All posts from 2019.

Rotations and balls in even dimensions

2019 December 28

This Christmas, while many of us may be admiring the 3-dimensional balls hung on Christmas trees, let us spare a moment to consider the properties of rotations and balls in even dimensions, and how their properties are simplified by pairing up dimensions. Along the way we will find a surprising identity involving the volumes of even-dimensional balls!

The normal distribution

The first place many students encounter the dichotomy between even and odd dimensions is in statistics class, when studying the normal distribution. Let

f(x) = \exp(-x^2)

which is the unnormalized normal distribution: we can think of this as a kind of “fuzzy” one-dimensional ball. To normalize f(x) we need to calculate its “volume”:

V = \int_{-\infty}^\infty f(x)\ dx.

The “Gaussian integral” V is a famous problem because the antiderivative of f cannot be expressed in elementary form, but there is a quite clever trick for finding V without using the antiderivative. The idea is to work in two dimensions: as the two-dimensional normal distribution is rotationally symmetric, we can use the symmetry to convert to polar coordinates and solve the problem immediately. We demonstrate this below, where we let x = r \cos \theta and y = r \sin \theta:

V^2 &= \int_{y = -\infty}^\infty \int_{x = -\infty}^\infty \exp(-x^2) \exp(-y^2)\ dx\ dy \\
&= \int_{r = 0}^\infty \int_{\theta = 0}^{2 \pi} \exp(-r^2) r\ d\theta\ dr \\
&= 2\pi \in...

Here the factor of r appears because it is the Jacobian of the coordinate transformation from Cartesian coordinates to polar coordinates: a “unit” cell with dimensions dr by d\theta in polar coordinates has an area of r. Formally,

\begin{vmatrix} \frac{dx}{dr} & \frac{dy}{dr} \\ \frac{dx}{d\theta} & \frac{dy}{d\theta} \end{vmatrix}
= \begin{vmatrix} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{vmatrix}
= ...

Thus we find that V^2 = \pi or V = \sqrt{\pi}. This proof is originally due to Poisson, although remarkably it has been proven that the method shown above (of multiplying a definite integral by itself) is totally useless for integrating any other function. This technique is also known in computer science as the Box-Muller transform, where it is commonly used to generate random numbers from a normal distribution: such numbers are generated in pairs, rather than one at a time.

We will see this several more times, where the mathematics can be simplified by taking dimensions in pairs instead of singly. The most obvious example of this – complex numbers – will be discussed near the very end.

Rotations in n dimensions

We have a very strong intuition for rotations in three or fewer dimensions which will not guide us well as we look at higher dimensions.

In one dimension, rotation is not possible: we think of a line as stiff against any possible internal rotation. A creature in one dimension cannot turn around without leaving the line, and is stuck pointing in one direction forever.

In two and three dimensions, a rotation goes around either a point or a line (the “pole”) by a particular angle. If you were to continuously and steadily rotate an object, then after some time (the “rotation period”) it would return to its starting orientation. This is not true above three dimensions!

Fundamentally, the simplest possible rotation takes place in a plane. In three dimensions, you can’t rotate in two different planes at the same time because those planes would intersect in a line, and rotations within a line are impossible. Thus rotations in three dimensions are essentially the same as rotations in two dimensions, except with an extra dimension (the pole) which doesn’t do anything.

In four or more dimensions it is now possible to combine more than one of these planar rotations at the same time. Thus a single rotation could involve one plane rotating by \alpha and another plane rotating by \beta. If you smoothly and steadily rotated a four-dimensional object, and chose rotation rates that were irrational multiples of each other, then the object would never return to its starting orientation.

Again, rotations in five dimensions are much like rotations in four dimensions with an extra dimension which is fixed.

In general, in 2n dimensions, for any given rotation R there exists an orthonormal basis (e_1, \ldots, e_{2n}) such that R = R_1(\alpha_1) \cdots R_n(\alpha_n) where R_i(\alpha) rotates the plane \langle e_{2i - 1}, e_{2i} \rangle by the angle \alpha. That is, any rotation in 2n dimensions is a composition of n rotations, each of which acts on only two dimensions while leaving the others fixed. These rotations R_i(\alpha) all commute with each other.

(In fact, letting the \alpha vary, the group generated by the R_i(\alpha) is isomorphic to (\mathbb R / \mathbb Z)^n, and is a maximal torus of the special orthogonal group \text{SO}(2n) of all rotations in 2n dimensions. In particular, it is a maximal commutative subgroup of \text{SO}(2n): there is no set of n + 1 independent rotations in \text{SO}(2n) which commute.)

Another way to express this same concept is that in 2n dimensions, any rotation matrix is similar to a block diagonal matrix where the blocks are 2 by 2 matrices of the form:

 \begin{pmatrix} \cos(\alpha) & \sin(\alpha) \\ -\sin(\alpha) & \cos(\alpha) \end{pmatrix}

This can be proven by noting that the eigenvalues of a rotation all have magnitude 1 (since they are volume-preserving) and come in complex conjugate pairs; so we can diagonalize the matrix in complex numbers and then make pairs of conjugate complex numbers into 2 by 2 blocks of real numbers of the above form.

What happens in 2n + 1 dimensions? Well, it is exactly the same, except the orthonormal basis has an extra basis element e_{2n + 1} which is fixed by R.

Crystallographic symmetry

A remarkable application of the mathematical theory of rotations arises in the field of crystallography, which curiously stradles the boundary of physics and math. A crystal is a solid object made of a small number of different types of building blocks which are arranged with no “disorder”; that is, their arrangement repeats in a regular way.

(“Disorder” refers to the entropy of the system, and the third law of thermodynamics is often stated that the entropy of a perfect crystal goes to zero as temperature goes to absolute zero. The most common form of ice found on Earth, called icosahedral ice or ice Ih, is actually not a perfect crystal because of the disordered nature of the hydrogen bonds which gives it nonzero entropy at absolute zero. As icosahedral ice is cooled, it can form ice XI, in which the hydrogen bonds are regular.)

The repetitive structure of a crystal is a lattice. Real-world crystal lattices often have some symmetry: for example, table salt has cubic symmetry, which is why chunks of salt tend to come in rectangular shapes. Ice has hexagonal symmetry, causing the six-sided shape of snow flake crystals. However, until 2007, no natural crystal had ever been found with a rotational symmetry other than 2, 3, 4, or 6.

This is a consequence of the crystallographic restriction theorem, which describes precisely in which dimensions it is possible to find a lattice with a specified rotational symmetry. Suppose we want to know in what dimensions there is m-fold rotational symmetry. We factor m into distinct primes as

 m = 2^{a_1} p_2^{a_2} p_3^{a_3} \cdots p_k^{a_k}

with a_i > 0 (except possibly for a_1); then the minimum dimension \psi(m) with m-fold rotational symmetry is

 \psi(m) = [a_1 > 1] 2^{a_1 - 1} + \sum (p_i^{a_i} - p_i^{a_i - 1}),

where the first term is 2^{a_1 - 1} if a_1 > 1 and 0 otherwise (this is Iverson bracket notation). Thus, \psi(m) is defined identically to the Euler totient function \phi(m) except with a sum instead of a product, and a special case for when m is twice an odd number.

(Why a sum instead of product? If you have two rotations with coprime orders that can be performed in n_1 and n_2 dimensions, then in n_1 + n_2 dimensions you can perform the first rotation in the first n_1 dimensions and similarly the second rotation in the other n_2 dimensions, resulting in a rotation whose order is the product of their orders.)

For example, \psi(5) = \psi(8) = 4 show that five-fold rotations and eight-fold rotations first arise in four dimensions, and \psi(7) = 6 shows that seven-fold rotations first arise in six dimensions.

