Try to get through 8 levels by moving in all 4 spacial [sic] dimensions and reach your spaceship.

A cube is one of the simplest solids one can imagine. Over the course of this article I'll try to explain how to expand it to the next dimension to obtain a tesseract -- a 4D equivalent of a cube.

I love spreadsheets. Spreadsheet programs like Microsoft's Excel, Apple's Numbers and Google Sheets are the secret heroes of our civilization.

I've also been interested in personal finance and the FIRE community for a while—not so much in the early retirement aspect but in the financial literacy it teaches its members. I have combined my passion for both into one mega-spreadsheet that I use to track my income, expenses, savings and investments in one overview. While creating this spreadsheet I got proficient in some new formulas, which I'll share here—and also write down for my own reference.

In a paper from 2012, Vanguard found that 66% of the time it is better to invest your money right away (“Lump Sum”) rather than buying in over 12 months (“DCA”). I don’t disagree with Vanguard’s results (my results were strikingly similar), but I don’t think they went deep enough in explaining why this is true.

The main reason Lump Sum outperforms DCA is because most markets generally rise over time. Because of this positive long-term trend, DCA typically buys at higher average prices than Lump Sum. Additionally, in those rare instances where DCA does outperforms Lump Sum (i.e. in falling markets), it is difficult to stick to DCA. So the times where DCA has the largest advantage are also the times where it can be the hardest for investors to stick to their plan.

This article explains Reed-Solomon erasure codes and the problems they solve in gory detail, with the aim of providing enough background to understand how the PAR1 and PAR2 file formats work, the details of which will be covered in future articles.

I’m assuming that the reader is familiar with programming, but has not had much exposure to coding theory or linear algebra. Thus, I’ll review the basics and treat the results we need as a “black box”, stating them and moving on. However, I’ll give self-contained proofs of those results in a companion article.

So let’s start with the problem we’re trying to solve! Let’s say you have n files of roughly the same size, and you want to guard against m of them being lost or corrupted. To do so, you generate m parity files ahead of time, and if in the future you lose up to m of the data files, you can use an equal number of parity files to recover the lost data files.

This text gives an overview of Gödel’s Incompleteness Theorem and its implications for artificial intelligence. Specifically, we deal with the question whether Gödel’s Incompleteness Theorem shows that human intelligence could not be recreated by a traditional computer.

It’s science’s dirtiest secret: The “scientific method” of testing hypotheses by statistical analysis stands on a flimsy foundation. Statistical tests are supposed to guide scientists in judging whether an experimental result reflects some real effect or is merely a random fluke, but the standard methods mix mutually inconsistent philosophies and offer no meaningful basis for making such decisions. Even when performed correctly, statistical tests are widely misunderstood and frequently misinterpreted. As a result, countless conclusions in the scientific literature are erroneous, and tests of medical dangers or treatments are often contradictory and confusing.

Although there are a few different public-key encryption algorithms, the most popular — and fortunately, the easiest to understand — is the RSA algorithm, named after its three inventors Rivest, Shamir and Adelman. To apply the RSA algorithm, you must find three numbers e, d and n related such that ((m^e)^d) % n = m. Here, e and n comprise the public key and d is the private key. When one party wishes to send a message in confidence to the holder of the private key, he computes and transmits c = (m^e) % n. The recipient then recovers the original message m using m = (c^d) % n.

The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.

I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.

As you can see, the tiles overlap and interact to generate new patterns and colors. And as we’re using magical prime numbers, this pattern will not repeat for a long, long time. § Exactly how long? 29px × 37px × 53px… or 56,869px! § Now this was something of a revelation to me. I actually had to triple-check my calculations, but the math is rock solid. Remember these are tiny graphics — less than 7kb in total — yet they are generating an area of original texture of almost 57,000 pixels wide.