Topology is a branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending. In mapmaking, topology has to do with how features like points, lines, and polygons share geometry.
Now, what do you suppose might happen if those two points of view of what topology is were combined?
You don't have to wonder, because Topology Fact has taken a map of the lower 48 contiguous states of the U.S. and revisualized it. All state borders have been either stretched, twisted, crumpled, or otherwise bent as needed into rectangles, then positioned to indicate if they border each other or an ocean. Here's the result:
The topologist's map makes it easy to see some neat features that would be harder to see on a more traditional map. Such as:
Maine is the only state that only borders just one other state.
Four states, Maine, Rhode Island, South Carolina, and Washington only border two other states.
Two states, Missouri and Tennessee, border eight other states each (and technically, each other).
No state is more than four states away from an ocean.
The downside to all the stretching is that the topologist's map doesn't do well in depicting how large the states actually are with respect to each other. Texas has the largest area of all the states shown, but is tiny on the map. States like California (second-largest) and Arizona (fifth-largest) look almost identical in size, while Oregon is stretched to enormous proportion.
The map also suggests that both Arizona and New Mexico border an ocean. Both however are landlocked. That could be easily fixed however, if the rectangles for California, Arizona, New Mexico, and Texas were extended to the bottom edge of the map and the rectangle for Oregon shortened to indicate California has a beach.
The map also leaves open the question of how Alaska and Hawaii might be added to this kind of map. As created, they would be isolated islands separated from the rest. But then, that's exactly how wall maps of the United States in schoolrooms often display them.
If you want to take the map of the lower 48 states to another level of abstraction, mathmaticians can also turn to graph theory to revisualize how the boundaries of states are connected to each other. If nothing else, it makes it easier to count the number of borders for each state, but fully disconnects each from their geographic area.
For mathematicians, prime numbers are special. Values that are only evenly divisible by themselves and one, they can be combined through either addition or multiplication to produce every positive integer greater than two, which makes them the building blocks of the set of all whole numbers.
But if you search for prime numbers, you'll find they're not evenly distributed among all those positive integers. While they are really dense at the low end of the positive number line, they thin out and appear less and less frequently as the count of numbers grows ever higher.
That leads to a natural question. If you count high enough, will you run out of prime numbers to discover?
Mathematicians are happy to confirm the answer to that question is no. There's always another larger prime number to be found!
They can confirm that characteristic of prime numbers with certainty thanks to one of the world's oldest mathematical proofs that was documented by the Greek mathematician Euclid around 2,300 years ago. James Grime talks through Euclid's proof in the following seven minute Numberphile video, in which the trick is to assume you can produce a list that contains all prime numbers, then prove that assumption is wrong:
Euclid's logic confirming prime numbers extend into infinity is an example of a proof by contradiction. It's a twisty-turny way of thinking, but one that works.
Here's something else that's neat about the prime numbers of Euclid's proof. If you remember that grid of numbers illustrating patterns of primes we introduced a while back, you'll always find prime numbers that satisfy Euclid's logic in the Set F column of the table.
For example, if we make a set of primes using just the first two in the table, the values 2 and 3, multiplying them together produces 6. Adding 1 to that value gives us 7, which is our first Euclid Set F prime number result if we treat Euclid's proof like an algorithm.
Let's take that algorithm one step further and include the third prime, 5, in the set of primes. Multiplying 2, 3, and 5 together results in a product of 30. Add 1 to that product, and we find 31 is our second Euclid Set F prime.
If we expand the list of primes to include 2, 3, 5, and 7, we find their product is 210 after multiplying them together. Adding 1, we reach the end of the presented table with our third Euclid Set F prime, 211.
We could continue, but since we've exceeded the listed values of the table, let's talk about why that works!
In the table, Set F values are always the result of taking a multiple of 6 and adding 1 to it. The way the table is set up, you can take the value 6, multiply it by the row number (ROW), and add 1 to produce the results shown in the Set F column.