In fact, one sees that each of the terms 2^{a_1 - 1} and p_i^{a_i} - p_i^{a_i - 1} are even, so their sum \psi(m) is always even. Therefore, lattice symmetries of order m first arise in an even dimension, and an odd-dimensional space doesn’t have any new lattice symmetries that were absent from the previous dimension.

A nascent development in crystallography is the discovery of quasicrystals, which are crystals that violate the crystallographic restriction theorem. These quasicrystals are described by quasilattices, which are not mathematical lattices, but share some of the same properties. Quasilattices are formed by taking an imperfect slice of a higher-dimensional lattice: in this way they acquire forbidden symmetries by taking them from a higher dimension, but the slicing process gives them a sort of regular pattern of imperfections that make them not a true lattice. The most popular example of such a quasilattice is the Penrose tiling:

The Penrose tiling. It has five-fold symmetry, which is forbidden in two dimensions for regular lattices. Image from wikipedia.

The first quasicrystal to be made in the lab was in 1982. The first naturally discovered quasicrystal was in a meteorite in 2007. The only commercial application of quasicrystals so far has been as a non-stick frying pan coating.

A Ho-Mg-Zn quasicrystal. Image from wikipedia.

Volumes of balls

Recall the definition of the unit n-ball,

 B(n) = \{ (x_1, \ldots, x_n) \mid x_1^2 + \cdots + x_n^2 < 1 \}

and let V(n) be its volume. Certainly the n-ball with radius r has volume V(n) r^n.

We can find V(n) as an integral iterated n times, integrating once over each coordinate, which can be readily solved using induction. Surprisingly (or not?), the induction is easiest when inducting by two dimensions at a time.

Given V(n), we aim to calculate V(n + 2). Let u = x_{n + 1}, v = x_{n + 2} be the two new coordinates, and let r^2 = u^2 + v^2 and r^2 + s^2 = 1 so that B(n + 2) consists of the points where x_1^2 + \cdots + x_n^2 < s^2. Then we have

V(n + 2) &= \int_{r^2 < 1} \left[ \int_{x_1^2 + \cdots + x_n^2 < s^2}\ dx_1 \cdots dx_n \right]\ du\ dv \\
&= \int_{r^2 < 1} V(n) s^n \ du\ dv \\
&= V(n) \int_{r = 0}^1 \int_{\thet...

where we converted from Cartesian (u, v) coordinates to polar coordinates by u = r \cos \theta, v = r \sin \theta as before. Now if r^2 + s^2 = 1, we have r\ dr + s\ ds = 0, so

V(n + 2) &= V(n) 2 \pi \int_{s = 0}^1 s^{n + 1}\ ds \\
&= \frac {2 \pi}{n + 2} V(n).

Since V(0) = 1, it follows that

 V(2n) = \frac {\pi^n}{n!}.

Thus the volume of the n-ball decreases faster than exponentially for large n, and so the sum of the volumes of the even-dimensional balls must converge. We find that

 \sum V(2n) = \sum \frac {\pi^n}{n!} = e^\pi,

and in fact if we use balls with radius r instead of unit balls, the sum of those volumes is e^{\pi r^2}. As a special case, when we use the radius \frac 1{\sqrt \pi}, the sum of the volumes is e!

For completeness sake, we can use V(1) = 2 to find

 V(2n + 1) = \frac {2 (n!) (4 \pi)^n}{(2n + 1)!}.

Why even dimensions? (Gaussian integral and volumes of n-balls)

The common cause for each of these phenomena is the repeated use of 2 in the exponents in several equations, such as the formula for the normal distribution and the Pythagorean formula for finding the distance a point is from the origin.

This becomes apparent by generalizing our definition of distance. In the \ell_p-norm, the distance |\mathbf x|_p of a point \mathbf x from the origin is

 |\mathbf x|_p^p = |x_1|^p + \cdots + |x_n|^p

with p = 2 giving us the ordinary \ell_2-norm. This is defined for 1 \leq p \leq \infty; in the case p = \infty we say that |\mathbf x|_\infty = \max (|x_1|, \ldots, |x_n|).

When p is an integer, we can evaluate the “Gaussian”-like integral \exp(-|x|^p) by taking the integral to the power of p (the particular case p = 1 can be evaluated directly with no special tricks), and the volume of the ball can be inductively calculated by inducting by p dimensions at a time.

We explicitly show how to do the latter calculation. Let

 B_p(n) = \{ (x_1, \ldots, x_n) \mid |x_1|^p + \cdots + |x_n|^p < 1 \},

the definition of the n-ball in the \ell_p-norm. Let V_p(n) be its volume. As before, we calculate V_p(n + p) from V_p(n): let x_1, \ldots, x_n, y_1, \ldots, y_p be the coordinates in n + p dimensions, with

 r^p = |y_1|^p + \cdots + |y_p|^p

and r^p + s^p = 1. Then, as before

 V_p(n + p) = \int_{r = 0}^1 V_p(n) s^n\ dy_1 \cdots dy_p.

Now the coordinate change is a little messier, because we are no longer using polar coordinates but rather converting to coordinates for the p-ball in the \ell_p-norm. Let A_p(n) be the surface area of the n-ball in the \ell_p-norm, so that

 A_p(n) = \frac d{dr} \left[ V_p(n) r^n \right] = n V_p(n) r^{n - 1}.

Then we have

V_p(n + p) &= V_p(n) \int_{r = 0}^1 s^n\ dy_1 \cdots dy_p \\
&= V_p(n) \int_{r = 0}^1 s^n A_p(p)\ dr \\
&= p V_p(p) V_p(n) \int_{r = 0}^1 s^n r^{p - 1}\ dr \\
&= p V_p(p) V_p(n) \i...

Thus in the same way we have a recurrence from V_p(n) to V_p(n + p) when using the \ell_p-norm. To actually use this, we need the value of V_p(p). I was unable to easily make progress computing this, but found a paper (Volumes of generalized unit balls) with the result:

 V_p(p) = (2 \Gamma(1 + 1/p))^p.

(The paper calculates V_p(n) by inducting over n one at a time. The resulting integrals are rather messy compared to the above, but with suitable changes of variable can be made into gamma functions.)

Combining with the inductive relationship, we get

 V_p(np) = \frac 1{n!} (2 \Gamma(1 + 1/p))^{np}.

Why even dimensions? (Rotations and crystallographic symmetries)

As lattice symmetries, and thus the crystallographic restriction theorem, are consequences of the properties of rotations in n dimensions, this leaves us only with the question of why pairs of numbers are important to rotations.

Rotations are isometries: they preserve distance. (They are also orientation-preserving, that is, don’t make mirror images, but that is not important here). “Distance” here means, as one expects, the ordinary distance in the \ell_2-norm.

Since rotations are defined with respect to the \ell_2-norm, it is natural that the pairing up of dimensions is important to rotations. However, unlike the case with volumes of n-balls, there does not seem to be any sensible way to define rotations with respect to the \ell_p-norm for p \neq 2. Such a space only has trivial isometries (those that permute or negate the coordinates), and it seems that any non-trivial way of creating isometries in \mathbb R^n essentially amounts to partially using an \ell_2-norm and getting some subset of the usual isometries.

Now what makes the \ell_2-norm so special that it is the only way to usefully define rotations? This brings us to the glaring omission in our discussion so far: the complex numbers. Complex numbers are the clearest example in mathematics of when taking a pair of real numbers can greatly simplify a situation that is challenging for single real numbers. While the link between complex numbers and isometries is not readily apparent, as what makes complex numbers unique is their algebraic properties rather than their metric properties, I hope to elucidate what I believe is the key connection.

