Risk in finance and in board games

I find it endlessly fascinating how risk is treated differently in finance, and in board games.

In finance, investments with higher risk usually also have higher average payoffs.  This indicates that risk has negative value which offsets the higher payoffs.  This makes sense, because after all, we like our security.  If you want to mitigate your risk, you can buy insurance, but note that insurance has a negative average payoff; insurance companies still need to make money.

In board games, risk may have positive value.  For the record, I'm thinking of the game I play most often, Dominion, but it works for any board game that has a blend of chance and decision-making, and which has an ongoing score to keep track of who is winning.  That's because in a board game, you don't really care about your average score.  You care about your likelihood of winning.

In a game, if you take a big risk and lose big, then all that happens is you lose until the next game.  If you take a risk and win big, then you beat out your opponent and win the game.  Flipping a coin to determine whether you win or not doesn't seem like a good deal at first, because at most it gives you even chances.  But the key point is that if you are already behind in the game, then you had less than even chances to begin with.

If you're losing, risk has positive value because it evens out your chances.  If you're winning, risk has negative value also because it evens out your chances.

If you're already winning, you want to play conservatively, and you might even sacrifice some of your average score to play safer (just like in finance).  If you're already losing, you want to play riskily, and you might even sacrifice some of your average score to increase your risk.

There is a slight complication that many people often care about their average score rather than their win percentage.  Even if you lost the game, having a higher score indicates that it was a "close game".  And if you have a very low score, that might indicate that you played poorly, not that you took a big risk and happened to lose.  So even in a board game, what counts as "good play" can be subjective, as some people value high scores while other people value winning.

Tying back to finance, for some one-percenters, it's not the money that matters (since they already have enough to live well), but how much status and power they have relative to other elites.  So perhaps there is also a realm in finance where "winning" is what matters the most.

In which I am mean to trolls

Recently, I got one of my first trolls since the comments RSS feed went down.  It was rather amusing.

They were saying, asexuality is bullshit.  So I responded in a particularly mean way: I asked them to explain their opinions, and then justify them with reasoned arguments.  It's funny because your average troll doesn't know how to offer reasoned arguments for shit.  This particular troll didn't even seem to understand why it was necessary.

A little background.  Given that my blog is not that widely read, and does not have particularly active comment sections, I generally don't attract trolls.   The trolls that I do attract, are typically wayward googlers (as I tend to think of them) who found some old page in my archives on the second page of some search term I don't care about.  Usually they're just incensed by the topic and don't respond in any way to what I wrote.

Before, other people could theoretically see the trolls' comments via the "recent comments" on the sidebar or following the comments RSS feed.  There weren't many readers tracking comments in the first place, but now that these features are broken, the only person who can possibly see the trolls' comments is me, because I alone have access to the moderation queue.  If the trolls are trying to get attention, they are doing it in about the least effective way possible.

In short, the only trolls I get are the ones who are thoroughly incompetent.

There are basically four reasons why I might care about a person's opinion:

1. The person has power.
2. I care about the person.
3. The opinion is backed up by arguments.
4. The opinion is succinct, eloquent, novel, or otherwise intrinsically interesting.

In the case of incompetent rando trolls, they don't have power, I don't care about them, they are too incompetent to argue, and too tedious to be interesting.  I like to poke at them until they either make coherent arguments or realize they've been wasting their time all along.

Of course, I'm much nicer to commenters who don't appear to be trolling.  Even when commenters don't offer supporting arguments for points of disagreement, I at least tend to care about their opinions by #2.

"Tenets"

Earlier, I read a reference to the "tenets" of atheism, and though it is not at all unusual to speak of tenets, I was struck anew how strange the idea is.

I have two issues with the idea of "tenets" of a social movement.  First, I don't think there is any non-arbitrary way to define what a social movement is.  It is not the case that social movements have specific and well-defined beliefs. There is no objective answer to the question of whether any particular individual is part of the movement.

