What is a symmetry group?

This is the first part of a series about symmetry in origami. Here I will explain what a symmetry group is through a series of examples.

This image is sourced from a video with folding instructions.

This heart illustrates one of the most basic forms of symmetry. A symmetry is a transformation that preserves the shape and orientation of the object. In this case, the transformation is a reflection. If you reflect the heart across a vertical line, you get back the same heart. But with further examples, we can see that this is not the only kind of symmetry.

Origami symmetry masterpost

When I was very young, I remember doing some math problems where I was given a shape, and asked whether there was a line of symmetry. This seemed very basic to me even at the time, and I thought that was all there was to it. But there is, in fact, much more. This has been particularly impressed upon me by my work in modular origami. For example, some of the most basic shapes I can make are the Platonic solids, which are very symmetrical indeed.

These are models I’ve folded for each of the platonic solids. From left to right, top to bottom: tetrahedron, octahedron, cube, dodecahedron, icosahedron.

Unfortunately, if you really want to understand the kind of symmetry extant in origami, you might need to take a course in advance mathematics. Specifically, this would be taught in Abstract Algebra, and even more specifically, finite group theory.

I intend to write a series explaining some of the basic concepts behind the symmetry of origami, but in a way that people can understand even without being into math. This isn’t necessary to creating or appreciating symmetrical origami, but you may find it helpful or interesting. For the readers who are into math, I hope you enjoy a more visually-oriented discussion of a topic that is typically discussed in rather abstract terms.

Articles in this series so far:
1. What is a symmetry group?

Biased Tests

[cn: Bayesian math]

Suppose that I create a test to measure suitability for a particular job. I give this test to a bunch of people, and I find that women on average perform more poorly. Does this mean that women are less suitable for the job, or does it mean that my test is biased against women?

Psychologists do this all the time. They create new tests to measure new things, and then they give the tests to a variety of different groups to observe average differences. So they have a standard statistical procedure to assess whether these tests are biased.

But I recently learned that the standard procedure is mathematically flawed. In fact, rather than producing an unbiased test, the standard procedure practically guarantees a biased test. This is an issue that causes much distress among psychometricians such as Roger Millsap.

Following Millsap, I will describe the standard method for assessing test bias, sketch a proof that it must fail, and discuss some of the consequences.

Gödel’s Second Incompleteness Theorem Explained

This is a followup to an earlier post where I talked about Gödel’s First Incompleteness Theorem. Here, I discuss the Second Incompleteness Theorem, and further implications.

Could you remind me what the theorem was?

The theorem states that a consistent formal system cannot prove its own consistency.

As previously discussed, there are a couple qualifiers. The formal system must include some amount of arithmetic, and must have a computable set of axioms.

What does consistency mean?

A system is consistent if it cannot prove any contradictions. A system is inconsistent if it can prove a contradiction.

Yes. The Explosion Principle states that if you can prove a direct contradiction, then you can prove absolutely any statement.

Here’s how the Explosion Principle works. Suppose A and not-A are both provable. Now consider statement B. “(A implies B) or (not-A implies B)” is a tautology. Since both A and not-A, that means we can prove B. Following the same procedure we can also prove not-B.

Gödel’s First Incompleteness Theorem explained

Once upon a time, mathematicians thought they would be able to prove everything. The endeavor was known as Hilbert’s Program. They would find a complete and consistent set of axioms, and on this foundation build all of mathematics. (Although to be fair, much of mathematics was already built and was to be placed upon on those foundations retroactively.) And then, if everything went well, they would generate an algorithm that could prove every statement either true or false.

To some extent, Hilbert’s Program was successful. We now have Zermelo-Fraenkel set theory, which is a solid foundation for the vast majority of mathematics. But there are two problems. First, set theory isn’t complete. Second, we can’t prove it’s consistent. And Gödel showed that these problems have no solutions.

Gödel’s First Incompleteness Theorem: No consistent formal system is complete.
Gödel’s Second Incompleteness Theorem: No consistent formal system can prove its own consistency.
(Both of these theorems have additional qualifiers that I’ll get to later.)

Here I will explain the proof for the First Incompleteness Theorem, and a few of its implications. In a later post, I will talk about the Second Incompleteness Theorem.

Kochen-Specker Theorem explained

I previously explained Bell’s Theorem, which is a “no go” theorem of quantum mechanics. In brief, Bell’s Theorem proved in 1964 that any hidden variable interpretation of quantum mechanics must be nonlocal.

Of course, you may be thinking, maybe the world just is nonlocal, and that hidden information is being passed around faster than light. Unfortunately, there’s another major theorem which makes hidden variable theories even more unpalatable. In 1966-1967, the Kochen-Specker Theorem proved that any hidden variable interpretation must be contextual.

To understand the meaning of “contextual”, suppose we have a quantum cat, and the cat has many possible states. It could be awake or asleep. It could be happy or unhappy. Or the cat could be none of those things because it is dead. Now suppose there are two possible measurements, which answer the following questions:

(1) Is the cat awake, asleep, or dead?
(2) Is the cat happy, unhappy, or dead?

This is a quantum cat, so you can only choose one of the two measurements. However, even if you can’t make both measurements experimentally, you might reasonably expect that the outcomes of the two measurements are related to each other.  Specifically, if measurement (1) would find a dead cat, then so would measurement (2), and vice versa. This assumption is called non-contextuality. This cannot be true of hidden variable interpretations of quantum mechanics! Such theories must be contextual.

Fig. 1: Cat of ambiguous state.  Credit: Visentico / Sento

Bell’s Theorem explained

In 2009, I wrote an explanation of Bell’s Theorem that could be understood by popular audiences. I wanted to repost it, but ended up rewriting it completely.

Although the predictions of quantum theory are well-understood, its interpretation is famously difficult. Because quantum mechanics only makes probabilistic predictions, many people have desired a “hidden variable” interpretation, where quantum objects have definite states, despite appearances to the contrary. But hidden variable interpretations are generally not accepted.

It is certainly possible to create a hidden variable interpretation that agrees with the all the predictions of quantum theory, and de Broglie-Bohm theory is an example of such a interpretation. However, de Broglie-Bohm has a number of unsatisfying properties. Indeed there are a few theorems that prove that any hidden variable interpretation must have unsatisfying properties.

The most important of these is Bell’s Theorem, formulated in 1964. What follows is an explanation of the thought experiment, the mathematical proof, and its implications.

The setup

Bell’s Theorem considers a particular thought experiment, in which two electrons are emitted simultaneously from a single source in opposite directions. This source emits electrons that are “entangled”, meaning that their quantum states are correlated with one another. If you perform the same measurement on both electrons, both measurements will always produce the same result.1