Legs in more than 3 dimensions

Importance: 2
Confidence level: 6
Mathematical: 6
Length: 2100 words

Cw: no mention of Cthulhu


The last post discussed why 3 is a terrible number of legs for a species to have, contra the Idirans in the prominent sci-fi novel Consider Phlebas. The physicist’s obvious next question is how we generalize expected number of legs to higher dimensions.

(Caveat 1: if inclined, stop and think about this for a bit now; untouched puzzles don’t come around all that often, especially this more-conceptual and less-mathematical kind. If you are the opposite and want only the answer, the abstract is at the bottom.)

(Caveat 2: there are usually reasons why life could not exist with very different physical parameters than we have, called the fine-tuning paradox in physics and heavily debated. This extends to number of spatial dimensions. So don’t assume life could actually exist in these other dimensions: this is a purely theoretical exercise.)

On the surface of a planet in 3-dimensional space, gravity pulls in one dimension through a 2-dimensional plane (the ground). A 2-d plane is defined by 3 points (a 1-d line defined by 2 points, and a 0-d point defined by 1 point, so we can say that an N-dimensional plane is defined by N+1 points). For an animal to stand straight up, it needs to keep itself on the defined plane of the ground, which means 3 points of contact. This is pretty close to the definition of stability, but we need an extra leg to move (for reasons explained in the Idiran post). For D-dimensional space, if we still imagine gravity as pointing in a single direction (which physically it should), we then have defined a plane with D-1 dimensions needing D legs for stability and D+1 legs on the most prevalent animals.

If you are checking me, you’ll notice that this gives 4 legs for our world, a prescient model given the number of earthly quadrupeds. However, strange life forms abound on earth: snakes have 0 legs, sea creatures have as many as they want to, some creatures—I kid you not—fly through gas, and humans themselves are bipedal. For simplicity, I generally ignore these. Gas- and liquid-dwellers have numerous forms, snakes I deal with in the comments, and humans are strange. We have freed up two of our legs to become arms, and might conjecture that the first “intelligent” or “environment-changing” organism would for this reason have (some?) arms and thus (a few?) less legs than the norm for the dimension. However, note that humans do touch their heels to the ground to establish third and fourth points of contact. Now try standing on tiptoe.

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Tripping on Tripedalism

Importance: 3
Confidence level: 6
Length: 1400 words


The Idirans are a powerful race from Consider Phlebas, the first book of Iain Banks’ “Culture” series. Each is very strong, several times larger than a human, and is tripedal. A creative choice, but three legs is the very worst number.

In the most basic sense, to make a free-standing structure stable requires at least 3 points of contact on the ground (if you try to balance a V upside down on its two points, it would just fall down to the side; this is why humans put cameras on tripods). But to move, an organism needs to be able to lift up at least one of its legs. Then four legs should be the minimum for most land-dwelling species.

Like a typical physicist, I’ve elided over several of the messier engineering aspects here. Some animals do not use four legs: humans and pangolins, the two most awesome species on the planet, both walk on two legs. They do this by actually putting their heelbone on the ground, in contrast to most mammals that walk on their toes (hooves are toes, etc). Doing this essentially provides them four points of contact for stability. Other animals use four or more legs to walk but move at speed on two, like kangaroos, lizards, and cockroaches, both in real life and in your nightmares from now on. Birds use two legs but frequently have toes pointing backwards, and also suck at walking [citation needed]. In all these cases, having less than four legs either comes at a disadvantage or is somehow worked around.

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Rations for Rationalists

Importance: 4
Confidence level: 8
Length: 1600 words
Tl;dr: Everyone is pretty overconfident about their pet nutrition theory (even you). We also trust announced panaceas too much, whether it’s a new diet today or bloodletting 2000 years ago.

I recently read The China Study, a book on nutrition. Or, rather, a bit over half. I stopped after a few hours because he was starting to repeat the narrative over and over, and I wanted to look at rebuttals before I put in the time to finish the book. I should clarify, at this point the book sounded pretty convincing.

The go-to rebuttal was written by Denise Minger, and took her a month to write and almost that long for me to read (over an hour). After reading it, it too sounded pretty convincing.

I aired my thoughts to a family friend, and his defense of The China Study sounded pretty convincing too.

This is not a healthy pattern. Luckily, at each point, I realized what was going on. I had gone into this endeavor with a highly, almost radically skeptical mindset, and it paid off. But many times, I and others aren’t so lucky.

One of my housemates sometimes complains that he believes too many things he reads. I completely empathize; humans evolved such big brains largely to make very compelling arguments. Nearly everyone disseminating information to us has an incentive to exaggerate as much as they can get away with, be it reporters, peddlers, researchers, or Donald Trump. Even researchers, whom we usually think of as trustworthy in cases other than climate change, are still motivated to exaggerate: if they can sway their field or the public to think their work more important than it is, it means more citations or media attention for them. Much of this is because good hypotheses are hard to come by; if a compelling one presents itself it may be your only chance for fame, and you certainly don’t want to pass that up because the data is unlucky. From HPMOR, ch. 78:

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Beyond News

Importance: 8
Confidence level: 6
Length: 3000 words

Tl;dr: most people are not wary enough of news’ negatives, are a little too nonchalant (not chalant enough?) about the material they choose to spend so many hours of their life reading, and may be lacking in good alternative sources.


True to the tagline, this blog is concerned heavily with learning about the world in a more effective, efficient manner. There are many good sources of information in the world; oddly, mainstream news, tasked solely with being a good source of information, is not one of them.

I originally wrote this as an invective against news before I realized that had been done already. No need to clutter the internet by repeating the argument again. But some things still need clearing up.

Dobelli’s conclusion is that you should quit the news “cold turkey.” But I think there is something to be said for searching for a replacement; surely something out there is better than chaff. Before we dive into improvements though, we best follow the rationalist’s tenet of checking his arguments for correctness. I broadly agree with him, but ultimately find a somewhat softer conclusion dependent on reader to be in order.

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Arche

“He was born with a gift of laughter and a sense that the world was mad.” —Raphael Sabatini, opening line of Scaramouche (1921)

On the other hand, this blog shall be born middling in laughter and with a sense the world is almost too sane.

Laughter is great. Wonder at the world is great. But these things are bombarding us daily and readily accessible on the internet, while I find it much harder to find sources for piecing together how the world works. Since I abide by the law of comparative advantage, I devote this space to that quest.


This blog’s first claim is we want to understand the world.

This blog’s second claim is that viewing the world as not mad, but in fact, quite sane, will give us significant power for discovering the underlying rules that make the world go round.

While I’ll deal with the second in future posts, the first is a clear prerequisite. Luckily, the legwork has already been done for us at the nexus of the rationality blogosphere, LessWrong. If you don’t want to understand the world, I won’t push you. But if you’re on the edge, I’d strongly recommend it.

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