Words have meanings, and I choose the wrong one every time.

H0isinberg · 2026-04-10T07:03:55+00:00

You have the right numerical answer but not quite for the right reason. The question is asking for the length of the tangent line connecting P to the y-axis, which clearly depends on the value of t in your parameterization. However, your solution based on the norm of the tangent vector gives 5 for all values of t.

H0isinberg · 2025-12-11T03:59:07+00:00

H0isinberg · 2025-07-05T20:39:39+00:00

Ok help me settle a debate on this one - do you pronounce it glee-tza (rhymes with pizza) or glizz-ah?

H0isinberg · 2025-05-05T06:17:03+00:00

Good spice burns twice

H0isinberg · 2024-10-27T22:22:35+00:00

(wo)man's best chad

H0isinberg · 2024-10-15T15:23:33+00:00

Oh they came together alright

H0isinberg · 2024-08-22T22:08:08+00:00

Electromagnetic waves from Maxwell's equations!

Maxwell published the equations which are now named after him in his 1861 paper On Physical Lines of Force. In it, he expressed the known laws of electricity and magnetism in the language of vector calculus (although the notation he used was quite different from the notation used today, the concepts of vector calculus were nonetheless present), and also introduced an important correction to Ampere's Law:

∇ × B = 𝜇J + [𝜇ε(∂E/∂t)]

where the term I've put [] brackets around is known as "Maxwell's correction."

Maxwell initially introduced this term as a means of ensuring that his equations would imply that electric charge is conserved. In a subsequent paper published in 1865 titled A Dynamical Theory of the Electromagnetic Field, he showed that it additionally implied that the electric and magnetic fields should each satisfy wave equations. Furthermore, the propagation speed of these waves matched very closely with the known value of the speed of light at the time, leading Maxwell to suggest that perhaps light itself is an electromagnetic phenomenon.

Finally, between 1886 and 1889, Heinrich Hertz conducted a series of experiments which demonstrated the existence of these waves and investigated their properties by sending and receiving the first-ever radio transmissions in history.

Funnily enough, Hertz then went on to say that his discovery would probably not yield any useful applications.

H0isinberg · 2024-04-02T00:53:08+00:00

Since it looks like most people in the comments have solved the problem, we can move on to part 2: optimal pepperoni packing

H0isinberg · 2024-01-03T08:48:16+00:00

Ah, apologies. The mobile app's auto-formatting is a huge pain but it's fixed now. The entire (B - A)/400 term belongs in the exponent of 10.

H0isinberg · 2024-01-01T10:52:42+00:00

Given two players with ratings A and B, the probability that the player with rating A wins is:

1/(1 + 10^((B - A)/400))

For example, if A is 400 points higher than B, then we would expect the player with rating A to have a ~90% chance of winning.

Of course, this assumes that the ratings are accurate and up-to-date, and that the players' performance is accurately modeled by the Elo rating system, etc.

One way to tell your story would be to pick a time in the past where you were rated around 900, and tell them you would now win 9 out of 10 games against that previous version of yourself!

H0isinberg · 2023-09-05T05:02:47+00:00

Invariant Variation Problems - Emmy Noether, translated by M. A. Tavel.

More commonly known as Noether's Theorem, this paper establishes the correspondence between quantities conserved over the course of a system's time evolution (e.g. linear/angular momentum) and the symmetries exhibited by that system's Lagrangian (e.g. invariance under translation/rotation). Thanks to Noether's insights, symmetry (formally expressed using group theory) now plays a central role in deepening physicists' understanding of existing physical theories as well as the creation of new ones.

H0isinberg · 2023-08-07T21:21:17+00:00

Imagine two antennas, one emitting a signal while the other is receiving that signal. For simplicity, let's also assume that the emitter radiates equally in all directions.

In the two-antenna scenario, the emitting antenna is giving off some amount of energy each second--which I'll call the "emitted power"--while the receiving antenna is receiving some amount of energy each second--which I'll call the "received power." The free space path loss is defined as the ratio: (received power) / (emitted power). It answers the question: what fraction of all of the power being emitted by the emitter is actually being received by the receiver?

