Things to Make and Do in the Fourth Dimension Quotes

“As the Nobel Prize-winning physicist Richard Feynman allegedly said of his own subject: ‘Physics is a lot like sex; sure it has a practical use, but that’s not why we do it.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Sadly, very little school maths focuses on how to win free drinks in a pub.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“the invention of the wheel is up there as the second greatest of all human inventions (slightly behind fire and just ahead of sliced bread).”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“For example, to a mathematician, the number 28 is really 2×2×7, which is known as the prime decomposition of 28. Prime numbers are, in a way, the atoms of maths, the components that make up all other numbers. The non-prime numbers are known as composite numbers.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The square of every prime number is one more than a multiple of 24.”
― Matthew Parker, Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, At Least Two Kinds of Infinity, and More

“Despite their ubiquity on the number line, transcendentals are surprisingly hard to pin down. It took until 1873 to prove that e was transcendental, making it the first number we knew for definite was. The poster-child of maths, pi, didn't join the transcendental fold until 1882. Even today, we know that at least one of e + pi and e × pi is transcendental, but we have no idea which. On David Hilbert's 1900 list of important maths problems to solve, one of them involved checking if e^pi is transcendental, and since 1934 we have known that it is. However, e^e, pi^pi, and pi^e are still open problems. Transcendentals are really hard to find in the wild.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“They had no idea how important they, and beer, ended up being for the development of human civilization. As I said before, living in cities was one of the things that caused humans to rely on math. But which part of city living is recorded in our longest-surviving mathematical documents? Brewing beer.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“We’re going to consider a perfectly circular pizza which has a uniform coverage of the same topping and is without a crust; also, its base is extremely thin (so you can’t cheat by cutting horizontally through it). To sum up, then, the pizza can be described as perfectly circular, homogenous, infinitely thin and (because it’s two-dimensional) terrible value for money. It can also be frictionless and in a vacuum if you like, but that would make it substantially harder to eat. Although possibly much easier to digest.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Twelve pentagons form a Platonic solid (the great dodecahedron), as do twenty triangles (the great icosahedron).”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“triangles will join in this way to give you an icosahedron. A sixth triangle on a vertex fits perfectly, leaving no”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The cube is a great 3D shape, and it’s so great partly because it’s so regular. In 2D, if you remember, we had our regular polygons which had all angles and edges the same, and now we can extend this to 3D. The 3D equivalent of a polygon is a polyhedron, and it’s made by joining polygons together in the third dimension. A cube is a polyhedron made by joining six square polygons together. A tetrahedron is a polyhedron made from four triangle polygons. Whereas polygons have only corners and edges, a polyhedron also has vertices where the polygon corners meet. A regular polyhedron is one made only from identical regular polygons and in which all the vertices are also exactly the same.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Infinity earned its rightful place as a legitimate mathematical concept towards the end of the nineteenth century. The two big players behind this were German mathematicians Georg Cantor and his champion David Hilbert. No longer just accepting infinity as a general notion but actually investigating it rigorously did not go down well. Contemporary mathematicians described Cantor as a 'corrupter of youth'. This had a detrimental effect on Cantor, who already suffered from depression. Thankfully, Hilbert saw the power of what Cantor had done. He described Cantor's work as 'the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity', and famously said, 'no one shall expel us from the Paradise that Cantor has created'.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Finding a method to categorize transcendental numbers was one of David Hilbert's great unsolved maths problems (along with the Riemann Hypothesis), and it remains justbas unsolved today.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The proof in 1882 that pi is a transcendental number put a 2,000 year old problem to rest: for any given circle, can you draw a square of the same surface area using a compass and a straight edge? Since 1882, we know that no, you can't. To draw a square the area of a circle you need to be able to draw a line pi units long, and you cannot draw transcendental numbers.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The compact disc may now be outdated technology, but a lot of music albums are still released on CD. A standard 700-megabyte CD is actually 703.125 megabytes (a rare case of the music industry giving something extra away for free), which is a total of 5,898,240,000 1s and 0s. By my calculations, the number of possible different CDs in base-10 would have 1,775,547,162 digits. Which is also the number of corners a hypercube would have in 5,898,240,000 dimensions. So whenever a musician claims they have written a new album, all they have really done is choose a corner on a very high-dimension hypercube.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“There are only a limited number of sporadic groups, and one of them does indeed have the geometric interpretion with the highest number of dimensions. It's the Monster Group, and the shape it corresponds to can exist only in 196,883 dimensions. This boggles my mind. As you travel up past hundreds of thousands of dimensions, with only a few predictable infinite families of shapes to keep you company, suddenly, out of the blurred monotony, a shape flashes into existence for a single dimensional space. It wasn't there in 196,882D and has gone again by 196,884D. In that one tiny window, a shape beyond any human comprehension exists. It is a real mathematical object, as much as a triangle or a cube. The title of Griess's 1982 paper gives the Monster its other, more affectionate name: the Friendly Giant. We will never be able to picture the Friendly Giant, but we know it exists.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“There are only three Platinic solids in 5D. No problem; we'll check 6D. In six dimensions, there are still exactly the same regular shapes: the hypercube, simplex and cross polytope. Now that wasn't very exciting. We'll go on to 7D. Same three regukar shapes again. Nothing changes in 8D either. Dimension after dimension, only those three Platonic-solid shapes seem to appear. The search is futile: for every dimension from 5D onwards there are always the same three regular shapes. ............ It was Schlafli himself who proved this bleak Platonic-solid landscape for all dimensions, back in 1852.