Everyday Life on a Hypergeometric Earth

[PatrickPowers]

I need readers for my book to make suggestions and find any mistakes, so as soon as it's done I'll post it somewhere you can get it. To get you excited here's a copy of the cover.

cover size 2.jpg

(290.39 KiB) Not downloaded yet

Hypergeometric is a contraction of hyperdimensional geometry. I was going to write hyperdimensional but a crackpot is already using "hyperdimensional Earth." By "dimension" most people mean some mystical place, so it confuses them.

[PatrickPowers]

The book may be downloaded from https://www.researchgate.net/publication/349832144_Elsewhere_--_Everyday_Life_DOCX/related, on the shaky assumption that their software works. Please inform me of any error/typos, subjects too difficult to understand, things you like, things you don't like, etc.

[Auden]

Hi. I'm new here. Maybe too new to be of a whole lot of use, but I started looking at it.

In the dedication: and of which in these pages reader shall encounter no trace (not sure if you want to say "readers" or "the reader" etc)
Same paragraph: While on a research expedition Clifford survived shipwreck off the coast of Sicily. (maybe "a shipwreck")

top of p 10: A turn could be somewhere in between right, ana, and forward, in which case we might say “veer leftana.” (did you mean to change from right to left? etc)

p12: How about though when a snake slithers? (How about when a snake slithers though? or- How about, though, when a snake slithers?)

[PatrickPowers]

That's helpful. I'm looking forward to the rest.

[Auden]

Just my impressions, which you are of course free to reject etc:

p22- We start with easy with sports, then go to houses, roads, and landscapes.

I see what you're saying, but I think it might read more nicely as

"We start with something easy like sports, then go to..."

or

"We start easy with sports, then go to..."

*

Bottom of p28- the heading "STADIA" got stranded.

Not sure if you can control that in the format you'll ultimately be using - maybe some sort of parent/orphan control?

*

p29- Maybe the simplest such is air hockey. (I would nix "such")

So in the 4D world, a 3D disc is the same thing as what we call 3D sphere in our 3D universe.

1. I would add "a" before "3D sphere"
2. I would either write "a 3D disc is the same as what we call..." or "a 3D disc is the same thing that we call..."

top of p31- I would contract "we have" to we've".

*

I'm not weighing in on things like human/animal anatomy and other 4D issues, partly because I'm new here and need to do some more reading first, but also because at a quick glance it seems like these types of issues have been discussed by people better versed than I, so I'm figuring you're solid on how you want to treat those things.

[PatrickPowers]

Please keep up the good work!

[Auden]

Okay - this is a nice way to work toward messaging privileges:)

p31: "all and all" - I always heard "all in all"

p33: "Now if draw that 3D disc in our world," - maybe "if we draw..." or "if drawing"?

p35: "I like to think this would be consider" - (considered)

"Lets draw a match between two wrestlers." - (Let's)

"When using this style we can no longer tell whether a wrestler has touched the surface with [do you want a word here?] other than [and maybe here, depending on your choice earlier] feet, thus losing the match."

p37: "Now recall that footprints are justfeet" (missing space)

"Let’s go [] beach."

[PatrickPowers]

I'll do "As is often said in the argot of that locale, let’s go beach."

[Auden]

p 38: You talk about not having a name in the English language for the (2D) shoreline in 4D (as opposed to "line"). Geometrically, is it not a surface? Or do you feel that that word has been overly imbued with the idea of a physical object as opposed to a set of locations(points)? (The non math world so often wreaks havoc on words, but I still like to use them(!)) Really, even in 3D, a shoreline technically isn't a line ether. More of just a path. I wonder if the more general term of "boundary" might be what you're reaching for. (I still want to call it a surface though.)