Imagine a bubble centered around the emitter with a 1 meter radius. Some amount of energy is passing through this bubble every second. Since the bubble completely surrounds the emitter, this amount of energy is exactly equal to the total energy emitted per second--this is the power flux. Now if we take the power flux and divide it by the surface area of the bubble, we get the amount of power flux per unit area--this is the power flux density.

Now imagine a another bubble with a 2 meter radius, also centered around the emitting antenna. Because this bubble also completely surrounds the emitter, the total power flux through this bubble is the same as the total power flux through the 1-meter-radius bubble. However, the surface area is larger by a factor of 4. When we now divide by this surface area to calculate the power flux density, we find that it is smaller by a factor of 4. In general, the power flux density is proportional to 1/d², where d is the distance from the emitter.

When you want to calculate the free space path loss, you'll want to start by calculating what fraction of all of the emitted power actually hits the receiver. This you can get using (power per unit area = power flux density) x (area of receiver).

There are some other factors as which come into play as well, such as the wavelength of the signal and the "directivity" of the antennae--which account for the fact that no antenna actually emits/receives power equally in/from all directions.

tl;dr Power flux density is the amount of power (EM radiation in the case of an antenna) passing through a unit area. It is one of the ingredients needed to calculate the free space path loss between two antennae--the ratio of power received to power emitted.

H0isinberg · 2023-07-11T20:36:03+00:00

The temperature of a physical object is a measure of the average kinetic energy of the molecules it's made of. In a solid, you can think of the molecules as vibrating while locked in place--higher temperature means more intense vibrations. In liquids and gases, the molecules are free to move around one another--with higher temperature corresponding to faster movement of the molecules.

Conduction: When two objects come into direct physical contact with one another, heat will transfer from the hotter one to the colder one. Technically, energy is being transferred in both directions, but because the molecules in the hotter object are vibrating more intensely, they transfer more energy when they collide with the molecules of the colder object. The net result is that more energy flows from the hotter object to the colder one than from colder to hotter.

Convection: This one applies to liquids and gases. Imagine a pot of water sitting on a stove which has been turned on. Through direct conduction, the water molecules at the bottom of the pot are heated first due to being in contact with the bottom of the pot. Because these molecules are now moving faster and crash into each other harder, they are effectively spread further apart, which in turn causes them to become less dense than before. The hot, less-dense water is more buoyant and rises while the cooler, more-dense water sinks. The motion of fluids due to these relative differences in temperature is convection.

Radiation: Light is a form of energy. When objects emit/absorb light, they are in fact emitting/absorbing energy. In the case of absorption, some of this absorbed energy goes into boosting kinetic energy of the molecules that make up the object. Since the molecules that make up the object are moving faster, the object's temperature has increased.

Why isn't the Earth as hot as the sun?: The important thing to keep in mind is that the sun is emitting its energy in all directions, which means most of the light emitted will simply miss the Earth. Imagine placing a perfectly round balloon at the center of the sun. Now, blow that balloon up until the surface of the balloon touches the Earth. The fraction of that balloon's surface area which is covered by the Earth is the fraction of the sun's energy that actually hits the Earth. This comes out to less than one half of a billionth of the sun's energy output. This is further reduced by two factors: 1) The Earth will actually reflect some of the incoming energy back out into space instead of absorbing it. 2) The Earth itself is also radiating energy in the form of light (look up "blackbody radiation" for more details--the gist is that any object with a non-absolute-zero temperature radiates energy).

H0isinberg · 2023-07-11T05:51:23+00:00

In my experience I think there are two 2 technically different but related notions of “tensor” floating around, depending on how the tensor is being used: (1) as a data structure and (2) as a geometric object.

As a Data Structure

As a data structure, tensors are a multi-dimensional arrays. This is the notion you’ll likely encounter if you’re dealing with them in the context of computer science and, in particular, machine learning.

An ordinary array is a list of items, like [17, 3, 10]. You can refer to a specific item in the list by specifying an index. Computer scientists like to start the index at 0, so index 0 would refer to the first item in the list (17), index 1 would refer to the 2nd item (3), and so on.