Second, I don't believe in a foundationalist worldview.  We do not start with assumptions and then build upwards.  Social issues are not mathematics.  Instead, we have a web of ideas which connect in all directions, up down left right in out.  We can easily observe that many people in a social movement share similar webs of ideas, but there is no "basis" which defines the movement.

Even Objectivism, which explicitly purports to be based on a few basic axioms ("existence exists" and other vapid statements), does not really have a basis which defines it.  If someone agreed with the axioms but disagreed with everything else, that does not make them Objectivist; if someone quibbled with the axioms, but agreed with the philosophy built on them, they're Objectivist as far as I'm concerned.

So it is true that most atheists don't believe in gods.  But there are also people who have complicated feelings about god beliefs, and yet still agree with the goals of the atheist movement or participate in atheist groups.

This is not hypothetical.  I've met plenty such people in meatspace groups, and often I have trouble remembering who exactly considers themselves "technically agnostic" or otherwise.  That they disidentify with atheism indicates a point of disagreement between me and them, but there are so many points of disagreement to speak of, and this one isn't special just because of its implications on the word "atheist".

Similarly, there are plenty of atheists who I don't consider to be part of the atheist movement or who are only distantly associated with it.

I would not speak of the tenets of a social movement.  Instead, I would speak of organizations, communities, media outlets.  I would speak of common (but not monolithic) beliefs, values, and goals.

Abusing skeptical tropes: case in point

The other day, I wrote something about the word "groupthink" and how it's often used in a rather meaningless way.  I didn't have any examples in mind, but what do you know, Ron Lindsay (head of the Center for Inquiry) provided an example just yesterday:

Unfortunately, at least in my experience, some humanists do treat certain views and principles as “sacred.” These principles appear to be adopted more out of reflex, emotion, or groupthink than evidence-based reasoning.

This also runs into what I said a couple months ago about how "sacred" is abused.  Over half of the essay appears to be abusing this sort of rhetoric, claiming his opponents are failing to question everything, blindly accepting ideological principles, using empty rhetoric etc. etc.  I could probably write a series about these sorts of tropes, but I would begin to repeat myself very quickly.

The worst part is that most of these are recognizably skeptical/atheist tools.  Part of skepticism 101 is learning about lots of logical fallacies.  Part of atheism 101 is questioning everything and rejecting faith and dogma.  The tools are meant to be applied across the board in hopes that we can better reach the correct conclusion of any argument.  But do they really work?  Or do they just lead to extraneous bloviation?

This, here, is why I all but stopped naming fallacies.  I use them to aid my thinking, but I try not to use them explicitly as argumentative shortcuts.  I am unconvinced that these skeptical tropes help me to avoid being wrong, and unconvinced that it's effective at persuading others.

It's not that fallacies and cognitive biases aren't good to know about.  They just really need some quality standards.  The practices of naming fallacies or accusing opponents of bias is not really conducive to maintaining standards. Anyway, those are my current thoughts, though I realize I'm just asserting an opinion at this time.


(Via Pharyngula.  In case you're wondering, I oppose the death penalty, but I mostly don't care because it affects a very small number of people relative to other problems with California's prison system.)

Deciding who dies in a story

If storytelling were itself a story, the main conflict would be between two beasts: Freedom and Constraints.

When it comes to action, battles, duels, and combat, they all grant a boon to Freedom.  Because, honestly, you can write a combat scene to come out almost any way that you want.  Who wins or loses, who lives or dies, often has little to do with who is more powerful or who has the greater numbers.  The Constraints are instead provided by what makes story-sense.

Sometimes, what makes story-sense also makes physical sense.  For instance, it makes sense for the greenhorn protagonist to lose at the beginning of the story, not just because they are inexperienced, but also because it sets up further conflict.

On other occasions, what makes story sense is the opposite of what makes physical sense.  One example is the law of conservation of ninjutsu (look it up on tvtropes), where the strength of an army is inversely proportional to its size.  In a story, the strength of a character is often proportional to how much we care about them, and we simply don't care about large numbers of faceless individuals.