Consider a linear transformation T : \mathbb R^n \to \mathbb R^n. For T to be an isometry it needs to preserve the length of any element of \mathbb R^n, including in particular its eigenvectors. Thus, its eigenvalues must all have magnitude 1. The eigenvalues of T can be found from its characteristic polynomial:

 P_T(\lambda) = \det (\lambda I - T).

The roots of P_T are the eigenvalues of T. Finding the roots of a polynomial is an essentially algebraic operation, and for P_T to always have n roots requires moving from \mathbb R to its algebraic closure \mathbb C: working in the real numbers, the most we can guarantee that P_T can be factored is into terms that are quadratic or less. These quadratic terms correspond exactly to the planes within which T rotates.

Each eigenvalue \lambda \in \mathbb C must have magnitude 1 for T to be an isometry. Which norm should we use to calculate the magnitude of \lambda? For the finite extension \mathbb C / \mathbb R there is only one natural choice of norm, given by taking the product of all of the conjugates of \lambda. If \lambda = a + bi, we have

 N(\lambda) = \lambda \lambda^* = (a + bi) (a - bi) = a^2 + b^2.

In this way, we see that the \ell_2-norm is inextricably linked to the complex numbers and to isometries in \mathbb R^n.

Even if we try to move from the real numbers to another number system, we run into the Artin-Schreier Theorem: if F is a proper subfield of the complex numbers with finite index, then the index is 2 and \mathbb C = F[\sqrt{-1}], so exactly the same norm N(a + bi) = a^2 + b^2 would be used. Furthermore, any finite extension of a field in its algebraic closure has this same property and is essentially like the real numbers in the complex numbers. Finally, since \mathbb C is algebraically closed, it has no finite extensions, so we can’t build larger finite extensions of the real numbers.

Thus in a fundamental sense all finite dimensional rotations involve repeated copies of the Argand plane representation of the complex numbers, where multiplying by e^{i \theta} rotates by an angle of \theta.


Another remarkable example of simplifying rotations by moving from an odd number of dimensions to an even number is the application of quaternions to simplify the computation of rotations in three dimensions.

As we saw above, any individual rotation in three dimensions can be described as a pole which remains fixed together with an angle in the plane of rotation. If two different rotations have the same pole, they commute with each other and we can compose them by simply adding angles: this is the simple circumstance of rotations in two dimensions. However the general problem of composing rotations in three dimensions requires some thought.

The space \text{SO}(3) of rotations has three dimensions, so we can describe any rotation with three numbers, and there are many ways to do so. The most obvious ways are the axis-angle representation described above, Euler angles which describe a rotation in terms of a succession of rotations around the three coordinate axes, and Tait-Bryan angles which are another way of composing rotations around the three coordinate axes.

Besides the difficulty of computing the composition of two rotations given in the above systems, any chart representing \text{SO}(3) with three numbers cannot cover the whole space of rotations without singularities. These singularities result in gimbal lock, which is a phenomenon in which the coordinate system becomes linearly dependent, and certain degrees of freedom cannot be expressed at the singularities. In addition to being a mathematical problem, this happens to real-world gyroscopes when only three gimbals are used: the positions of the three gimbals is a representation of a rotation with three coordinates, and their exists orientations in which some of the gimbals become redundant, and the gyroscope locks up against rotation in a specific direction. This physical limitation meant that (some of) the Apollo spacecraft would lose orientation information if pointed in specific directions. In fact, when the damaged Apollo 13 was descending to Earth its astronauts were forced to jetison the lunar module (see also: part 1) prematurely because its automatic attitude adjustment was sending the command module too close to gimbal lock, which would have endangered the final descent trajectory.

A gyroscope with three gimbals. Image from wikipedia.

The clever trick to solve this problem is to use four coordinates instead of three. The unit sphere in four dimensions is a double-cover of \text{SO}(3), so that we can represent a rotation as a vector of four numbers (a, b, c, d) subject to the condition that a^2 + b^2 + c^2 + d^2 = 1; any particular rotation has exactly two such representations. The surprising feature of this representation is that if we view the vector as a quaternion a + bi + cj + dk, then composition of two rotations is exactly the same as quaternion multiplication.

Quaternions thus give another example of simplifying a problem involving rotations by moving from an odd number of dimensions to an even number. However, any deeper link between the quaternions and the examples given above has eluded me, and for now I can only explain it as a coincidence.

Addendum: reddit user AntiTwister suggested this accessible and informative article which gives the insight to explain why quaternions represent rotations and the appropriate generalization rotors (see geometric algebra) to rotations in any number of dimensions.

Mormon church discovered to fraudulently keep more than $100 billion tax-free

2019 December 20

A whistleblower has revealed that a covert tax-exempt fund of the Mormon church has a value of $100 billion, and does not meet the requirements to maintain its tax-exempt status.

Ensign Peak Advisors, Inc.

Mormon members in good standing are required to pay a “tithing” of 10% of their income to the church. (Failure to pay results in being banned from certain church services including Mormon weddings, as well as facing social stigma. Members are often required to produce financial documents to prove that they are paying sufficiently.) The excess of these tithes over the church’s operating expenses, estimated at $1-2 billion per year, has since 1997 entered the Ensign Peak Advisors, Inc. (EPA) fund, where it has accrued 7-10% return on investments to accumulate over $100 billion by 2019.

The EPA fund is a supporting organization of the Corporation of the President of Church of Jesus Christ of Latter-Day Saints (i.e., the Mormon church, organized as a corporate sole owned by the current Mormon prophet), and claims a 501c3 exemption from taxes as it purports to spend its resources on religious, educational, charitable, or other exempt purposes. Note that “investing” and “saving” are not considered exempt purposes.

However, since 1997 EPA has spent absolutely none of its money on exempt purposes: in fact it has only made outlays twice. In 2009, it spent $600 million to bail out a for-profit insurace company owned by the Mormon church, and from 2010 - 2014 it collectively spent $1.4 billion to cover cost over-runs of the construction of the for-profit City Creek Mall, also owned by the Mormon church. In particular, the $1.4 billion came directly from tithing cash that had not yet been invested, despite claims from the church that no tithing money had contributed to City Creek Mall.

More specifically, the IRS requires private foundations to spend a minimum of 5% (follow links to “distributable amount” and “minimum investment return”) of its assets on charitable purposes each year, or face penalties of 30-100% of the amount that should have been spent. EPA does not meet this requirement. (There are limited exceptions for setting aside funds for up to 60 months for large projects, but it has been 22 years since EPA’s founding, and such projects must be approved by the IRS in advance.)

EPA employees are told (among other reasons) that the purpose of the EPA money is to save for armageddon or the second coming. These are not tax-exempt purposes, and one hardly expects money to be especially relevant at such a time anyhow.


What makes EPA “covert”? It was founded in 1997 one month after a Times magazine cover story “Mormons, Inc.” about the extensive business holdings of the Mormon church, for the purpose of concealing these holdings in the future. It has a bare-bones staff of 75 people organized into teams whose data is “siloed” from each other; only 4 EPA employees are authorized to know the complete financial information. EPA has no sign in its lobby. All EPA employees are required to be tithe payers who hold “temple recommends” (i.e., obedient Mormons): until 2017, the head of EPA “would go desk-to-desk requiring employees to show the temple recommend that they were required to keep current. Now, temple-recommend checks are conducted electronically, automatically, and periodically. In addition, the COP implemented in 2019 a means for local clergy to electronically surveil and report to church headquarters the weekly church attendance and temple worthiness of church employees who attend that specific congregation.”.

BYU owns a small amount of funds managed by EPA, and until 2015 auditors of BYU accepted EPA’s refusal to allow any examination of EPA financials. At that point pressure from auditors eventually led EPA to form a separate account for assets owned by BYU (and other arms of the church) so that auditors can examine the limited account without getting access to full EPA financials. Similarly, when financial partners request financial information from EPA to comply with the Patriot Act’s disclosure requirements, EPA simply gives a letter saying that “Ensign Peak does not distribute financial statements” but assets are above “$5.0 billion”.