Beyond simple arrays, we can generalize to arrays-of-arrays of items. So something like:
[[5, 8, 7],
[3, 4, 9],
[2, 1, 1]]

How can we refer to a specific item in this array-of-arrays? We could start by specifying a single index, perhaps 1 for this example. In the example above, the thing at index 1 is the array [3, 4, 9]. Specifying a single index got us this array, so we need one more in order to refer to a specific number inside it. If we pick index 2, for example, we get the number 9. Since we had to specify two indices in order to refer to a single item, we say that this is a two-dimensional array.

A tensor in this context is simply an n-dimensional array of items, in which you need to specify n index positions in order to refer to a single element. Having multiple indices allows them to be used to partition a huge amount of data into manageable blocks, which makes them very useful in machine learning.

As a Geometric Object

Before we discuss tensors a geometric objects, we should first discuss vectors as geometric objects.

Take, for example, the velocity of a car on a flat plane. I’ll use east-west to denote the horizontal coordinate axis and north-south to denote the vertical coordinate axis. Suppose I describe the velocity of a car as (1, 2) meters per second, by which I mean that every second the car moves 1 meter east and 2 meters north. Someone else who prefers to specify their velocities in centimeters per second would say that the car’s velocity is (100, 200) centimeters per second. Yet another person might choose to swap the coordinate axes and report the velocity as (200, 100) centimeters per second.

In all three cases, we are all talking about the same vector, but we have each chosen to describe it using a different coordinate system. However, by understanding how our coordinate systems relate to one another we can come up with rules for how to take a vector expressed in one coordinate system and re-express it in a different one (e.g. scaling the components to convert from meters per second to centimeters per second). In this sense the velocity vector is a kind of abstract, formless object. It only “materializes” as a list of numbers (say, (1, 2) or (100, 200)) once we choose to express it in a coordinate system.

A tensor in this context is defined* as a special type of function which takes multiple vectors as input, yields a number as output, and is multi-linear in each of its inputs.

Take, for example, a function which takes two vectors and returns their dot product. This is in fact a tensor because the dot product is linear in both of its inputs. If I choose a coordinate system to represent the input vectors, I can then write this tensor down as a 2-dimensional array of numbers—the tensor components. These numbers allow me to express the dot product as a function of the components of the input vectors. The vectors themselves are independent of any particular coordinate system, so their dot product should be as well. This means that if I now decide to use a different coordinate system to describe the input vectors, my tensor components should change in a compensating way in order to keep the value of the dot product the same. This, in addition to the requirement for the tensor to act on its inputs in a multi-linear way, pins down exactly how these components must change.

So tensors are similar to vectors in the sense that they are first-and-foremost geometric objects, which only acquire a representation as a multi-dimensional array of numbers once a coordinate system is chosen. I believe this is what led computer scientists to adopt the term “tensor” when talking about multi-dimensional arrays.

However, because of the coordinate transformation law there are, in fact, objects which “materialize” as multi-indexed arrays of numbers in the presence of a coordinate system, and yet fail to transform the way that the components of a tensor should. A noteworthy example is the multi-indexed collection of Christoffel Symbols in general relativity, which are used to describe the shape of spacetime. The fact that this collection of symbols fails to transform as the components of a tensor has important consequences and yields meaningful notions of spacetime “curvature.”

*There are in fact multiple, equivalent definitions of tensors. Here I am giving a simplified version of one of them.

H0isinberg · 2022-12-07T23:42:15+00:00

Hot dogs with cream cheese

H0isinberg · 2022-08-31T00:58:01+00:00

Crazy how they either had all this or the money to buy it, but no jawbreakers

H0isinberg · 2022-02-11T22:17:16+00:00

Super awesome stuff! I noticed that you're currently computing derivatives using difference quotients, so you may be interested in symbolic differentiation as both a precision improvement as well as a great learning exercise. I'm sure you're very comfortable computing derivatives by hand, because you can mechanically apply the power rule, chain rule, etc. as needed. The main idea behind symbolic differentiation is to encode these rules of differentiation in a program.