Other examples left as exercise to the reader: Why are love interests so frequently captured or killed?  Why is the protagonist's mentor always fated to die?  Why are the first ones to die in a horror film always the hot girl, black guy, and gay guy?

There are of course other means to add Constraints.  For instance, if you really wanted to be realistic, you could have the winners determined at random.  I would call this "aleatoric storytelling", named after aleatoric music.

From XKCD

For obvious reasons, aleatoric storytelling isn't very popular, except in sports and D&D.  You'd end up with a lot of things that don't make story-sense at all, like having the protagonist die or having them achieve victory at a random point in time.

Another way is to establish a set of rules about how your universe works.  This is the path followed by Death Note and HPMOR, for instance.  On the other hand, this is usually something a writer will attempt only if they think themselves clever enough to get around the constraints.  It seems you really need that Freedom.

"Groupthink"

One word I never use against opponents is "groupthink", fundamentally because I do not know what that means.  What does groupthink look like from the outside?  What does it look like from the inside?  What are its operating principles?


Alas, these are questions without answers, because groupthink is not a thing, it does not have an essence, it does not have a definition with necessary and sufficient conditions.  It is just a word, and a rather typical one: you see the word used in a few situations, then you form a mental model of when it's most appropriate, and later you use the word yourself.

My mental model of "groupthink" is "generic insult used against large groups that uniformly disagree with me."  And as a generic insult it's fine, but it seems void of any real explanatory power.

Given its common use as an insult, we know what groupthink looks like from the outside.  However, if we wanted to actually "solve" the problem of groupthink, we'd try to appeal to the people with the most power to change it, i.e. the insiders.  So we need a model of what groupthink looks like from the inside.  If there is no way to identify it from the inside, then it hardly seems like we can hold people morally responsible for what they cannot identify.

To the extent that groupthink is a thing (which it isn't), we can offer various theories about why it is a thing:

• People who are part of a group mostly associate with each other.  There's a cognitive bias to agree with views that we perceive as common.
• We get exposed to our group's ideas more often and with more favorable framing.
• We like the people in our group, and we're more inclined to agree with people we like.
• People who disagree with the group on a particular issue are less likely to voice their opinion.
• People who disagree with the group leave the group or aren't part of it in the first place.

Each theory offers a possible way to identify groupthink as an insider.  Do I ever associate with outsiders, or read about the issues as framed by them?  For those issues on which I disagree with the group, how do I treat those opinions within the group?

On the other hand, these theories do not offer very compelling solutions.  Just because I have some biases does not mean I'm wrong about any particular issue.  There isn't necessarily anything wrong with a group selecting for particular beliefs.  And though we can partially reduce biases by transforming the way we socialize, it's not clear that this is a good tradeoff.

I offer a concrete and feasible solution.  If you're part of a movement, a community, or some other kind of group, obviously that's something that you like.  So join a second one.  Take careful note of the different norms and worldviews in the groups, and acknowledge the dissonances.

Aesthetics in the ontological argument

This is part of my series on debugging the ontological argument.

Let's take a little break from ontological arguments, and switch to math.  Consider the following proof.

Theorem: Given a square, opposite sides are parallel.
Proof:
By definition, a square is a rectangle with all equal sides
By definition, all the corners of a rectangle are right angles.
In the following figure of a rectangle, line AB and CD are cut by transversal BC.
By the consecutive interior angles theorem, AB and CD are parallel.

There's something very wrong with this proof.  Logically speaking, it's completely valid and sound, and I can't really think of a better way to prove the theorem it intends to prove.  But think about it:  Why are we only proving the theorem for squares?

The entire proof rests on the fact that squares are a special kind of rectangle. It's obvious that if we just replaced "square" with "rectangle", we could prove a more general theorem, and use fewer steps.  "Given a rectangle, its opposite sides are parallel."