In 2017, EPA ended a financial relationship with a bond broker when it was realized that they were on the board of Open Stories Foundation, which had supported the Mormon Stories Podcast, which was run by a prominent exmormon John Dehlin. In 2018, one day after MormonLeaks had revealed the existence of 13 LLCs worth billions of dollars and secretly owned by the Mormon church (and which we now know were owned specifically by EPA), EPA held an internal meeting to discuss the formation of new holding companies to better conceal their ownership; it was decided that this would be too obvious if done immediately, and would be better done over several years.


The Mormon church has a very long history of fraud, starting with its founder Joseph Smith’s treasure hunting. The whistleblower recounts an early such incident, when Smith illegally started a bank and illegally printed money, which bore the mark “THE KIRTLAND SAFETY SOCIETY ANTIBANKING CO”, with “anti-” and “-ing co” written in very tiny font.

EPA continues this tradition. It submits essentially no financial information to the IRS and is thus effectively immune to oversight. It does, however, submit a form 990T to the IRS, which requires the disclosure of “Book value of all assets at end of year”. In 2007, 2008, and 2010-2015, EPA has each year claimed “$1,000,000.” as a book value, with some years prepending the word “over”; most recently in 2017, EPA has just left the field blank.

It is possible, however, that the EPA financials contain something much more serious – or rather, don’t contain. The whistleblower discovered that EPA has a deliberate policy of deleting from its records unpaid accounts receivable: that is, when investing in another entity, if the investment doesn’t come back, then to delete the missing investment from its assets. This allows payments to other entities to be concealed as loans which never get repaid. However the whistleblower has no knowledge of such a payment taking place.

Why break the law just to hoard cash?

If the whistleblower’s understanding of the church’s motives are correct, then the Mormon church does not even want the money (or so much, anyhow): it is a curse.

Certainly all this money is not benefiting the church: EPA does not spend its money, only accumulates it indefinitely. What the church (allegedly) wants is not money, but secrecy. By registering as a 501c3 organization and concealing its holdings behind fraudulent tax documents and layers of LLCs, the church is able to hide how much money it has. If EPA followed the law by paying taxes, or by charitably disbursing funds in accordance with its tax-exempt status, it would be required to report how much money it has and spends; furthermore, it would be difficult to charitably spend tens of billions of dollars a year without visibly partnering with major international secular organizations.

By concealing the value of its assets, the church is able to represent to its members that it is much poorer than it is and thus credibly demand crippling tithes and volunteer labor:

“If paying tithing means that you can’t pay for water or electricity, pay tithing. If paying tithing means that you can’t pay your rent, pay tithing. Even if paying tithing means that you don’t have enough money to feed your family, pay tithing.” - official Mormon church newsletter, 2012

The demands made on its membership are a means of control. It is therefore critical to the church that the membership do not become aware that their tithe is unnecessary. Not only does tithing exceed operating expenses by more than $1 billion per year, it is estimated that the returns on investment from EPA alone exceeds tithing: thus the Mormon church could cease collecting tithing and survive its current operating expenses in perpetuity without even drawing down on the balance of EPA.

Mormons are told to donate exclusively to the church to avoid the operating overhead of secular charitable organizations. Were the church to perform charitable work in partnership with other organizations (and how else could it spend nearly $10 billion a year?), this lie would be revealed.

When combined with the real estate holdings that EPA manages (which includes 2% of the land area of Florida, making the Mormon church the largest land owner in Florida), EPA has approximately $124 billion in funds: only 10 countries have soverign wealth funds larger than this. Would Mormons still feel beholden to the church if this were known?

(Note that EPA does not own all Mormon church assets, but only owns highly liquid assets – in fact it holds $7 billion in cash, and estimates it could liquidate 85% (!) of its holdings within 3 months. The church has many other subsideries, both non-profit and for-profit, which hold their own extensive reserve funds, together with extensive non-liquid holdings in Mormon temples and other physical property. Because many of these assets don’t go to market for many decades, it is difficult to value them, but crude estimates of the total valuation of the Mormon church go above $200 billion.)

What this means for US citizens

There are a number of ways that restitution, if it takes place, could be calculated: the whistleblower presents calculations for several such methods. What I found interesting was the calculation of the EPA’s value had it maintained the minimum annual disbursement of 5% of assets required to maintain tax-exempt status. In this scenario, it would be worth an estimated $67 billion less than it is today. (Note that such a scenario imposes no practical burden on the Mormon church: it did not use any of this $67 billion, had no plans to use it in the future, and would still be left with a $33 billion reserve that it doesn’t need.) This difference represents what it is has collectively cost the US to allow EPA to get away with failing to follow the tax-exemption requirements. Between 327 million US residents, each US resident has effectively subsidized (on average) $202 into growing EPA’s funds.

For one US resident, EPA’s fraud may be worth quite a bit more. Whistleblowers who reveal to the IRS the fraudulent omission of tax payments receive 15% (or even up to 30%) of the amount that the IRS recovers. Under reasonable estimates of the amount of taxes that EPA owes to the IRS, such a payment could make the whistleblower a billionaire – with enough money leftover to make charitable donations exceeding the Mormon church’s entire charitable work for the last 22 years.

Why can you see your breath in winter? And can you breathe clear air in hot, foggy conditions?

2019 December 05

Experience tells us to expect that exhaling into sufficiently cold, clear air creates a visible cloud. However, our experience does not guide us in the reverse question: can exhaling into sufficiently hot, cloudy air create a pocket of clear air? (I encourage you to guess before reading further!) It turns out that a subtle aspect of how water behaves in air is responsible for both of these phenomena (or their absence).

Background: evaporation and condensation of water

First, what is “foggy” or “cloudy” air? The atmosphere is a gas that contains a mixture of different types of molecules, including a variable amount of gaseous water, called water vapor. At any given temperature, there is a saturation pressure, which is the maximum water vapor pressure for that temperature. Any excess water vapor past this maximum condenses into liquid water: this can happen on solid surfaces exposed to the air (which is why water droplets appear on cold surfaces) or around tiny dust particles or other nucleation sites suspended in the air, forming microscopic water droplets we call a cloud. An air mass with more water than the saturation pressure is foggy, whereas an air mass with less water is clear.

To a first approximation, we can ignore all other constituents of air but water. The partial pressure of water vapor is the air pressure times the proportion (volumetric, or molar) of air that is water; this is the pressure that the water vapor would have if the other constituents were removed. For a given temperature and pressure, we can see on a phase diagram of water what phase of water is preferred: but in the absence of other constituents, what is providing this pressure except water vapor?

Phase diagram of water, from Wikipedia. Much more information about water, including details about its phase diagram, can be found here.

So, the way to interpret the “liquid” region of water’s phase diagram is that if the partial pressure of water is so high that the pressure reaches the liquid region, then condensation of water vapor into liquid water is favored. If the partial pressure drops until it reaches the gas region of the diagram, then evaporation of liquid water into water vapor is favored. At the boundary between these regions, which is the saturation pressure, the rate of condensation equals the rate of evaporation. The “relative humidity” is defined as the partial pressure of water divided by the saturation pressure: whenever relative humidity is less than 100%, any standing bodies of water will tend to evaporate over time. The competing processes involved in determining this boundary are the latent heat released by water condensing from gas to liquid, versus the entropically favored expansion of a gas to fill available space.