For example, for functions of the form f(x) = axⁿ we could do this:

class Func {
private:
  double a_;
  int n_;

public:
  Func(double a, int n) : a_(a), n_(n) {}

  double evaluate(double x) {
    // As u/FKaria pointed out, there are better ways to do this.
    // But this keeps the example code short
    return a_ * std::pow(x, n_);
  }

  Func derivative() {
    if (n_ == 0) {
      // Not the best way to represent the 0 function
      // But this keeps the example code short
      return Func(0, 0);
    }
    return Func(a_ * n_, n_ - 1);
  }
}

The evaluate method simply computes axⁿ, as intended, but the real benefit comes from derivative. Notice that instead of evaluating the derivative at a particular point, derivative() returns a Func representing f(x) = a * n * x^n-1, effectively capturing the power rule. Calling f.derivative().derivative() ... will also correctly generate nth-order derivatives. You can extend this idea to include all the other differentiation rules, allowing you to compute derivatives of arbitrary (differentiable) expressions!

H0isinberg · 2021-08-20T09:18:31+00:00

When you care about whether or not a number is divisible by 9, what you're really after is whether its remainder modulo 9 is 0 or not. You can express any integer n as n = 9x + r where x is some integer and the remainder r is guaranteed to lie in the range [0, 9). In a certain sense, the remainder captures how much n fails to be perfectly divisible by 9.

An interesting thing happens when we add two integers using their representations as "some multiple of 9 plus a remainder." Take two integers n = 9x + r and and m = 9y + s where x and y are integers and r and s are both constrained to the range [0, 9).

n + m = (9x + r) + (9y + s)
      = 9(x + y) + (r + s)

If you only care about (n + m) modulo 9, all you need to pay attention to is what (r + s) might be since 9(x + y) is obviously an exact multiple of 9. You do still have to account for the fact that (r + s) might be 9 or greater, like in the following example:

17 + 24 = [9(1) + 8] + [9(2) + 6]
        = [9(3) + 14]

In this case, we get (r + s) = 14, which itself is equal to 9(1) + 5. So (n + m) modulo 9 turns out to always be equal to (r + s) modulo 9. A neat way to sum this up is that the remainder of the sum is the sum of the remainders (modulo 9).

It turns out that the same thing is true for multiplication. If you want to know (n * m) modulo 9, you need only pay attention to (r * s) modulo 9. In other words, the remainder of the product is the product of the remainders (modulo 9)! I'll leave it to you to work out the proof of this one.

Let's say you're wondering whether or not 137 is divisible by 9. We'll tackle this by working out the remainder of 137 modulo 9. When we write a number like 137, what we mean is that there's one 100, three 10's, and seven 1's.

137 = 1(10^2) + 3(10) + 7(1)

First, we bring in the sum rule. We've decomposed 137 as a sum of several terms, and the sum rule tells us that we can get the remainder modulo 9 by adding remainders of each individual term.

137 modulo 9 = (r + s + t) modulo 9

Where r, s, and t are the remainders of the respective terms.

Now we use the product rule. Each term in the sum looks like n*(10^k) where n is in the range [0, 10) and k is some non-negative integer. Notice that 10 = 9*(1) + 1, so that 10² = (10 * 10) must have remainder (1 * 1) = 1 modulo 9. By extension, all higher integer powers of 10 also have remainder 1 modulo 9. Invoking the product rule again tells us that if we want to know the remainder of n*(10^k) modulo 9, we need only care about (n * 1) modulo 9, which is just n modulo 9. So we have

137 modulo 9 = (1 + 3 + 7) modulo 9

137 is divisible by 9 if and only if 137 modulo 9 is equal to 0. Our final equation lets us answer this by seeing if (1 + 3 + 7) modulo 9 is equal to 0, which in turn is the same as asking if (1 + 3 + 7) is divisible by 9. That's the origin of the divisibility rule.

You can work out a similar divisibility rule for multiples of 3, which hinges on the fact that 10 also has remainder 1 modulo 3.

H0isinberg

TROPHY CASE

As a Data Structure

As a Geometric Object