Reading further, you describe this partition as something we don't have in 3D, but I'd say we have it in spades, just not as a shoreline. Any time we have two distinct 3D bodies of matter (that don't mix) coming into contact with each other in that "smashed up against" way you mentioned, we get that. Dig into the dirt and hit a layer of rock, there's a surface in between. Pour water into a glass of oil, etc. If I'm reading something and the author says we don't have something in our world (and I disagree) and so they're going to use a word that has a 1D meaning but just stick "2D" in front of it, that's sort of going to use up some points with me as a reader. Too many of those and inclination to keep reading wills start to erode.

Again though, I'm new here, and I could be missing something about the culture and language that has grown around and within 4D (yikes pls excuse the failed pun->) circles. So I may be missing something, and if I am, I'm sorry for that, and for my intense wiring and delivery, which, as far as I can tell, is a permanent feature.

p39: When you say a 3D 10-mile shoreline might be 10 square miles in 4D, is there a reason you're using a mile in the ana/kata dimension to arrive at your rule of thumb? My mind first went to 100 square miles, figuring the default estimate for the measurement in the ana/kata dimension would just be to match the original length in 3D.

[quickfur]

Very interesting document!

I'm having my doubts about birds having only 2 wings, though. First, supposing 4D aerodynamics works analogously to 3D, we need an aerofoil of some sort. The airfoil needs to span enough area in order to produce sufficient lift to stay airborne. In 4D, this means it must span a 3D hyperarea. No problem, any section of a 3D hyperplane would work, assuming the requisite airfoil cross-section when cut perpendicular to the w direction. (We're also handwaving away any issues with vortex instabilities when a 3D aerofoil converges to the trailing edge of the wing. Let's leave that to another discussion.)

Now, next to having the aerofoil itself, the next most important thing in flight is the maintain the correct upright orientation of the aerofoil. Once it loses its horizontal orientation, it can no longer produce the lift required to stay airborne. The bird would stall and fall. In order to maintain upright orientation, though, it would require stability in all lateral dimensions. The forward/backward direction is taken care of by the direction of the flight itself -- the bird presumably would have the means to maintain its pitch, thus leaving (N-2) lateral dimensions to be stabilized. In 3D, that leaves just 1 horizontal dimension, which is taken care of by the extension of the two wings. In 4D, you need stability in at least 2 dimensions. If you only have a pair of wings opposite to each other, that only grants stability in 1 dimension. You'd roll over in the other dimension and thus lose the horizontal orientation of the wings.

Now, in 4D you could in theory have a single wing that wraps around the body of the bird (and still have room for legs to attach ), that would grant stability in all horizontal dimensions. But such a wing would require enough mass to span the volume of a cylinder wrapping around the bird: it would be massive and heavy, too inefficient for the job.

A better solution is 3 wings, in a triangular formation. They don't have to wrap around the bird, so they can have minimal mass and require minimal energy to maintain and move. All they need is to extend outwards sufficiently to grant horizontal stability in all 2 lateral dimensions -- the tripod being the most stable construct in the 3D hyperplane along which the bird flies.

So I postulate that 4D birds would be 3-winged. Maybe 4-winged, but an extra wing means extra weight and extra energy to maintain and move it, whereas for flight it is best to conserve as many resources as possible. Two wings would be too unstable, so that leaves 3 as the best solution. So 4D birds would have trigonal symmetry, an elegant and aesthetically-pleasing configuration.

[quickfur]

Enjoyed the bit about spinning pizza dough. Just wanted to say that spinning 5D pizza dough would essentially amount to setting the dough in a 4D clifford rotation, so the pizza maker would have to spin it in two orthogonal planes simultaneously. Would take some skill, but I'd wager that with practice, this could be done easily. You'd see the dough spin in a Clifford isoclinic rotation, spreading outwards along the spiralling fibres of the Hopf fibration. The resulting dough would likely retain a spiralling texture, reflecting the fibers of the Hopf fibration. It'd have one of two orientations, corresponding with the two senses that an isoclinic could happen in ("left-handed" or "right-handed"). But this being 5D, you could simply flip it over to switch between orientations, something impossible in 4D. Perhaps there'd even be a 5D tradition where pizza is always made with the toppings on the left-handed side, and pizza made in the other orientation could be considered as a sign of inferiority or amateurism. Or perhaps there'd be a competition between the two schools of pizza-making, with one school insisting that left-handed pizza tastes better and the other school countering that right-handed pizza is more aesthetically-pleasing.