Set F Value = 6*ROW + 1
Euclid's proof is doing identical math! The product of the first two prime numbers, P₁ (2) and P₃ (3), is 6, so we can extract them from the rest of the prime number multiplications:
Q = P₁*P₂*P₃*...*Pₙ + 1
Q = (P₁*P₂)*(P₃*...*Pₙ) + 1
Q = (2*3)*(P₃*...*Pₙ) + 1
Q = 6*(P₃*...*Pₙ) + 1
In this formulation, the product of the prime numbers P₃ through Pₙ is equivalent to the row number of the table. Recognizing that equivalence, we get:
Q = 6*ROW + 1
Where the Euclid prime Q will therefore always be a Set F value from the table. Specifically, they're the ones that correspond to the row numbers that are the cumulative products of prime numbers greater than 3.
It's a typical day for any scientist or engineer. You're doing some calculations while playing around with the fundamental forces of nature and... you suddenly realize that what you're doing could have fatal consequences. As in the potential death toll starts in the dozens, if not hundreds or thousands. Worst case, you find what you're doing could cause the apocalypse.
Movie fans who hung around after watching Barbie to also watch Oppenheimer as part of the "Barbenheimer" social media-inspired double feature craze recently become aware that's a thing the people that do this kind of work deal with on a daily basis. Admittedly, for many of these fans, seeing the main character of Oppenheimer getting all worked up over results of math that suggested a runaway nuclear reaction could end the world could be a real consequence of detonating an atomic bomb caused a real head rush.
So what's in the math that Robert Oppenheimer did that made him think the proverbial end of the world was something that had a non-zero chance of happening because of the work he was doing? Welch Labs explains the math Oppenheimer did in the following short video:
We now return you to the world that continued muddling through when the invention of the atomic bomb didn't cause it to end.
Earlier this year, big news broke out in the world of geometry. A unique shape was discovered that could tile an infinite plane, creating a geometric pattern that would never repeat.
But that discovery left a problem for geometric purists. The 13-sided "hat-shaped" aperiodic monotile that had been identified could do the job, but required the use of "reflected" tiles to continue the tiling. For the purists, that was like using regular glazed square tiles to cover a floor, but sometimes flipping the grout-side of some tiles up to complete the pattern. They wanted a completely unique and uniform tile shape that would create a pattern that never repeats, but only using the same-shaped tiles whose "glazed"-side would always face upward.
So David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss went back to their drawing board and tweaked their monotile design. At the end of May 2023, they posted a new preprint paper announcing their new discovery. They had defined the shape of a unique monotile capable of fully tiling the infinite plane without ever repeating a geometric pattern. Interesting Engineering reports the development:
Not wanting to leave this problem unsolved, the team kept working. They found a different shape closely related to the hat but could cover the surface without needing to be flipped over. It still did the aperiodic tiling, but now with only a single shape, no mirror images were needed.
“I wasn’t surprised that such a tile existed,” said the co-author Joseph Myers, a software developer in Cambridge, England. “That one existed so closely related to the hat was surprising,” he added.
The new shape, which they called "Spectre," was discovered by tweaking an "equilateral version" of the hat, a shape that didn't initially seem to have the aperiodic tiling ability. By modifying this shape a bit, they found it could do the non-repeating tiling without mirror images.
The paper introduces three different variations of the new "strictly chiral" aperiodic monotile, which has 14-sides:
Here's how the geometers describe their new achievement from their paper:
The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.
To really appreciate what they've done, we need to see the new chiral monotiles laid out together on a two-dimensional plane. The following illustration shows the non-repeating pattern that can be created using two of their monotile variations, with the straight-sided polygon on the left-hand side, which transitions into the arc-sided version of the monotile on the right-hand side.
To us, the right-hand side of this figure suggests a devilish application lies ahead for this new geometry. Imagine a jigsaw puzzle where every piece inside the edges uses that exact same shape. *Every* interior piece would fit together regardless of where it might supposed to be within the puzzle!
That may add a whole new level of difficulty to a popular pastime.
