly perpendicular to Erfworld's surface.
2. Erfworld may actually have relativity and the Sun's position is lying about Time. The sun's position simply keeps track of turn order/stuff for each individual unit but has nothing at all to do with "real time." This unfortunately makes perfect sense ... (I added the italic stuff in parentheses)
   

   Parson remembered time zones and the weirdness of the international dateline. Watching the Olympics at 2 am, people running in the hot sun on the other side of the planet, where it was tomorrow. Erfworlders carried their own personal time zones around. Was that odder than anything else about this place?
   
   Yes, yes it was. It meant the whole universe worked differently here, and he wasn't prepared to let it go at that. He leaned forward even farther and put his elbows on his knees. "Aha! What if I come back?"
   
   Sizemore looked at him blankly. Maggie watched them both, a dreamy grin on her face.
   
   "It's four hours later to me, but maybe four minutes to you." (in real time)
   
   "Yes?" said Sizemore.
   
   Could he really not see the problem? "Where is the sun?" asked Parson in exasperation.
   
   Sizemore squinted at him. "In that hex? Four minutes further along in the sky." (the sun simply shows the turn order of the dominant Observer in that hex, NOT anything to do with "real time")
   
   "It jumps backwards when I enter the hex?!"
   
   "Of course! To you," said Sizemore. "Because you traveled and were observed from that hex. If you weren't observed, you would find the sun in the same position as the previous hex."
   
   "Ridonkulous!"
   
   Maggie slumped backward on the blanket and giggled helplessly up at the sunlit sky.

   - Erfworld Summer Update - 029

[moose o death]

right see the word count you just did, thats why erfworld is flat, and that your trying too hard. there isn't a NEED for it to be a sphere as it's just over complicating a legitimately simple problem.

[Hari Seldon]

Sounds good to me. So Erfworld's surface is flat and the 3D hexes are stacked below and on top of that "tabletop." The surface being a sphere needlessly complicates things.

Back on topic we still don't know if those 3D hexes are hexagonal dipyramid or hexagonal prisim, or how tall they are (some people say they are infinitely tall and the 3D cubes above they eyemancer table were just for effect).

Just remember if Parson later finds out that units going down a hill or otherwise going downwards gain some extra move (because they released potential energy which fractionally decreased their move cost through that terrain) I won't be the one who's surprised that the reverse is true and Erfworld counts going upwards as costing some move per distance (which would mean there are stacked 3D hexes or some similar way to keep track of and limit move).

And I still think that the Sun only keeps track of the turn order of the dominant Observer in that hex, and is lying about "real time" as a unit sees it track across the sky as "real time" goes by.

[moose o death]

i doubt they have any height, just layers, your in airspace, on ground or in a dungeon. i'm quietly confident the hexes are all different sizes as well, allowing for cities to occupy as much space as they need for their contents. or cities are multi hex but all count as one hex...or else garrisoned units would be stuck in certain areas.

i think you'll find 3d space is relative in erfworld...as evidenced by their magical items featuring a resize ability.

[moose o death]

Sounds good to me. So Erfworld's surface is flat and the 3D hexes are stacked below and on top of that "tabletop." The surface being a sphere needlessly complicates things.

Back on topic we still don't know if those 3D hexes are hexagonal dipyramid or hexagonal prisim, or how tall they are (some people say they are infinitely tall and the 3D cubes above they eyemancer table were just for effect).

Just remember if Parson later finds out that units going down a hill or otherwise going downwards gain some extra move (because they released potential energy which fractionally decreased their move cost through that terrain) I won't be the one who's surprised that the reverse is true and Erfworld counts going upwards as costing some move per distance (which would mean there are stacked 3D hexes or some similar way to keep track of and limit move).

And I still think that the Sun only keeps track of the turn order of the dominant Observer in that hex, and is lying about "real time" as a unit sees it track across the sky as "real time" goes by.

you said this whil i was responding else where

you will find this is handled by units having a move penalty over certain terrain, ground units CAN'T cross water hexes, what you need to consider is that a mountains hex will have just as many downhill walks as uphill walks, but a rough terrain penalty of -1 move would surely apply. this isn't particularly hard to accomodate at all.

[Hari Seldon]

you said this whil i was responding else where

you will find this is handled by units having a move penalty over certain terrain, ground units CAN'T cross water hexes, what you need to consider is that a mountains hex will have just as many downhill walks as uphill walks, but a rough terrain penalty of -1 move would surely apply. this isn't particularly hard to accomodate at all.

But a mountain's rather big (ex: Gobwin Knob's Volcano) ... how can that be only 1 hex?! This isn't Wesnoth ... plateaus exist (Transylvito is high elevation plateau where Gobwin Knob is lower) so your equal uphill and downhill walk was a good idea but doesn't quite explain everything. And when they were explaining Transylvito's defenses didn't they say it was a hard uphill climb which slowed enemy infantry way down?

The stack swept in from the south, hugging the mountaintop and approaching the city along the old aqueduct trail. Non-fliers could only reach the city by this trail, and the move penalty was murder. Low-move units (like, say, most siege engines) might need three or even four turns just to climb the mountain. The capital had not been seriously assaulted by ground in thousands of turns.

- Summer Update 2009 - 017



I agree that buildings/fortifications/cities have some special rule that "makes them their own hex"

I agree with the layers idea ... the problem is though that might have been a special case of the dungeon being part of the Gobwin Knob City - which is special because it's "all one hex" - whereas in other situations you'd have to keep track of the "3D movement cost" of going up and down ladders ... like when units try to climb ladders to scale a wall (that's before they enter the "one hex city") to kill the defenders (and weaken the wall so Siege can destroy it). And of course the quote above which pretty much proves 3D movement cost (but Parson didn't know about this he only found out at the end of that turn after he rode Banana).

And I didn't go into the shape of Erfworld originally in this thread since we just have no idea if it's a sphere or not (though it being flat would be cool, too).



It'd still be ridonculous though if Gravity could deform hexes .

[moose o death]

well if i was designing the place the rough terrain bonus would apply no matter if it was moving up or down. you can't descend everest quickly ...safely, so moving down a mountain is still slow even with the gravity assist.

have you ever sprinted down a hill? at a certain point you realise that was a very bad idea. your legs are moving faster than your brain can process the instructions and you become aware that you may not make it to the bottom in your current upright state. at least that's what happened to me the few times i've done it.