[quickfur]

Some further thoughts about body plans:

In 4D, having two perpendicular pairs of shoulders are not a problem for embracing at all. Imagine the body of a 4D person as approximately a spherinder. Reserve the front and back for ... well, the front and back, which leaves an approximately cylindrical cross-section where shoulders may be attached. So say we attach 4 arms on the corners of a square, around the rim of this cylinder. The 4D body can rotate around axis of this cylinder without changing the orientation of the front/back axis, so two embracing persons could simply rotate slightly relative to each other (22.5°) and the shoulders of one person would be in between the shoulders of the other, and they would still be chest-to-chest. No trouble at all.

Now about beasts: 6 legs seem to be the optimal number for stability and economy. The question is, how should they be oriented? In the document it's proposed that there are two front legs and two hind legs, with two middle legs in perpendicular orientation for stability. That's an octahedral arrangement, with two opposite edges aligned to the front/back of the creatue. Other orientations are possible, though. One possibility is to align one leg to front, one to the back, and have 4 on the sides. This seems a little awkward for walking, though. Another possibility is to orient three legs in the front and three in the back, i.e., a triangular antiprism. (Alternatively, a triangular prism would also work, a non-octahedral arrangement.) The triangular prism/antiprism configuration would seem ideal for galloping, so I'd imagine that equine beasts would have this arrangement of limbs. Slow-moving animals like cows would prefer a more stable arrangement; the 2-front, 2-back, 2-middle arrangement proposed by the document seems to be a good solution.

Another interesting possibility is 4 legs, but not in a square formation; instead, in a tetrahedral / disphenoid arrangement, with a pair of legs in the front and another in the back rotated 90° relative to the front legs. This arrangement is the minimal for 4D stability, corresponding with a 5-cell, akin to the tripod arrangement in 3D (corresponding with the tetrahedron). Furthermore, the two back legs can cross over and land ahead of the front legs (i.e., inverting the tetrahedron / disphenoid), then the front legs cross over the back legs and land ahead of the back legs again. This sort of gait seems suitable for small animals like rabbits and squirrels, and having less limbs to maintain seems better suited for small animals with simpler body structures. So I'd venture to speculate that 4D rodents would have 4 legs in a disphenoid arrangement, and walk by crossing their hind legs ahead of their front legs. Such a gait is also suitable for scurrying away from predators and angry humans. Attach some sharp claws that can hold on to a wooden surface, and they could climb trees and furniture too.

[PatrickPowers]

p 38: You talk about not having a name in the English language for the (2D) shoreline in 4D (as opposed to "line"). Geometrically, is it not a surface? Or do you feel that that word has been overly imbued with the idea of a physical object as opposed to a set of locations(points)? (The non math world so often wreaks havoc on words, but I still like to use them(!)) Really, even in 3D, a shoreline technically isn't a line ether. More of just a path. I wonder if the more general term of "boundary" might be what you're reaching for. (I still want to call it a surface though.)

Reading further, you describe this partition as something we don't have in 3D, but I'd say we have it in spades, just not as a shoreline. Any time we have two distinct 3D bodies of matter (that don't mix) coming into contact with each other in that "smashed up against" way you mentioned, we get that. Dig into the dirt and hit a layer of rock, there's a surface in between. Pour water into a glass of oil, etc. If I'm reading something and the author says we don't have something in our world (and I disagree) and so they're going to use a word that has a 1D meaning but just stick "2D" in front of it, that's sort of going to use up some points with me as a reader. Too many of those and inclination to keep reading wills start to erode.

Again though, I'm new here, and I could be missing something about the culture and language that has grown around and within 4D (yikes pls excuse the failed pun->) circles. So I may be missing something, and if I am, I'm sorry for that, and for my intense wiring and delivery, which, as far as I can tell, is a permanent feature.