Generally speaking, air found near the surface of the Earth has moderately high relative humidity. Cooling air causes relative humidity to go up (because the amount of water remains fixed, but saturation pressure goes down) until it reaches 100% (this temperature is the “dew point”) and liquid water condenses out. So as a quick rule of thumb – though not necessarily helpful for the problem we are considering – cooling air causes condensation: for example, cold surfaces gain water droplets by cooling the air near them; rising air in the atmosphere condenses to make clouds; and visible “steam” (actually water droplets) appears above hot beverages when the hot humid air coming off the surface cools.

What makes boiling different from regular evaporation? This is the part where the non-water constituents of the air matter a lot. Consider a pot of water under standard atmospheric conditions. In the interior of the water, the pressure equals the air pressure. However, because of the physical separation of liquid water and the air, the partial pressures within the interior of the water is dominated by water, with small contributions from dissolved gases, while within the air the partial pressures reflect the amount of water vapor or other gases in the air. So evaporation within the interior of the water is only favored when the total pressure of the air is below the saturation pressure of water: when this happens, the water is boiling, and it starts evaporating from its whole volume all at once. If the partial pressure of water in the air is below the saturation pressure, but the total air pressure is above, then evaporation only takes place at the surface, and the water is not boiling.

Another way to understand boiling is that when a bubble of water vapor forms in the interior of the liquid, the pressure within the bubble equals the total air pressure outside; so boiling is only favored if saturation pressure is above total air pressure. The bubble displaces an equal volume of the surrounding air (via displacing some liquid water), but doesn’t immediately gain the entropic benefit of mixing with the surrounding air: the latter only happens if the bubble reaches the surface.

You’ve likely heard that, when exposed to a vacuum, liquid water both boils and freezes. This is true: the low air pressure causes water to boil, cooling the water until it freezes. The process continues with the solid water, which sublimates, cooling it further, and given a perfect vacuum eventually the whole mass will turn to vapor (as can be seen from the phase diagram). However sublimation mostly only happens from the surface, and at low temperatures the saturation pressure of water falls super-exponentially with temperature, so the sublimation can take a very long time.

Why can or can’t you see your breath?

Now, the resolution of our questions can be found in the phase diagram of water if we know where to look. When you exhale in winter, the warm, humid air in your lungs mixes with the cold, dry air outside. The two initial air masses are clear (there is no liquid water), but under certain conditions their mixture can go above the saturation point (causing a cloud to condense). This is the same process that happens above a hot beverage: hot air touching the beverage can carry a lot of water vapor, creating hot, humid air. When mixed with the ambient air, this can result in saturated air, which we describe as “steam” (although the visible condensate is actually made of liquid droplets).

The converse question of breathing clear air into a hot fog imagines warm, saturated air in your lungs mixing with very hot, saturated ambient air. Can mixing two saturated air masses result in an unsaturated air mass?

The answer to both of these questions can be found in the fact that the liquid-gas boundary in water’s phase diagram is concave-up. (Note that in the above diagram it looks concave-down due to the log scale of pressure.)

Let us examine this boundary a little more closely. The Clausius-Clapeyron relation describes how the saturation pressure e_s of water varies with temperature T (in Kelvin):

\frac {de_s}{dT} = \frac {L_v e_s}{R T^2}

where R is the specific gas constant of water vapor and L_v is the latent heat of vaporization of water (which itself depends on temperature). Thus, crudely speaking, e_s is roughly exponential in temperature. A high-accuracy approximation for e_s is given by Teten’s equation:

e_s(T) = 610.78 \text{ Pa } \exp \left( \frac {17.27 T}{T + 237.3} \right)

where now T is given in Celsius.

Detail in phase diagram of water; the preferred phase is liquid in the saturated region, and gas in the unsaturated region. Example air masses are marked on the diagram, representing cold, clear air in winter; air exhaled from the lungs; and hot, foggy air. The grey region represents ambient conditions where one’s breath will condense on exhalation.

A mixture of two air masses falls somewhere on the line connecting them, with the location depending on the mixing ratio. Note that any mixture of saturated air masses gives a saturated air mass, while some mixtures of unsaturated air masses can be saturated.

When two air masses mix, the resulting temperature and pressure is roughly the average of the two initial masses’ temperatures and pressure, weighted by the relative sizes of the initial masses. Thus, in a plot of temperature and pressure, the final mass should fall on a straight line connecting the initial masses (or more generally, lie in the convex hull if there are more than two initial masses). Shown are hypothetical temperatures and pressures representing cold, wintery air; air as exhaled from the lungs; and hot, foggy air.

For a healthy person, air exhaled from the lungs is saturated, with a temperature slightly below body temperature. When mixed with cold, unsaturated air, we see that the resulting air mass falls on a line that passes through the saturated region of the plot; the location on this line depends on the amount of each initial mass. A typical exhalation has a volume of about 0.5 liters; the amount of ambient air that gets mixed increases with time since exhalation, so the cloud of visible condensate increases in volume over time until so much ambient air is mixed in that it falls back below saturation pressure, and the exhalation disappears. The colder and wetter the ambient air, the more ambient air can be mixed to maintain saturated conditions, and the larger one’s visible breath is. (At cold temperatures the relative humidity of the ambient air matters very little because the ambient air contributes very little water regardless.)

The grey shaded region shows ambient conditions where mixing some ratio of exhaled air and ambient air results in saturated air. Thus it would appear that you should always see your breath at a temperature of 10 C or below, regardless of how dry the air is. However, near the boundary of the grey region, condensation can only occur if a very small amount of ambient air is mixed with your exhalation: to achieve a 10:1 mixing ratio for example, requires only 50 mL of ambient air. Furthermore, the amount of condensate that would (very briefly) form is extremely small, possibly far too little to be visible. In practice, you will only see your breath above 10 C if the air is very humid.

Hot, foggy air

Now we consider the converse question of breathing clear air into a hot fog. First, we emphasize that such conditions do not occur naturally on Earth, and are exceedingly hostile to human survival. Ordinarily it is possible for you to briefly survive high temperatures (like the dot shown in the diagram) through sweating, as the evaporation of water from the skin is endothermic. However, effective cooling through sweating requires that the ambient air be very dry – specifically, dry relative to the water’s saturation pressure at body temperature, not at the ambient temperature.

In our hypothetical, the hot air is not only wet compared to the saturation pressure at body temperature, but even to the much higher saturation pressure at ambient temperature. Instead of evaporation, water would condense quite rapidly onto any exposed skin. Exposure to 60 C water causes serious burns within approximately 5 seconds, but since condensation is exothermic, the condensed water would actually be hotter.

So, if you were to exhale in such conditions, could you breathe clear air into a fog? (We suppose that you stand upside-down to slow the accumulation of hot water in your lungs.) No. Because the saturation pressure is concave-up, there are no combinations of saturated air masses in any mixing ratios that can result in an unsaturated air mass. That is to say, foggy air mixed with foggy air will always make foggy air.

If the saturation pressure were concave-down, the reverse would be true: any clear air masses would always mix to make a clear air mass, and it would be possible to mix foggy air masses to make clear air.

Welcome to this blog! (start here)

2019 December 02

I hope you find some interesting things here. You can navigate between posts using the left and right arrow keys. Click on images to enlarge them to fullscreen. Follow RSS/Atom feed or twitter for updates.

The blog entries older than this one are items I originally posted to social media; most of these are just links with minimal commentary. So that you have no reason to go digging through the older posts, I’ve collected the most worthwhile bits and put them all in one place. You can find the most interesting links here. The most interesting non-link posts or posts too long to go into the list of links can be found below.

Calculating pi in Factorio

“2001: A Space King” and other videos I made

From the index of a book: “Bears, combats with”

Knife money

Volcanos with unusual lava

The planet Mercury

Non-zero electromagnetic effects even with zero electromagnetic field

The unreasonable effectiveness of recurrent neural networks

Lead poisoning part 1 part 2

Mormon senator discusses how he is loyal to the church before the US, and used his position to further church interests above his constituents.