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“For our 2D square, this diagonal length is sqrt 2 = 1.414; ....The diagonal from the centre of each padding sphere to the centre of the cube is sqrt 3 = 1.732, which allows our specimen sphere to expand to a radius of .732:.....The distance from the centre of each padding 4-sphere to the centre of the 4D box is a very exact and tidy 2 (i.e. sqrt4), givimg our centre specimen sphere a radius of 1....Onwards to 5 dimensions and things start to get a bit strange: our specimen sphere has continued growing and now has a radius of 1.236, bigger than the padding spheres around it.....The big surprise is when the box enters ten dimensions and the inner sphere's radius hits 2.162, which means that it is actually reaching outside the box....From twenty-six dimensions onwards, the sphere is more than twice as big as the box it's inside.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The problem is that many mathematicians have done the same thing Riemann himself did: surged ahead using the prime counting method , assuming that someone would prove it later on. It looks like a safe bet: as we know, using computers, the first 10 trillion zeroes have been checked, and all of them are on that line. That said, mathematical theories have been disproved with numbers bigger than that, so there could be a zero off the line that we've simply not reached yet.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“To this day, the Clay Mathematics Institute's bounty of $1 million for anyone who can prove that all the non-trivial zeroes of the zeta function are on that line has gone unclaimed.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“There has been some progress on proving the Riemann Hypothesis, but it remains unsolved. In 1914, Hardy managed to prove that there are infinitely many zeroes on that line, but he couldn't prove that there aren't any extra
zeroes off the line. We currently know that 40 percent of the non trivial zeroes are definitely on that line, but we need to know it's true for 100 per cent. So much as a single zero somewhere else, and the Riemann hypothesis would be disproved, causing our apparent understanding to come crashing down. But it hasn't. Everything has uncannily indicated that we are on the right track, but we can't yet prove it for sure.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The Riemann Hypothesis states that all the non-trivial zeroes of the zeta function are on this line. If we can prove the Riemann Hypothesis is true, then we'll also have proved the method for counting the prime numbers. In some weird twisted act of mathematical logic, at a fundamental level the alignment of these zeroes stems from the same logic as the density of the primes. It doesn't seem to make sense. But if we can understand this mysterious alignment, we understand where the numbers are hiding their primes.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Part of Ramanujan's genius was to find a way to get values out of the zeta function for negative values. As we saw before, these sums diverge and go racing off to infinity, but Ramanujan was able to extract the bit of the answer which explodes and leave the important bit behind. Using Bernoulli numbers, he could produce values for the negative half of the zeta function-which, eventually, gives us a complete plot. This is the zeta function in graphical form.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Investigating the sum of reciprocal powers gives insights into the product of all the primes (in fraction form).”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Along with working on the Basel problem, Euler realized that adding an infinite sequence of reciprocal powers for all whole numbers will give you the same answer as multiplying together an infinite sequence of fractions which use only the prime numbers. So the zeta function can be written as two different equations, one of which relies only on the prime numbers. The one which uses all the whole numbers gives the same result as the prime fractions, but it's easier to work with. We know what all the whole numbers are, but we don't know what all the primes are. So we can substitute one for the other.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The zeta function is the sum of an infinite sequence of inverse powers.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“In the chapter on prime numbers, I mentioned Bernhard Riemann's 1859 paper 'On the Number of Primes Less than a Given Magnitude'. In it he found a method of calculating how many primes there are below any given number. This would give mathematicians an amazing insight into the distribution and nature of prime numbers. The only problem was that he couldn't prove that this method definitely worked. He did, however, prove that if an apparent alignment in the zeta function was real, then the prime counting method was real. Then he failed to prove that too.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“If you recall, in Ramanujan's letters to English mathematicians, he claimed that 1 + 2 + 3 +...= -1/12. He was so surprised when Hardy took him seriously that he replied on 27 February 1913 in the following words: 'I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study Infinite Series and not fall into the pitfalls of divergent series. If I had given you my methods of proof I am sure you will follow the London Professor. I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 +...= -1/12 under mu theory. If I tell you this, you will at once point out to me the lunatic asylum as my goal.' It turns out that not only had Ramanujan independently rediscovered the Bernoulli numbers, but he may have found more than one way to prove that 1 +2 +3 + 4 ...= -1/12. This is now called Ramanujan summation and gives us an insight into the ways in which the sum of a sequence can be divergent. Of course, the sum of all the positive whole numbers is infinite, but if you can somehow peel that infinity back out of the way and look at what else is going on, there's a -1/12 in there.”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“Oresme's genius was to make a new series which was definitely smaller than the harmonic series. He took the list of all unit fractions, and for any of them which did not have a power of two as a denominator, he replaced it with a smaller fraction which did. As all these new fractions were either the same or smaller, the total of this new series had therefore to be smaller than the sum of the harmonic series. But when Oresme grouped these fractions into runs, each of which added up to 1/2, he was left with a sum of an infinite sequence of 1/2s, which definitely diverges. This meant in turn that the greater harmonic series must also diverge. Oresme had proved that a sequence of ever-decreasing numbers could still be divergent. (His proof was lost for a while, and the same result was independently rediscovered in the 1600s.)”
― Matt Parker, Things to Make and Do in the Fourth Dimension

“The Lucas-Lehmer primality test also uses a recursive function to produce a sequence of numbers (the Lucas-Lehmer sequence). This sequence starts with 4, and each number after that is the previous number squared, minus 2 (see below). For any Mersenne number (such as 2^3 - 1 = 7 and 2 ^ 8 - 1 = 255), you take its power of 2 ( so, 3 and 8 in our examples) and, if the Lucas-Lehmer number in the position one less than that power ( so, second and seventh) is an exact multiple ( no remainders allowed) of the Mersenne number, then it is definitely prime.”
― Matt Parker, Things to Make and Do in the Fourth Dimension