Thank you very much, this is what I need to know -- what is it that people find hard to understand? This is where you have a big advantage. People who already know the stuff aren't much use for that.

The four dimensional world has 4D volumes with 3D surfaces. Their 1D things are pretty much the same as ours, so that leaves 2D things as being the odd balls. I am hoping to convince you that most 2D things in the 4D world have more in common with our 1D things than with our 2D things.

Let's say that here in our 3D world you have rope. You would still be able to grab that rope. You can grab that rope no matter how long it is. Let's say that in the 4D world you had a 2D object. You could grab that object no matter how big it was in those two dimensions. Weird, eh? It's always possible for a 4D person to grab a 2D object no matter how large it is as long as the other two dimensions are small enough for your hands to encircle.

Let's say they had a 2D tightrope. It's 50'x50'x1"x1". It would be just as hard for a 4D person to balance on that 2D tightrope as it is for us to balance on a 1D tightrope. The 2D shoreline is like that tightrope.

Or let's say that you are in 4D and come to a door with a rope stretched across it. You can just walk around that rope. No need to go over or under it. Wild! If 4D people want to block off a door, they have to use a 2D rope. You could sit on that rope and use it as a swing. If you on the other hand wanted to pull something with a rope, a 1D rope would be better. As you can see, some of our 1D things split into 2D and 1D versions.

You are correct as to in our world 2-D surfaces form partitions between 3D volumes. Our shoreline is the 1D intersection between that 2D partition surface and the 2D surface of the Earth. That intersection is one dimensional and forms a crooked line. In the 4D world, 3D surfaces are partitions between 4D volumes. The intersection of that 3D surface with the 3D surface of the Earth is 2D. That's the equivalent of our shoreline. It has no extent in one of the sideways direction, so it has no surface area. A 4D person standing on a 2D shoreline seems quite analogous to a 3D person standing on a 1D shoreline.

Let's say in the 4D world you had a 2D pizza pie crust. It has no surface area so it couldn't hold any cheese or tomato sauce. It would have more in common with a strand of sphaghetti in our 3D world.

In our world we have 2D road surfaces which are partitioned by 1D dividing lines. In the 4D world they have 3D road surfaces which are partitioned by what I could have called 2-lines or 2-planes. It seems to me that they have more in common with lines, so I called 'em 2D lines.

How easy would it be to see a 2D thing? Easier than 1D, harder than 3D, that's all I could say for sure. Since letters would be 3D, you wouldn't be able to write on a 2D thing. It would be like us trying to write on a strand of sphaghetti. (Letters have to be 3D so you can see them from all directions. 2D letters would be invisible from certain angles.)

All in all this is one of the hardest things to get used to. My shoreline concept was all wrong for at least a year. Then one day I thought, uh oh.

p39: When you say a 3D 10-mile shoreline might be 10 square miles in 4D, is there a reason you're using a mile in the ana/kata dimension to arrive at your rule of thumb? My mind first went to 100 square miles, figuring the default estimate for the measurement in the ana/kata dimension would just be to match the original length in 3D.

Oops. This is why we need proofreaders.

[PatrickPowers]

Some further thoughts about body plans:

In 4D, having two perpendicular pairs of shoulders are not a problem for embracing at all. Imagine the body of a 4D person as approximately a spherinder. Reserve the front and back for ... well, the front and back, which leaves an approximately cylindrical cross-section where shoulders may be attached. So say we attach 4 arms on the corners of a square, around the rim of this cylinder. The 4D body can rotate around axis of this cylinder without changing the orientation of the front/back axis, so two embracing persons could simply rotate slightly relative to each other (22.5°) and the shoulders of one person would be in between the shoulders of the other, and they would still be chest-to-chest. No trouble at all.