Maximally symmetric 4-colorings of a pentagonal hexacontahedron

Governing the commons

Anti-nuclear hysteria

Mathematics made difficult

A little philosophy about beliefs

Markov chain text generator

Links to the past

2019 December 01

I’ve collected into one place the more interesting links I’ve previously posted, so that you have no reason to go hunting through the older posts. For slightly longer posts, see here.

Division of three for cardinals without the axiom of choice.

In 2017 a neutron star merger was observed by both gravitational and electromagnetic waves! Aside from the usual confirmations, in particular this confirmed that such mergers are the origin of short gamma ray bursts and are the origin of most heavy elements (heavier than iron). It is estimated that the gold and platinum produced was “vastly” more than the mass of the Earth.

“The worst result in a simultaneous [chess] exhibition given by a master occurred in 1951, when International Master Robert Wade gave a simultaneous exhibition against 30 Russian schoolboys, aged 14 and under. After 7 hours of play, Wade had lost 20 games and drawn the remaining 10”

Two interesting sequences: Gijswijt’s sequence contains every positive integer, but grows so slowly that the first 5 appears around term 10^{10^{23}}. The Golomb sequence grows asymptotically at a rate of \phi^{2 - \phi} n^{\phi - 1} where \phi is the golden ratio.

Near Xenon’s critical point at 58 atm and 17 C, Xenon gas has a density of 1100 kg / m^3, greater than liquid water at those conditions. So what happens if you try to float liquid water on Xenon gas at those conditions? You get a solid.

There is a layer of sodium at an altitude of ~100 km in the Earth’s atmosphere caused by meteors.

“They lifted half a city block […] and an estimated all in weight including hanging sidewalks of thirty five thousand tons. Businesses operating out of these premises were not closed down for the lifting; […] One patron was puzzled to note that the front steps leading from the street into the hotel were becoming steeper every day and that when he checked out, the windows were several feet above his head, whereas before they had been at eye level. […] the practice of putting the old multi-story, intact and furnished wooden buildings – sometimes entire rows of them en bloc – on rollers and moving them to the outskirts of town or to the suburbs was so common as to be considered nothing more than routine traffic.”

An economics paper: “Japan’s Phillips Curve Looks Like Japan”.

A data structure that takes advantage of uninitialized memory to use less time than space!

I wrote a Newton fractal generator. See animation.

Special phenomena in 4 dimensions

Tarski’s theorem about choice. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest.

A timeline.

Measurements of the gravitational constant vary sinusoidally. “Fisher’s analysis was only possible because Darwin had designed his experiment so well. In fact, Fisher was often frustrated with the quality of other people’s experiments. ‘To call in the statistician after the experiment is done,’ he said, ‘may be no more than asking him to perform a postmortem examination: he may be able to say what the experiment died of.’”

Grid cells

If a CPU cycle is one second, then accessing L1 cache is 3 seconds and accessing the disk takes 1 to 12 months.

“Viktor Klee, in a parody of the usual optimistic prophecies people like to make, wrote that: ‘…progress on this question, which has been painfully slow in the past, may be even more painfully slow in the future.’”

The third-closest known star system to the Sun was discovered only in 2013.

Let’s play… identify the city from its satellite picture! (To get the highest zoom, click on the icon in the upper right of each picture.) Total population between all the pictures is roughly 333 million people.

“Paper and Pencil: a Lightweight WYSIWYG Typesetting System” (page 88)

A review of What We Know About Climate Change by Kerry Emanuel.

Apparently x-rays are visible to the naked eye.

A remarkable collection of unique chess problems (with solutions). Problems 3, 4, 6, and 10 are easier.

A bug in “file” that made it unable to print on Tuesdays.

“An honest examination of the sky forced him to displace the center of the spiral pattern away from us by more than half the galactic radius in the direction of Cygnus, and his drawing gives the impression of a man struggling with the truth and losing.” An article about the 1920 debate on the existence of other galaxies

“Although the new calendar was much simpler than the pre-Julian calendar, the pontifices apparently misunderstood the algorithm for leap years. They added a leap day every three years, instead of every four years. According to Macrobius, the error was the result of counting inclusively, so that the four-year cycle was considered as including both the first and fourth years; perhaps the earliest recorded example of a fence post error.”

Natural nuclear reactors have occurred on Earth before, with a cycle time of about 3 hours, and lasting for hundreds of thousands of years before exhausting their fuel.

Vlad the astrophycisist performed by Peter Mulvey.

Parable of the broken window: war does not help the economy.

Death of an ant colony (fiction)

3 of 6 found this book review useful: “If you are looking for practical advice on the avoidance of large sea-faring vessels, this is not the book for you. I neglected to read the subtitle, as it was set in very small type and seemed utterly unimportant beneath the huge title. And so it was that I came to a bad pass, colliding with the tanker Condoleezza Rice some 150 miles east of the Bay of Fundy.”

“Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell’s 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world’s record for this famous theorem.”

Java 4 ever

“This phenomenon shows how, after the more complex visual system is damaged, people can use the latter visual system of their brains to guide hand movements towards an object even though they cannot see what they are reaching for.”

2019 November 17

originally posted on facebook

Here is an excellent timeline of past and future impeachment hearings in the house:

2019 November 07

originally posted on facebook

We’ve all seen how the executive branch has been gutted of qualified people from the top-down, with agency heads vowing to destroy their own agencies and senior-level officials with decades of experience being fired or quitting. Less visible, however, is that this gutting has extended deep into the executive branch to employees usually far removed from politicking.

As an example, here is a scientist at the USDA’s Economic Research Service explaining how their agency was relocated from DC to Missouri in retaliation for their scientific research (eg, their findings that food assistance is beneficial economically and that the 2017 tax cut disproportionately benefited wealthy farmers). This relocation caused more than 2/3 of the agency to quit, with whole research teams disappearing.

Below is video of Mick Mulvaney admitting that the goal of the relocation was to make people quit.

Washington Post


2019 October 23

originally posted on facebook

In a dramatic new low for US Congress, 30 Republicans raided the confidential Laura Cooper hearing, bringing electronic devices into a secured area where none are permitted. Here is a photo of Alex Mooney recording video while inside of the secured area:

Matt Gaetz and Mark Walker tweeted that they were inside the secured area:

Matt Gaetz also raided the Fiona Hill hearing and committed witness tampering for the Michael Cohen hearing. He is also known for having DUI charges against him dropped under mysterious circumstances.

Calculating pi in Factorio

2019 October 04

originally posted on facebook

Calculating pi in Factorio (I suggest watching at 1.5x speed as my typing is slow at real-time speed):


The method used to calculate pi is the Bailey-Borwein-Plouffe_formula, which can find the nth bit of pi in O(n \log n) without needing any information from the earlier bits of pi. It also does not require multi-precision arithmetic: when implemented with 32-bit integers, the probability it will get a bit wrong is about 1 in 2^{30} (and it would be possible to detect if there is a potential error for a given bit, although I did not implement this). This feature makes it well-suited for implementing in Factorio.

“2001: A Space King” and others

2019 September 18

originally posted on facebook

I did a bit of video editing, see links… the first three videos require headphones. See video descriptions for info.

2001: A Space King YouTube

Willy Wonka and the Stargate Sequence YouTube

The Great Patton YouTube

Blade Matrix Knight YouTube

2019 September 16

originally posted on facebook

Something you can do right now: Public comments are being solicited on a proposed federal rule that would remove many existing restrictions on government contractors from discriminating on the basis of religion/gender/orientation/etc. The public comment period ends today!