This isn't an issue of logic.  It's an issue of aesthetics.  My mathematical training taught me to make the fewest assumptions necessary.  My training taught me to prove more general cases when I am able.

And that's why looking at the definitional ontological argument, I boggle.
$$\text{God is defined to have all the perfections.}\tag{1a}\label{1a}$$ $$\text{Existence is a perfection.}\tag{1b}\label{1b}$$ $$\text{If something exists by definition, then it exists.}\tag{1c}\label{1c}$$ $$\text{Therefore, God exists.}\tag{1d}\label{1d}$$
Earlier, we placed all our focus on the inference \ref{1c}, because that's the most questionable step.  But let's also take a moment to focus on the abysmal aesthetics of \ref{1a} and \ref{1b}.

The entire proof is based on the fact that God exists by definition.  Why, then, is it necessary to assume that God has all the perfections?  It is sufficient to assume that God just has one perfection--existence.¹  If \ref{1c} were a correct inference, then we could prove a more general theorem, not just the theorem about God.

This is a problem for most ontological arguments.  They define God to have all the perfections, or to be the most supreme being imaginable.  In Alvin Plantinga's version, he goes so far as to define a "maximally great being" as omnipotent, omniscient, and morally perfect in every possible world.

Any decent mathematician would look at these words and say, "Why bother?"

Florid prose, or obscurantism?

I don't know what runs through the minds of philosophers who state ontological arguments with patently extraneous assumptions.  However, the florid definitions of God do serve a purpose, whether intentional or not.

The purpose is to prove only the special case of God, and draw attention away from the more general theorem.  Like with the theorem about squares, it's as if the mathematician wants you to only think about squares, and wants you to conveniently forget that the same theorem also holds for rectangles.

If people realized that the ontological argument works equally well for a large class of objects and not just God, then they might just realize that the ontological argument proves some things which are completely ridiculous.  For example, The Flying Spaghetti Monster is the most perfect cluster of noodles imaginable.  Therefore it exists.

I've made such reductio ad absurdum arguments before, and I find that ontological argument proponents dismiss them for various reasons.  For example, they would say the Flying Spaghetti Monster is inconceivable, or that it is conceptually inconsistent.

This is all missing the point.  When I look at the ontological arguments, it's clear that all the premises about God being the greatest and most perfect being are completely extraneous.  If ontological argument proponents truly think that the ontological argument only works to prove the existence of God and no other objects, then they clearly have some unstated premises.

In the next part of the series, I will be considering Plantinga's modal ontological argument.  However, the part of the argument where he talks about omnipotence and omniscience, that part will be dismissed and erased with contempt.

Maximization arguments

I should mention, for technical completeness, that sometimes the "florid" definition of God does some work after all, creating what I call the Maximization Ontological Argument.

In the Maximization Ontological Argument, God is specifically defined as the greatest thing that is conceivable.  Then it's asserted that a thing would be greater if it also exists.  I suppose this draws on the intuitive assumption that given any ordered set of things, there must be a maximum.  Of course, this assumption is just mathematically wrong (e.g. take the set of all integers, or the set of all irrational numbers less than one).

Even putting aside the fact that not all ordered sets have maximums, you can't simply assert things about a maximum on the basis that it would make it the maximum greater.  For example, consider the greatest even number less than 10.  Your first guess might be that the answer is 8, but consider that any number would be greater if you added 3 to it.  Therefore, 11 is the greatest even number less than 10.  I feel this argument is just flatly invalid, and doesn't really dignify in-depth discussion, so I'm going to move on.

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1. It's also worth noting that the extraneous assumption actively makes the proof weaker, because now we need the additional assumption that existence is a perfection.  

Review of Harry Potter and the Methods of Rationality

Harry Potter and the Methods of Rationality (HPMOR) is one of the best-known pieces of fanfiction ever written, meaning it was even read by people like me, who otherwise despise fanfic.  This is my (spoiler-free) review.