References
David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. A Chiral Aperiodic Monotile. [Preprint: PDF Document]. DOI: 10.48550/arXiv.2305.17743. 28 May 2023.
If you ever want to go down a very deep rabbit hole, and you're mathematically minded, searching out patterns in prime numbers is a great place to begin.
In fact, if you just want to start somewhere simple, try this. Set up a table with six columns. Starting in the first row of the table, with the first prime number, 2, number all the boxes of the table from left to right. Then, move down to the next row of the table and continue the numbering process. Add as many rows as you like.
When you've made a decent-sized table, stop adding rows and go back to visually distinguish the prime numbers among all the numbers you wrote. You should pretty quickly find your first patterns.
We did that exercise, using just the numbers from 2 to 211. Here's the table we generated, along with some of the patterns we were able to identify:
Except for the first row, which has four prime numbers (2, 3, 5, and 7), each subsequent row contains no more than two prime numbers in it. Each of those prime numbers appear in just two columns that we've identifed as Set D, which starts with 5 in the first row, and Set F, which starts with 7.
By setting up the list of numbers this way, we've effectively sieved the full set of integers to exclude multiples of 2 (Set A, Set C, and Set E) and multiples of 3 (Set B and Set E). We find that we can define all potential prime numbers larger than three as a multiple of six (Set E), where we either subtract one to obtain a potentially prime value in Set D or add one to obtain a potentially prime value in Set F.
We can then look at how frequently primes appear in these columns. When both the Set D and Set F values in the same row are prime, these are what mathematicians call twin primes. Whenever Set F in one row is prime and is followed by another prime in Set D in the next row, that's what mathematicians call couple primes. And whenever prime numbers appear next to each other in the same column, either in Set D or in Set F, mathematicians call "sexy" primes.
That last kind of prime is actually a bad mathematical pun. You'll notice that the minimum difference between any two sexy primes is six. In Greek, the word for six is εχι. If you're the kind of mathematician who slurs their words after consuming certain beverages, you might say that the Greek word for six issexi....
Unfortunately, not every value in the Set D and Set F columns are prime numbers. Some are composite numbers, which is to say they are found by multiplying prime numbers together, and even they have a pattern. If you multiply two primes from Set D together, you'll find their product in the Set F column. That's also true if you multiply two primes from Set F together.
But if you multiply a prime from Set D with a prime from Set F, you'll only find the resulting composite product in the Set D column.
These are far from the only patterns going on with the distribution of prime numbers. Back in 1963, Stanislaw Ulam found that when he wrote numbers out in a spiral order on a square grid, the prime numbers would appear in diagonal patterns.
The Coding Train's Daniel Shiffman explains how to generate Ulam's Spiral and develops code to generate it in the following 24 minute video:
The Ulam (or Prime) Spiral forms when numbers are written in order in a spiral pattern on square grid paper, but since we already know that the distribution of primes is linked to multiples of six, what would we find if we repeated that exercise using a hexagonal grid?
The following image illustrates some new patterns that emerge in a hexagonal spiral. Just for fun, we've indicated the prime numbers with boldface font and have colored the hexagons that correspond to Set D and Set F in our original table and have used red lines to connect the twin primes:
Using a hexagonal grid like this, additional relationships can be identified to map out where prime numbers may occur. In practice, that means we can break the Set D and Set F values into subsets that can be independently explored. In 2011, Dmitri Tishchenko mapped 90,000 numbers onto a hexagon grid like this, which gives some tantalizing hints at the underlying structure of primes. He had left the task of visualizing the Set D (6n-1) and Set F (6n+1) on the grid for a future project, which is what inspired us to create our hexgrid chart.
Then again, if you really want to things to the next level, you could write out the numbers using polar coordinates on an Archimedian spiral. The patterns that arise when you do that leads to a whole different story for another long holiday weekend.
References
José William Porras Ferreira. The Pattern of Prime Numbers. Applied Mathematics, Vol. 8, No. 2. DOI: 10.4236/am.2017.82015. [PDF Document]. February 2017.