You are 100% right about that. 4D people with four shoulders have no trouble embracing. The trouble is drawing them here in the 3D world. If we draw them here with two pair of perpendicular shoulders then there is no room left for a huggable chest. That's one reason I have drawings of swimmers. The huggable chest isn't missed. Only a drowning man wants to hug someone who is swimming. With people standing on dry land we mostly went with the two shoulders.

Now about beasts: 6 legs seem to be the optimal number for stability and economy. The question is, how should they be oriented? In the document it's proposed that there are two front legs and two hind legs, with two middle legs in perpendicular orientation for stability. That's an octahedral arrangement, with two opposite edges aligned to the front/back of the creatue. Other orientations are possible, though. One possibility is to align one leg to front, one to the back, and have 4 on the sides. This seems a little awkward for walking, though. Another possibility is to orient three legs in the front and three in the back, i.e., a triangular antiprism. (Alternatively, a triangular prism would also work, a non-octahedral arrangement.) The triangular prism/antiprism configuration would seem ideal for galloping, so I'd imagine that equine beasts would have this arrangement of limbs. Slow-moving animals like cows would prefer a more stable arrangement; the 2-front, 2-back, 2-middle arrangement proposed by the document seems to be a good solution.

Among living things here on 3D Earth we see an overwhelming preference for symmetrical pairs. I've gone with that. Partly it's for balance, partly its because it is far easier for a cell to divide into two copies than into three. Odd numbers are few and far between. Starfish have five limbs, that's the only exception I can think of. It appears I should mention this in the book.

Machines don't have that bias. I used the 3 in front, 3 in back arrangement for race cars. It seems a clear winner for speed and stability, it's just not much good for carrying groceries. Tricycles would be quadcycles, with a tetrahedral arrangement.

Another interesting possibility is 4 legs, but not in a square formation; instead, in a tetrahedral / disphenoid arrangement, with a pair of legs in the front and another in the back rotated 90° relative to the front legs. This arrangement is the minimal for 4D stability, corresponding with a 5-cell, akin to the tripod arrangement in 3D (corresponding with the tetrahedron). Furthermore, the two back legs can cross over and land ahead of the front legs (i.e., inverting the tetrahedron / disphenoid), then the front legs cross over the back legs and land ahead of the back legs again. This sort of gait seems suitable for small animals like rabbits and squirrels, and having less limbs to maintain seems better suited for small animals with simpler body structures. So I'd venture to speculate that 4D rodents would have 4 legs in a disphenoid arrangement, and walk by crossing their hind legs ahead of their front legs. Such a gait is also suitable for scurrying away from predators and angry humans. Attach some sharp claws that can hold on to a wooden surface, and they could climb trees and furniture too.

That has real possibilities. Squirrels. Yeah.

[PatrickPowers]

Enjoyed the bit about spinning pizza dough. Just wanted to say that spinning 5D pizza dough would essentially amount to setting the dough in a 4D clifford rotation, so the pizza maker would have to spin it in two orthogonal planes simultaneously. Would take some skill, but I'd wager that with practice, this could be done easily. You'd see the dough spin in a Clifford isoclinic rotation, spreading outwards along the spiralling fibres of the Hopf fibration. The resulting dough would likely retain a spiralling texture, reflecting the fibers of the Hopf fibration. It'd have one of two orientations, corresponding with the two senses that an isoclinic could happen in ("left-handed" or "right-handed"). But this being 5D, you could simply flip it over to switch between orientations, something impossible in 4D. Perhaps there'd even be a 5D tradition where pizza is always made with the toppings on the left-handed side, and pizza made in the other orientation could be considered as a sign of inferiority or amateurism. Or perhaps there'd be a competition between the two schools of pizza-making, with one school insisting that left-handed pizza tastes better and the other school countering that right-handed pizza is more aesthetically-pleasing.

I got a real kick out of this one. Wild!

[PatrickPowers]

I'm not sure about the two wing thing. I will say that every time I take a walk I see little kids on Ripstiks. I figure if they can ride a two-wheeled skateboard then a 4D bird can fly with two wings.