I should begin with the caveat that I hardly remember most of HPMOR.  Like much of internet fiction, it has updated very slowly over a long period of time.  I started reading HPMOR over three years ago, and I know because there's something in my blog archives about it.  Frankly it would have been better suited to reading over a short period of time rather than a period of years.  But this is hardly relevant now, because the fanfic has now been completed and you can read it at your leisure.

HPMOR begins as a light-hearted parody of Harry Potter as well as a tutorial on rationalist ideas.  It takes place in an alternate universe where Harry Potter is an incredible genius.  Rather than unquestioningly accepting the magical world revealed to him, Harry applies scientific thinking to it, revealing many absurdities.  Over and over again, he has realizations that apparently nobody else in the history of the wizarding world has thought to consider.

In these early chapters, Harry is obviously an author insert and a Mary Sue.  Despite being 11 years old, he can do no wrong.  Nonetheless, if you simply accept the premise that Harry is unrealistically smart, the same way we accept the premise that there are wizards, HPMOR takes that and goes interesting places with it.  It's okay for Harry to be really smart, and the rest of the world to be really stupid, because we get a lot of laughs, and the rationalist lessons are effective.

As the story moves on, it becomes more serious and enters a thriller cycle. Harry repeatedly gets into impossible predicaments, and the joy is in finding out how he gets himself out.  The rationalist themes also become more mature.  For example, one set of chapters was on the theme of taboo tradeoffs, such as making trades where lives are on the line.  Rather than Harry didactically delivering lessons, he has arguments with other characters, and it's not always clear who is right.  Although at times one suspects that the author still believes Harry is always right.

I consider this gradual maturation to be one of the most appropriate characteristics of HPMOR.  It mirrors the intellectual development of someone who has encountered rationalist/skeptical ideas for the first time (as many in fact do when reading HPMOR).  At first, it's exciting to learn about all these fallacies and cognitive biases.  Everything seems so straightforward, and everyone else seems blind.  But then over time you realize, stuff is complicated, and maybe you don't really know what you thought you knew!

Now for the bad stuff: Wizard battles.  These battles are deliberately riffing on the part of Ender's Game where all the kids at the military school have team battles in zero G.  But as I saw it, the point of those battles was to distract all the kids with pointless masculinity contests as a twisted way to turn kids into military generals.  None of the details of the battles actually mattered.

The wizard battles in HPMOR read like someone who adored Ender's Game for all those details.  The battles occur repeatedly, and every time as multi-chapter epics.  It was lots and lots of tactical details, with hardly any thematic content except the glorification of competition.

The emptiness of these sections I felt also poisoned the rest of the fic, as it became clear that none of Harry's trials really matter.  So Harry plays escape artist by transfiguration again by transfiguring X into Y.  So what?  What do I get out of it?  (You might be starting to see why I despise fanfic.)

I have very little connection to other readers of HPMOR, but my sense is that many fans were disappointed with the ending.  Why?  According to Hallquist, the particular way Harry gets out of the final predicament is unsatisfying because it involves the evil overlord behaving like a typical evil overlord (i.e. stupidly).

This can be seen as a failure of HPMOR to do what it was trying so hard to do.  But I feel that only highlights how little I cared about what it was trying to do.  The tactical details of how Harry wins in the end is not important to me at all.  That some of the "smart" characters sometimes behave stupidly is not at all surprising, and if anything, it should have happened far more frequently.

Thematically speaking, I felt the ending had a lot going for it.  As you may know, one of the fatal character flaws of the canonical Voldemort is that he utterly fears death.  The author of HPMOR, Eliezer Yudkowsky, is also known for fearing death, and is entirely serious about advocating immortality technology, such as cryogenics.  This leads to some good dialogue between HPMOR and the canon about death.

I also like how Voldemort is portrayed as a dark reflection of Harry's rationalism, with all the intelligence but without the morals.  This seems like the most fitting end to a fic about rationality.

Disqus and Blogger hate each other

In bloggy news, it seems that Disqus and Blogger have stopped syncing with each other.  That means that the comments RSS feed will not work, nor will the "most recent comments" on the sidebar.

