Comments on: Polyhedral Maps

Polyhedral Maps

Tube — Sun, 01 Jun 2008 10:33:53 -0800

Polyhedral Maps is a website that explores unconventional methods of mapping the surface of the earth. The most famous of these unusual maps was Buckminster Fuller's Dymaxion map, which used the net of an icosahedron. Da Vinci had experimented with this technique in his "Octant" map of 1514, which used Reuleaux triangles as map elements. This process is now being used by photographers and artists in manipulating panoramic images. A good example is Tom Lechner's The Wild Highways of the Elongated Pentagonal Orthobicupola.

By: nebulawindphone

nebulawindphone — Sun, 01 Jun 2008 11:12:47 -0800

The Wild Highways of the Elongated Pentagonal Orthobicupola. Ha! Right up there with "The Second Dream of the High-Tension Line Stepdown Transformer."

By: nebulawindphone

nebulawindphone — Sun, 01 Jun 2008 11:13:53 -0800

So I collect strange titles. Good art. Good post. Thanks.

By: farishta

farishta — Sun, 01 Jun 2008 11:29:28 -0800

Wonderful post! Thanks so much for this, I have a real love of maps, and now I can't wait to get this Waterman Projection when it comes out next year. Great stuff.

By: churl

churl — Sun, 01 Jun 2008 11:44:45 -0800

This post made my day. I especially like when the particular projection is a perfect fit for the photograph, such as with Truncated-icosahedral rind, You can't make an omelette without breaking some eggs and Ball!

By: silence

silence — Sun, 01 Jun 2008 12:15:07 -0800

Without meaning to de-rail the thread - if there's a geometer out there who would be interested in giving me a little advice for a project I'm working on I would be extremely grateful if they would get in touch. I'm looking for ways to tile a sphere with the minimum number of differently shaped units.

By: owhydididoit

owhydididoit — Sun, 01 Jun 2008 12:56:13 -0800

Thanks, Tube! I'm a map fan too.

By: Ynoxas

Ynoxas — Sun, 01 Jun 2008 13:50:53 -0800

Are these supposed to be in any way useful? Or are they just novel?

By: mismatched

mismatched — Sun, 01 Jun 2008 14:06:32 -0800

I wonder if anyone has used Google Earth or other satellite images to make one. I'd love to see Earth at Night or something similar, though that'd end up more decorative than anything else, probably.

By: Sys Rq

Sys Rq — Sun, 01 Jun 2008 14:08:19 -0800

Tutorials, please!

By: BrooklynCouch

BrooklynCouch — Sun, 01 Jun 2008 14:17:37 -0800

This post again raises an old question: are there any maps that translate mountainous (and, I suppose valley-ous) terrain into surface area? In otherwords, how much larger would, e.g., Switzerland or Nepal look if you smooshed the slope of mountains into flat distance?

By: Jimbob

Jimbob — Sun, 01 Jun 2008 14:58:50 -0800

There would be a way to do that, BC. It's a technique called "Area Cartograms", where shapes in the map (countries, counties, whatever) are warped to take up a different area. It would be possible, using a digital elevation model, to calculate the "surface area" of countries, then scale them appropriately using this method.

By: hortense

hortense — Sun, 01 Jun 2008 15:00:12 -0800

Time cube guy's globe.

By: BrooklynCouch

BrooklynCouch — Sun, 01 Jun 2008 15:18:46 -0800

Thanks Jimbob. It has just always struck me that "two inches" on a map of New Hampshire take a lot longer to go across than "two inches" in Iowa, and that the places with "non-flat" terrain are not getting the credit they deserve. Or something...

By: rokusan

rokusan — Sun, 01 Jun 2008 15:59:49 -0800

"two inches" on a map of New Hampshire take a lot longer to go across than "two inches" in Iowa Not to mention the motion of the ocean.

By: empath

empath — Sun, 01 Jun 2008 18:11:28 -0800

By: Jimbob

Jimbob — Sun, 01 Jun 2008 18:48:06 -0800

bc, part of the problem with what you're proposing is that mountainous terrain is fractal, so if you flattened it out, it could be nearly infinite Not a new problem - coastlines are fractal, and if you look for estimations of the length of the coastline for a given country, you will find the number will vary widely. So, you have no option but to set an arbitrary spatial scale at which to work, and ignore any variation smaller than that scale.

By: Ynoxas

Ynoxas — Sun, 01 Jun 2008 20:55:37 -0800

bc, part of the problem with what you're proposing is that mountainous terrain is fractal, so if you flattened it out, it could be nearly infinite, depending on how precisely you mapped it (to the mile? Foot? Centimeter?) posted by empath at 8:11 PM on June 1 A mountain surely has a definite "size". It is just a surface, isn't it? Take a piece of paper, and crumple it into a nice mountain shape. If you pull it taunt and flat, it has a definite, finite, size. Or am I thinking about this wrong?

By: Jimbob

Jimbob — Sun, 01 Jun 2008 21:01:06 -0800

Yes, but what if you take into account the texture of the individual fibers that make up the paper? A 10cm x 10cm piece of paper will have a surface area of 100cm² at the macro scale, sure, but it's surface area will be much more if you start looking at it under a microscope. As it is with mountains. If we measure the elevation of a mountain range with, say, 1km x 1km pixels, it will have a different shape than if we use 100m x 100m pixels. If we use 10m x 10m pixels we will start to pick up crags and caves that wouldn't be apparent with the larger pixels. If we use 1m x 1m pixels, we will start measuring the surface area of individual boulders. If we measure the elevation of a mountain at a resolution of centimeters, we will have to add in the surface of every pebble we come across. A mountain has a finite volume, but a potentially infinite surface area.

By: empath

empath — Sun, 01 Jun 2008 21:05:38 -0800

Not really. There's surely a finite surface area to a real mountain range, because of atomic limits, but if you were to 'flatten' a mountain range on a map, the area would depend on the resolution that you used to measure the various peaks and valleys. Think of a mountain range as a 3 dimensional koch curve. In the same way that a koch curve has a finite area, but infinite circumference, an (ideal) mountain range has a finite volume but infinite surface area. If one were to 'flatten' a koch curve into a straight line, the length of the line you would end up with wouldn't be the 'true' circumference (which is infinite), it would merely be a function of how many iterations of the curve generation you include in the measurement. In the same way, the area of a flattened mountain range would depend on the resolution of the measurements you used to determine it's 'roughness'. I'm just saying, it would be a really imprecise map, though it would still be interesting to see an attempt made.

By: empath

empath — Sun, 01 Jun 2008 21:06:27 -0800

jinx.

By: BrooklynCouch

BrooklynCouch — Sun, 01 Jun 2008 21:16:43 -0800

Precision may be an issue, but it would be nice have something better; that at least acknowledges the parameter.

By: Jimbob

Jimbob — Sun, 01 Jun 2008 21:21:34 -0800

Well in terms of your original question, BC, if we assume you want to compare walking across Iowa to walking across New Hampshire, an elevation model on the order of the size of your foot (say, 30cm) or of a pace (say, 1m) would give an adequate representation of the different distances. If we want to look at the distance an ant would have to travel in these two different environments, we might be looking at needing a model of the environment with a 1mm resolution. The ant would have to walk over every pebble, every grain of sand. 30cm elevation models are rare and expensive to make, but it would be possible.

By: ikahime

ikahime — Sun, 01 Jun 2008 22:01:42 -0800

Thanks for this post, if for nothing more than the great addition to my vocabulary: orthobicupola! PS - the maps are great, too.