Graham Wheeler's Random Forest

In the past few posts I have discussed a diverse set of topics, from number representation to recurrence relations, Pythagorean triples. All of these topics are related, and this is particularly illustrated when we consider the work of Leonardo of Pisa, better known as Fibonacci (from the Latin “de filiis Bonacci”, or “of the family of Bonacci”). Fibonacci was born in 1175, in Pisa, Italy, shortly after the start of the construction of the famous leaning tower.

“And the seventh angel sounded [his horn]; and there were great voices in heaven, saying, The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever.” (Revelations 11.15, King James Edition) In Christian and Islamic folklore, it is the angel Gabriel who is considered to be the seventh angel, who announced the coming of judgement day. An angel with such a huge responsibility clearly needs an instrument worthy of the task, and indeed there is one: Gabriel’s horn, also known as Torricelli’s trumpet, after its discoverer, Evangelista Torricelli, a student of Galileo.

Pythagoras is known for two great contributions to mathematics – he established the need for formal proofs instead of just conjecture and rules of thumb, and he established the existence of the irrationals. In popular culture of course, Pythagoras is more well known for the Pythagorean theorem – that the square of the hypotenuse of a right angled triangle is the sum of the squares of the other two sides – but this was one of the oldest known results in mathematics and in fact predates Pythagoras by as much as a thousand years.

Many people have heard that the number zero was introduced later than the other digits. Upon hearing this it seems fantastical - how could people have managed even elementary arithmetic without zero? However, it is not the concept of “nothing” that was missing, but rather the use of a zero in other places when representing or recording numbers. For example, in the number 101, zero is used as a separator, and it is this concept that was lacking in early number systems.

Number theory really began with Euclid, around 300BC, in books 7 through 9 of his masterwork, The Elements. It is here that we find the original definitions of odd and even numbers, prime and composite numbers, perfect numbers (numbers which are the sum of their factors, e.g. 6 = 3 + 2 + 1), and more. But the greatest achievements of all were his proofs that composite numbers are the products of primes, that this factorization is unique, and that there are an infinity of primes.

In a previous post I described the Babylonian/Sumerian sexagesimal (base 60) counting system. Unlike this system, most cultures adopted a base 10 counting system due to the natural inclination to count with the fingers (leading to the term digits). Less common were quinary (base 5) systems, but vigesimal (base 20) were not unusual - for example, this was widespread in native American culture, including (with a novel variation) the Mayans. Duodecimal (base 12) has also played a role; we still have remnants in the groupings of dozens and gross.

In a previous post I wrote about how Fermat scribbled his famous “last theorem” in the margin of Diophantus’ Arithmetica. This is called Fermat’s last theorem not because it was the last thing Fermat wrote but because of all the incomplete theorem’s we know were left by Fermat it was the last to be proved, taking about 350 years. The section of the book where Fermat wrote his comment was on finding Pythagorean triples: square numbers whose sums also form squares.

Imagine an infinite grid filled where each square is initially either black or white. On this grid is an ant, which can face either north, south, east or west. The ant moves over the grid according to the following rules: it it lands on a black cell it turns left 90 degrees; if it lands on a white cell it turns right 90 degrees in each case the cell it just left changes color to its opposite (white to black or vice-versa) Running a computer simulation of this system, which was invented by Christopher Langton in 1986, shows that after a while the ant gets stuck in a cycle of 104 moves which move it two squares diagonally, after which point it continues building this diagonal “highway”.

In the late 1980s I was contracted to write some software for a company that produced video-based educational systems. They had video cassette machines that had been modified to interface with a PC, which could send instructions to the VCRs such as play, stop, fast forward, and rewind. The educational programs consisted of short recorded segments which typically ended with a question, and based on the answer the user provided they wanted to continue play at different portions of the tape.

My 5 year old daughter wanted to heat something up in the microwave. I suggested she use a minute, and she asked me how to enter that. I said enter 1, 0, 0, Start. “But when mom tells me to do a minute she says 6, 0, Start”, she responded. I wasn’t sure how to explain to her that 1:00 minute is the same as 60 seconds; microwaves are confusing enough to her as is.