I think that’s a face centered cubic lattice. It’s still a three dimensional space. I’m rather rusty, but I think you can specify any point in space as a sum of unit vectors using only three of the four “axes”.
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I think you’re right! It looks like a face centered cubic lattice to me. The 4 axes come from the vertices of a cube.
It’s still 3D space, but I had fun trying to figure out how to add a 4th axis. I had a bunch of failed attempts before I figured it out.
I took off the 4th axis, so now it only has 3. I agree you could specify any point with it since they’re all equidistant units of measurement. At least, I think it’s possible. I’m not the best at mathematical geometry, but I think it’s entertaining nonetheless.
I thought you could make some cool shapes with a 4D lattice. The problem is everything is angled, so the gravity would make it hard to walk on. It might be a cool shape for a space station though. I don’t know.
I played around with such a grid in high school when I wanted to create a Star Trek (board) game, and wanted to create a space in 3D that did what hexagonal grids did for simulation/war (board) games in 2D at the time. This lattice was the answer. You can slice it an any of four different planes and get a hexagonal grid.
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Maybe this book will be interesting for you.
Flatland: A Romance of Many Dimensions
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To project more than the space dimensions, you should include movement of the 3D objects, typically just a time- bar at the bottom:
https://images.app.goo.gl/U5fSooL2unA2pVuD6
The fourth axis is placed outside the (allready projected) three Axis of the dimension space…
This might be of interest, too:
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Yeah, the problem with my graph is you can’t graph certain points like 1,0,0,0. If you move in one direction, it turns out to be 1,1,1,0 or something because the axes are the diagonals of a cube. The graph is so abstract and dense that it’s hard to even count along the axes. So I’m pretty sure this isn’t a true 4D lattice. The 4th dimension would be outside the 3D space. You’re right.
CMYK color space is another example of a space you might think has four dimensions, but color is still just three dimensions as it is in RGB. For a given point in RGB, there are more than one combination of C, M, Y, and K that describe the same point. That’s why when people pick on color based on CMYK and come back later only to find the numbers changed by the computer. It was translated into RGB and then back again, which could be a different combination of the components.
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I’ve been modeling this “flying city” for a few days now. It’s still a work in progress. There are six double decker baskets connected to an incomplete hexagonal central basket.
The hot air balloons use solar panels to power electric burners (similar to an electric stove) to heat the balloons. The tops of the balloons are black to increase the temperature of the balloons for extra lift. Ideally, it could fly as long as there’s sunlight or enough charge in the battery without the need to land. So it could truly be a flying city.
Each basket is connected by a, hopefully lightweight enough, rope bridge.
The floors are connected with a rope ladder.
Here’s up the ladder.
I’m thinking about adding inflatable furniture or rope furniture because it’s so lightweight.
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That’s a fun idea, but at some point you wonder, why not just make one big dirigible? You need something to keep them all together. How about a dirigible equivalent to a catamaran or trimaran?
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That’s a really good idea! Maybe after this project I’ll try it out. I guess I just like the look of hot air balloons because it’s kinda modular looking. And you can always add more balloons to the structure.
Currently, the rope bridges hold it together, but I’ll probably need to attach the balloons together too.
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I always enjoy you projects. They are fresh, and a bit outside of the box for the most part. This one reminded me of “Up”.
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Great post! Clearly at present the solar / heat output ratio is difficult, but who’s to say it will not improve? They flew a solar plane around the world after all. One could imagine floating above the clouds indefinitely on solar power or dropping slowly, hoping morning comes soon enough!
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This is the great post. You have shared valuable sketches to us. Sure It would be helpful to me.
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