You can learn more about the proposed rule from the Opening Arguments podcast (skip to 37:50, the segment is about 18 minutes long)

Opening Arguments

The proposal itself and the form to submit a comment on it can be found here:

Here is my own comment, which you are welcome to crib from, and hopefully serves the dual purpose of explaining (within my limited understanding) what the proposed rule is and its effects:

The proposed rule purports to “clarify” the interpretation of the EO by adding definitions and a rule of construction. However, these additions do not clarify but greatly alter the meaning of the EO by grossly expanding the religious exemption to apply to almost any discrimination performed by almost any organization that self-avows some connection to some religion.

The definitions added to 60-1.3 are inconsistent with the plain meaning of the terms they claim to define, and therefore clearly alter the intentions of the existing EO. Most notably “Religious corporation, association, educational institution, or society” is defined in such an expansive way that any organization that self-avows a connection to religion would qualify, and the text of the executive summary suggests that the government would be impotent to challenge any such self-avowal. The removal of the common expectation that such an organization is non-profit (or has only nominal financial transactions) weakens the regulation further.

No attempt is made in the executive summary to justify the addition of 60-1.5(e), perhaps because no justification is possible. The executive summary explains that the EO’s religious exemption has long been interpreted to only apply to discrimination in favor of coreligionists. This is a simple and understandable interpretation that required no clarification. 60-1.5(e) instead specifies that the exemption is “broad” to the “maximum extent permitted by the… law”. This is a plain alteration of the existing EO that, unlike the established interpretation, would be impossible for organizations to know the bounds of and anticipate if their actions are permissible. This runs contrary to the stated objective of the alterations to simplify the interpretation.

The effect of these alterations to the EO are to permit government contractors to broadly discriminate and to render the government incapable of regulating this discrimination.

I propose, as an alternative to the OFCCP’s proposal, to remove the religious exemption. This is a smaller alteration than the OFCCP’s proposal, better accomplishes the stated goals of bringing clarity to the EO, and has the effect of empowering the government to more strongly regulate government contractors which discriminate. Government contractors act as an extension of the federal government itself and should be subject to the same standards.

While no single government contract by itself constitutes the government establishment of religion, the overwhelming tendency of religious government contractors to be Christian (to the near exclusion of other religions), together with permitting these organizations to use government funds to preferentially hire coreligionists, amounts to the overall effect of a government establishment of religion in violation of the first amendment. The DoL should publish data on the religion of religious contractors, and their use of government funds to hire coreligionists. My proposal would end this violation of the first amendment.

I oppose the OFCCP’s proposal.

2019 September 03

originally posted on facebook

George Mason: “The President ought not to have the power of pardoning, because he may frequently pardon crimes which were advised by himself. … if he has the power of granting pardons before indictment, or conviction, may he not stop inquiry and prevent detection?”

James Monroe: “if the President be connected, in any suspicious manner, with any person, and there be grounds to believe he will shelter him, the House of Representatives can impeach him; they can remove him if found guilty; they can suspend him when suspected, and the power will devolve on the Vice-President. Should he be suspected, also, he may likewise be suspended fill he be impeached and removed, and the legislature may make a temporary appointment. This is a great security.”

“And so, Mason lost the argument.”


2019 July 31

originally posted on facebook

“In your otherwise beautiful poem, one verse reads, ‘Every minute dies a man, Every minute one is born;’ I need hardly point out to you that this calculation would tend to keep the sum total of the world’s population in a state of perpetual equipoise, whereas it is a well-known fact that the said sum total is constantly on the increase. I would therefore take the liberty of suggesting that in the next edition of your excellent poem the erroneous calculation to which I refer should be corrected as follows:

‘Every moment dies a man, And one and a sixteenth is born.’

I may add that the exact figures are 1.067, but something must, of course, be conceded to the laws of metre.” - Charles Babbage to Tennyson

“Passages from the life of a philosopher” is the highly eccentric quasi-autobiography of Charles Babbage which dedicates a chapter – 25 pages, with tables – to how much he detests street musicians, and another to an analogy between the origin of the universe and the manufacture of Gloucester cheese. He discusses descending into the fuming caldera of Mt. Vesuvius, his ambitions to solve chess by searching its entire game tree using the Analytical Engine, and attempts to summon the devil as a youth. [html] [pdf]

2019 July 23

originally posted on facebook


2019 July 15

originally posted on facebook

Observations and predictions of global temperatures under the four main emissions scenarios considered by the IPCC.

2019 June 15

originally posted on facebook

Division of three for cardinals without the axiom of choice. Formally, given 3x = 3y, the goal is to prove that x = y. (With the axiom of choice, you can prove that x = 3x for any infinite cardinal x, so the result is trivial in an uninteresting way.)

2019 May 15

originally posted on facebook

Seen in the index of a book: “Bears, combats with, 15, 26, 62, 75, 78, 90, 95, 106, 113, 118, 154, 169, 181, 188, 207, 213, 216; two men killed by a bear, 62; the crew made ill by eating a bear’s liver, 183”

Browsing through the book this seems to be the great minority of the bear encounters, which is apparently the main activity of interest.

From page 62: “a great leane white beare came sodainly stealing out, and caught one of them fast by the necke, who not knowing what it was that tooke him by the necke, cried out and said, Who is it that pulles me so by the necke? Wherewith the other, that lay not farre from him, lifted vp his head to see who it was, and perceiuing it to be a monsterous beare, cryed out and sayd, Oh mate, it is a beare! and therewith presently rose vp and ran away.”

Illustration from a different bear attack:

2019 May 12

originally posted on facebook

“Knife money” was used for much of 400 years in ancient China until round coins were finally invented. (Before, and concurrent, with knife money was “spade money” which were bronze shovelheads, usually too thin to be functional for digging.)

Coins were independently invented in Lydia (modern-day Turkey), India, and China, which used three different manufacturing techniques: stamping, punching, and casting respectively. Modern-day coins (mostly?) are derived from the Lydian invention, with China switching to the Lydian style of coin with the 1911 revolution that overthrew the last emperor.

Knife and spade money were far from the only forms of money to precede coins: cowry shells (or metal imitations) were used worldwide and arguably had the greatest geographic reach of any currency; metal ingots were also popular; and the archaic Greeks may have occasionally used cauldrons. Shells remain legal currency in East New Britain, an island of Papua New Guinea, which has a circulation of about $2 million worth of shells.





2019 May 03

originally posted on facebook

There is a peer-reviewed paper for which this is the sole figure:

2019 April 24

originally posted on facebook

Fun fact: The cornea receives most of its oxygen directly from the air. The cornea’s endothelium actively pumps water out of the cornea, but its metabolic activity is reduced at night because it has less access to oxygen when your eyes are closed. As a result, your cornea is slightly swollen when you wake up.

Less fun fact: the cornea has the greatest density of sensory nerves in the body (except the retina? I’m not sure how these things are counted), 400 times denser than skin, and 20 times dental pulp. These sensory receptors are almost all pain receptors. As a consequence, poking your eye hurts.

Bonus fact: There are an estimated 1 million Hannover canals in Hannover (population: half million).

2019 April 21

originally posted on facebook

NHC’s post-season analysis of the 2018 hurricane season upgraded Hurricane Michael to category 5, but their report on Michael incorrectly gave context for its strength. They write:

“In terms of wind velocity, Michael is tied with the San Felipe Hurricane of 1928 as the fourth strongest hurricane to strike the United States (including Puerto Rico) since 1900, behind the Labor Day Hurricane (1935), Camille (1969), and Andrew (1992). […] Additionally, Michael marks the latest date of a category 5 hurricane landfall in the United States.”

Both of these are false, as two weeks later Typhoon Yutu struck the Northern Mariana Islands at peak strength, making it the second strongest cyclone to have struck the US by either windspeed or barometric pressure. While forecasting Yutu fell under the responsibility of the Japan Meteorological Agency, not the NHC, it would be nice if they did not forgot about the US’s outlying territories in their public-facing reports. This particular error was repeated by CNN, the Washington Post, and NPR.