Fixed wing aircraft though, you have made me doubt that. It took me a while but now I see what you mean about that single wing. That's a fresh concept but as you say seems too heavy. I'm starting to think the three-wing aircraft is the way to go.

[PatrickPowers]

On second thought, I think two wings would be it not matter how many dimensions. It's a lot like two legs with feet. The feet have to have extent into N-2 dimensions. Wings would work almost the same way. The wings have to curve so they have extent into the extra N-3 dimensions.

[quickfur]

[...]
Among living things here on 3D Earth we see an overwhelming preference for symmetrical pairs. I've gone with that. Partly it's for balance, partly its because it is far easier for a cell to divide into two copies than into three.
[...]

Hmm, you got me thinking there. Trigonal symmetry would seem to be unlikely, or at least, require much more complex processes than simple cell division building up to a macroscopic bilaterally symmetric structure. One possibility for N-gonally symmetry that I can think of is if a group of cells replicates itself on two sides, but the two sides are at an angle, and there's some kind of process where crowding would inhibit further replication. So starting from the initial group of cells, it'd grow two copies of itself at an angle, resulting in 3 copies in an L-shaped configuration. The two new groups would replicate again, producing a 5-membered polygonal arc. When the ends of the arc start touching and crowding each other, further replication is inhibited. The groups themselves, however, would continue growing, and pressure from cell density would force the groups into approximately regular N-gonal configuration.

But such a complex process would probably be unlikely to happen for simpler beasts, so I'd expect this to be rarer. Probably the more common arrangement would be something with bilateral symmetry, or at least, local bilateral symmetry. So something with cubical or square antiprismatic symmetry would seem more likely than something with trigonal symmetry. Having 8 limbs seem costly, though. But perhaps slow-moving creatures like cows would be OK with that; they could have shorter, stubbier legs to offset to cost of maintaining 8 limbs, and they move slowly so do no need to expend too much energy moving 8 limbs. (Besides, I'm not 100% convinced about the efficiency argument; spiders have 8 limbs and get by just fine. Things like centipedes and millipedes have a lot more limbs and don't seem hampered by the energy required to maintain them.)

One interesting possibility is the spiral segmented centipede: each segment grows two legs, but the segments are slightly rotated relative to each other, so globally along the length of the worm the legs are positioned along two intertwining spirals. Each segment on its own is unstable in 4D, but the worm is stabilized by the other segments that are oriented differently from that segment. There's also the possibility of left-handed / right-handed spiral centipedes. Perhaps only one kind grows in 4D, for analogous reasons as in 3D molecular biology, where only one stereoisomer of amino acids of two possible mirror-images exists in organisms.

For cattle, though, perhaps your 2 front + 2 back + 2 middle arrangement might actually be the most likely candidate. And squirrels and rodents would be disphenoidal. Disphenoid squirrels! Now, that's something. And perhaps there'd also be capybara, overgrown rodents that behave like cattle. They'd be disphenoidal cattle.

[PatrickPowers]

(Besides, I'm not 100% convinced about the efficiency argument; spiders have 8 limbs and get by just fine. Things like centipedes and millipedes have a lot more limbs and don't seem hampered by the energy required to maintain them.)

Yes, but here on Earth the fastest runners have four limbs. The more limbs beyond four, the more slowly the creature can run, with millipedes being particularly slow. The more limbs you have the smaller they have to be so as not to get in one another's way. Small limbs are short and hence slow.

Millipedes have maybe a hundred legs but they are very small. It's the total mass that matters when we consider the energy needed to maintain them, not the number. The angular inertia of their legs is low, so while slow their movement is efficient. This gives them an advantage as scavengers. Instead of running away they defend themselves with chemicals, as do centipedes.

Humans have two legs but the feet needed for balance add angular inertia. That slows one down. The muscles needed for balance also add mass. More angular inertia, more slow. 4D people would need longer feet and more balancing muscles. That's even more angular inertia. They would not be able to run as fast as do we.