If my metrics are to be believed, this affects about three people, but one of those people is me!  Now I have to track comments by other less convenient means.  I may not see or respond to comments consistently, which I suppose should provide even less incentives to trolls, although the trolls I get don't read my front page anyway.

Apparently, the problem was caused by a recent Blogger update which killed a feature that Disqus uses to sync comments.  Fuck you, Google, next time I make a blog I'm using different software.  Disqus is looking for solutions but I have no idea if one will be found soon.

Feedback on Feedly plug

Last month I was talking about selling RSS to people on Tumblr, so that they might follow things that aren't on Tumblr.  This is a fairly small project.  I'll simply make a post on Tumblr, and hope that it reaches a lot of people.

Here is a draft of the post.  I would appreciate feedback, particularly from people who are new to Feedly.  Since I've been using Feedly for years, I don't have a good sense of what might be confusing.

Follow Aces in All the Places!

Isn't it great that Tumblr lets you follow all your friends in one place?  Don't you wish you could also follow Wordpress, Youtube, and webcomics in the same place?

There is a way!

Try Feedly.  It's a place where you can gather subscriptions to all kinds of websites.  You'll need to create a new account, or connect to your Google account, but it's well worth it.  (Also see iOS and Android apps.)

FAQ Below
[cut]

How do I add subscriptions?

On the left bar of Feedly, click on "add content".  Type the url of the website you want to follow.  Then click on the + button to subscribe.  It's that simple!

Alternatively, on the website you can look for a link with the word "RSS" or "subscribe", and often the link will let you subscribe on Feedly.

It's too hard to add all these subscriptions one by one!

It's easy to import all the Tumblrs you follow.  Go here and download the OPML file.  On the left bar of Feedly, click on "Organize".  Then click on "Import OPML", and give Feedly the file you just downloaded.

I don't even know who to subscribe to!

Here are a few startup packages.  Any of these can be imported by going to the left bar of Feedly, clicking on "Organize" and then "Import OPML".

Terrific Tumblr Aces (thanks to Arf)
Wonderful Wordpress Aces
Yes! Youtube Aces (thanks to Ivy)
Webcomics with Ace Characters
All of the above!

I don't like how everything's mixed together!

On the "Organize" page, you can create separate "collections" for different kinds of things.  Then when you browse Feedly, you can read one collection at a time, or everything together.  For instance, I keep separate a separate collection of websites that I mostly just skim.

I don't like some of those subscriptions!

On the "Organize" page, you can unsubscribe to subscriptions by clicking on the x next to it.

Feedly is ugly!

Feedly's appearance is more customizable than Tumblr!  On the left bar, go to "preferences" and try a different default view.  The options are "Titles only", "Magazine", "Cards", and "Full Articles".

Additionally, on the left bar you can try looking at "Themes" to change the colors.

Feedly isn't showing old articles!

That's because Feedly won't show articles you've already read.  You can change this in the preferences.  Turn off "Hide read posts".

Feedly isn't showing the entire article! And how do I reblog stuff?

If you click on the title of an expanded article on Feedly, it will take you to the original page.  You can reblog from there.

I use Tumblr Savior or XKit.  Can I use it on Feedly?

For Feedly, you can try an add-on called SPOI Filter.  I don't think you can block based on tags, unfortunately.

How do I follow Tumblr tags, reblogs, and asks?

I'm afraid you can't, except on Tumblr.  No one ever said you should stop looking at Tumblr.  It's just that if you also use Feedly, you can easily follow things outside of Tumblr in the same place.

Wow!  How does it work?

Feedly is an RSS reader.  RSS stands for "really simple syndication."

You can search for RSS readers and you will find many others with the same function.  Some people have suggested Bloglovin, NetNewsWire (for Macs), or Newsify (a Feedly app for mobile).