(Typhoon Karen was also officially stronger than Michael when it struck Guam in 1962, although there is some reason to think it may have been weaker.)

Washington Post



2019 April 21

originally posted on facebook

More Mueller report commentary:

  1. Volume 2 of the report has been condensed into one handy chart by an editor of Lawfare:

Note that “nexus”, as I understand it, can apply to likely future proceedings, not just active proceedings (the heading of the chart was not made by the Lawfare editor).

  1. A lawyer I am unfamiliar with has made a similar chart:

  1. Marcy Wheeler has blow-by-blow comments on the report in twitter thread format (ugh) which at least has now been assembled as a collection of links:

  1. Lawfare has a podcast with an analysis of the report. This is a little heavier on the legal stuff than the other commentaries I’ve been linking.


2019 April 19

originally posted on facebook

Best quote from the Mueller report: ‘[Trump] then asked, “What about these notes? Why do you take notes? Lawyers don’t take notes. I never had a lawyer who took notes.” McGahn responded that he keeps notes because he is a “real lawyer”’

Don’t have time to read the whole report? Here are some summaries:

  1. There are short, readable executive summaries at the beginning of each of the two volumes.

  2. Lawfare Blog has a lengthy article discussing the report. Take-away quote: “the more time one spends with the obstruction section of the report, the more it suggests that the Mueller team believed the evidence of obstruction to be very strong.”.


  1. Opening Arguments has an episode on the report, where they break down some of the legalese and explain some details including: Mueller used a very high legal bar for what is considered criminal conspiracy; Trump Jr and Kushner were considered to be too stupid to charge; Trump’s written response to Mueller’s questions was considered insufficient and inadequate, but Mueller was unable to get an in-person interview to resolve the matter; some of the key redactions by Barr, such as why Trump refused to testify, were clearly inappropriate. Take-away quote: “Trump was helped by the fact that only a handful of people who worked for him were remotely competent and experienced at any of the things they’ve ever done before”. I suspect OA will have a more thorough breakdown of the report in a future episode.

Opening Arguments

  1. My own impression (without having read most of the report) is that it doesn’t contain any new smoking gun to add to the pile of already known smoking guns, but the investigation would have found a great deal more if not for the extensive obstruction of justice.

2019 April 02

originally posted on facebook

Chess site “lichess” does not properly validate chess games imported into the site in pgn notation. This permits odd behavior such as moving into check (but only if the king itself is moved) or promoting pawns into kings. If you have multiple kings and one of them is in check, sometimes the wrong one will be displayed as being checked. You can take your opponent’s king, but if it is their last king the website crashes. If a player has more than one king, or is in check on the opponent’s turn, it is considered “game over”, but if you capture the excess kings or leave check it returns to a non-game-over state.

After confusing the game like this, the engine will suggest crazy moves like moving the opponent’s pieces or taking its own pieces and seems to try to “helpmate” or something.

To test for yourself go to:

2019 March 29

originally posted on facebook

Quote from a recent Supreme Court decision regarding a case in which a judge with the deciding vote died before the court’s decision was announced: “Because Judge Reinhardt was no longer a judge at the time when the en banc decision in this case was filed, the Ninth Circuit erred in counting him as a member of the majority. That practice effectively allowed a deceased judge to exercise the judicial power of the United States after his death. But federal judges are appointed for life, not for eternity.”

2019 March 12

originally posted on facebook

A college admission cheating ring involving 50 people was busted. From the indictment’s affidavit: “Caplan: if somebody catches this, what happens? Cooperating Witness 1: The only one who can catch it is if you guys tell somebody. Caplan: I am not going to tell anybody. CW-1: Well (laughing) Caplan: (laughing) CW-1: Neither am I. And, neither am I.”

US Attorney Andrew Lelling: “There can be no separate college admission system for the wealthy, and I’ll add that there will not be a separate criminal justice system either.”



After the head of the fraud ring, William Singer (CW-1), began cooperating with the FBI he called each of his prior co-conspirators to get them to verbally admit to specific crimes and payment amounts. Only one set of parents was suitably suspicious, saying on the phone “It’s really hard to hear” and asking to meet in person, wherein they said “I am so paranoid about this fucking thing you were talking about. I don’t like talking about it on the phone, you know” and correctly guessing “I can’t imagine they’d go to the trouble of tapping my phone – but would they tap someone like your phones?”. Later they remarked “if they get into the meat and potatoes, is this gonna be the front page story with everyone […] getting these kids into school” and, referring to their third child, “I think we’ll definitely pay cash this time and not run it through the other way.”. Unfortunately for them Singer was already wearing a wire at this point.

Also in this in-person conversation Singer messes up and says “I was just told to call everybody.”.

Another co-conspirator caught on to what was happening too late: the fourth time Singer called him to chat about their fraud, he started denying any involvement: “you did what you did and so I’m not going to take accountability for your actions […] I don’t want to talk about this any more because, you know, I think there were two separate things […] We donated as a charity, and it was a good charity and we were excited we could help you and […] if you’re trying to turn something around in terms of, you know, what you did and how you did it then I don’t want to be a part of that.”.

The dumbest co-conspirator award probably goes to the defendant who asked his executive assistant to write the bribery check and put in the memo line what the bribe was for.

2019 February 24

originally posted on facebook

In 1971 the proof below was given that x^3 + 117 y^3 = 5 has no integer solutions, using some difficult computations in the field \mathbb Q(\sqrt[3]{117}).

In 1973 it was observed that the equation reduces to x^3 \equiv 5 mod 9, which has no solutions.

Finkelstein, R., & London, H. (1971). On D. J. Lewis’s Equation x^3 + 117y^3 = 5. Canadian Mathematical Bulletin, 14(1), 111-111. doi:10.4153/CMB-1971-020-x

2019 February 21

originally posted on facebook

In exciting news today, the Supreme Court ruled that the states may not take “excessive” civil forfeiture (it was already unconstitutional for the federal government to do so). Every year the federal government takes about 1 to 2 billion dollars in civil forfeiture; figures for the states are not easily available. There are endless examples of people having their property wrongfully stolen by the police through civil forfeiture, and hopefully this decision will provide a recourse in the most egregious cases to argue in court that the property taken was excessive. This is a small but welcome relief from the burden of civil forfeiture, and optimistically may be a stepping stone to eliminating it entirely.

Here is the decision:

The majority decision was written by Ginsburg and joined by everyone except Thomas; there are also concurring decisions by Gorsuch and Thomas. (Thus continues a tradition of Gorsuch concurring with himself.)

Of interest is Gorsuch’s, the shortest at five sentences. First he agrees with Ginsburg; then he cites Thomas a bunch and agrees with him; then he says he doesn’t know whether Ginsburg or Thomas was right but since they got to the same conclusion, he may as well concur as well. As part of his ongoing efforts to pretend to be an originalist he manages to use the word “original” twice in five sentences, compared to none in Ginsburg’s 9 pages or Thomas’s 13 pages.

I expect Friday’s episode of Opening Arguments will have more to say about Gorsuch’s decision and how it somehow opposes Chevron deference – if so I might add some better informed commentary then.


2019 February 13

originally posted on facebook

This is old news, but I only just found out: in 2017 a neutron star merger was observed by both gravitational and electromagnetic waves! Aside from the usual confirmations, in particular this confirmed that such mergers are the origin of short gamma ray bursts and are the origin of most heavy elements (heavier than iron). It is estimated that the gold and platinum produced was “vastly” more than the mass of the Earth.

2019 January 23

originally posted on facebook

Today I wrote a python program called “” that writes a lua program that writes a python program.

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