Humans run slowly so early on they had to develop weapons to hunt and to defend themselves. Thus weapons are very basic to their psychology. I used to live in northern Michigan. There was so much interest that the supermarket had fifty different weapons magazines, more than all other magazines combined. (Printed magazines, not ammunition containers.)

The success of arachnids is puzzling. Eight legs seems like a waste. Crabs seem to share this view, having transformed two of those legs into useful claws. But spiders do all right with the full eight. That I don't understand.

[quickfur]

[...]
Millipedes have maybe a hundred legs but they are very small. It's the total mass that matters when we consider the energy needed to maintain them, not the number. The angular inertia of their legs is low, so while slow their movement is efficient.
[...]

There's also another factor at work here. The mass of a body in 3D is approximately proportional to n³, but in 4D this would be n⁴. Meaning, legs will have to be stockier in order to carry this extra weight. Or bodies will have to be smaller, in general. But if legs are stockier, that means they are also more massive and expensive to maintain. Or, there could be more legs but each leg is thinner, then the weight can be spread across more legs. I didn't work this out precisely, but I think proportionally, this works out to be more efficient than to have a few huge, heavy legs. So I'd expect that as the number of dimensions increase, things would shift in favor of having more (but thinner) limbs.

[PatrickPowers]

[...]
Millipedes have maybe a hundred legs but they are very small. It's the total mass that matters when we consider the energy needed to maintain them, not the number. The angular inertia of their legs is low, so while slow their movement is efficient.
[...]

There's also another factor at work here. The mass of a body in 3D is approximately proportional to n³, but in 4D this would be n⁴. Meaning, legs will have to be stockier in order to carry this extra weight. Or bodies will have to be smaller, in general. But if legs are stockier, that means they are also more massive and expensive to maintain. Or, there could be more legs but each leg is thinner, then the weight can be spread across more legs. I didn't work this out precisely, but I think proportionally, this works out to be more efficient than to have a few huge, heavy legs. So I'd expect that as the number of dimensions increase, things would shift in favor of having more (but thinner) limbs.

What I did in the book was figure out what would happen given two very dubious assumptions. The first is that 4D people would have the same number of atoms as we, the second that their atoms have the same radius as ours. Then a big person would be about 3 millimeters tall.

One can then also compute surface areas in units of atoms. I was surprised to find that 4D objects of the same mass have much larger surface areas than ours. So while that 3mm person might weigh 150kg, the soles of their feet might have more surface area than ours. I haven't done that calculation though so I'm not sure.

As far as airplane wings go, on second thought it seems to me that the single annular tutu wing is far superior. The shorter the wing the stiffer and lighter it is, and an annular wing would be naturally stiffer and lighter as well. The only reason our airplanes have long linear wings is because lacking a fourth dimension we have to do things that way. Long wings have more leverage, but this is a disadvantage. Air moving at hundreds of miles per hour generates so much force in a surface that little leverage is necessary. All that that excess leverage does for you is make it more likely the wing will break off.

Could birds get propulsion from a flexible tutu wing? I guess not, but a couple of sections of an annulus might be the ticket. Birds at slower speed also need more leverage. It might work out to some complicated compromise.

[quickfur]

Hmm. From what I understand, the amount of lift a wing generates is proportional to its surface area. In order for an aircraft to remain airborne, the amount of lift generated must be sufficient to counteract the effect of gravity on its body, which is proportional to its mass. So it would follow that we want to maximize surface area while minimizing body mass. To know for sure whether a tutu-wing is more efficient or a bunch of linear wings, we'd have to calculate the surface area in cubic units and compare that with the mass of the wings plus some nomimal mass for the fuselage. I'll have to sit down and work this out sometime.

[PatrickPowers]

Version 0.1 of the book is available at https://www.researchgate.net/publication/359213812_Elsewhere_Everyday_Life_On_A_Hypergeometric_Earth.

It is about 99% the same material, though massively rearranged. Now in PDF format.

Researchgate says that 33 people have read at least part of version 0.0 the book. Should I believe this? Why not. I wonder though how many got past page one.