Many people consider RSS to be old fashioned, and it is.  But it's practical!  One of the reasons Tumblr is great is because it borrowed ideas from RSS, simplifying it for the masses.  My only complaint is that Tumblr is a bit selfish, and only lets you follow Tumblrs.  That's why I'm advocating RSS.

I hate Feedly!  But I want those startup packages.

Well I tried.  You can find the startup packages here [link to be added].  You'll have to follow them one by one.

And here are the startup packages I'm using:

Terrific Tumblr Aces

Concept Awesome

Notes of an Ace Muslim

What makes you Acey?

Asexuals involved in BDSM

Everyday Ignorance

Queer As Cat

Resources for Ace Survivors

Tristifere

Draw from the Deck

Rotten Zucchinis

Ace Admiral

Demi Gray Speaks

Becoming a Person
TL;DR

Wonderful Wordpress Aces

The Asexual Agenda
The Asexual Census Blog
Asexuality Archive
Prismatic Entanglements
The Ace Theist
The Notes Which Do Not Fit
A Life Unexamined
Acing History
Next Step: Cake
Cinderace blogs
Critique of Popular Reason
Cake at the Fortress
YAPBNWECA
The Minus Roots
From Fandom to Family: Sharing my many thoughts
FISTFELT
Asexual Artists

Yes! YouTube Aces

Swankivy
AmeliaAce
Aces Wild
QueerAsCat

Webcomics with Ace Characters

Shades of A
Supernormal Step
Ignition Zero
David Doesn't Get It
Girls With Slingshots

 Since I'm clearly such a Feedly shill, do you think I could get them to pay me anything?

ETA: This project is done.  See here.

Five Intersecting Tetrahedra

Five Intersecting Tetrahedra by Thomas Hull.  Click for a larger version.

Today I bring one of the most inspiring origami models ever created.  It's Five Intersecting Tetrahedra, and the instructions are online.  It consists of five tetrahedra which interlock in a symmetrical way.  Thomas Hull says the following:

People's first reaction, when being shown the object, is usually to stop and stare at it for a few hours in fascination. Try it!

Don't you think that's odd?  How do we make a symmetrical figure with five tetrahedra?  Five, of all numbers?  A tetrahedron has four faces, four vertices, and six edges.  So where does the five come  in?

If you count all the vertices in the model, there are 5 * 4 = 20.  Now that's a more reasonable number, and it's the same as the number of vertices in a dodecahedron.  Indeed, the vertices of this model are arranged the same way as they are for a dodecahedron.  But still, it's strange that you can divide up the 20 vertices in this particular way.

The Five Intersecting Tetrahedra also has some profound things to say about group theory.

For one, this model single-handedly proves that the A₅ group is isomorphic to the group of dodecahedral rotations.  As you can see in the photo, the tetrahedra have five different colors.  Each of the "faces" of the dodecahedron looks like a five pointed star with five colors.  If you look at the way that the colors of the star are ordered, you find that every even permutation of colors is realized somewhere in the model.  So rotating the dodecahedron is equivalent to making an even permutation.

Incidentally, the A₅ group is really important in Galois theory.  Certain properties of the group prevent there from being any general algebraic solution to quintic equations!

And here's another thing.  In group theory, there's the idea of a quotient group.  For instance, say that we start with the group of all 3D rotations (the SO(3) group).  Also consider the smaller group of all 3D rotations that leave a regular tetrahedron unchanged (the T_d group).  The group of all unique rotations of a tetrahedron is the quotient group SO(3)/T_d.

The Five Intersecting Tetrahedra basically consists of five points in the SO(3)/T_d group.  I have a hard time wrapping my head around this group, but apparently it is possible to select five points in the group such that they obey some sort of symmetry.

This is an inspiring way to look at modular origami shapes.  Most shapes merely consist of points arranged symmetrically in the SO(3) group, but we can also consider quotient groups.  Upon reflection, I realize that the planar models really exist within RP³ (which is the quotient group SO(3)/Z₂), which is what makes those models cool.

Anyway, yeah, maths.