Once, I was in Princeton and attended a lecture by Persi Diaconis. John Conway was also there. During his presentation, Persi mentioned John by referring to him as Wonderful John Conway. John couldn’t resist and said, “Thank you, you are one of not many people who remember my real first name.”Share:
Archive for the ‘John Conway’ Category.
My favorite of the Escher plane tessellations is the one with shells. It is stunning, and the mathematics behind it is beautiful. I want to thank the late John Conway for teaching me the secret of this drawing.
Mathematicians are interested in tessellations because of the symmetries behind them. This tessellation has translational and rotational symmetries. Can you find them?
When I ask my students to find the rotational symmetries, they immediately tell me that they see two different 4-fold points, aka points where 90-degree rotations preserve the drawing. One point, I call G, is where four greenish shells meet, and one point, I call R, is where four reddish shells meet.
As you might have guessed, the students’ answer is not quite correct. There is more to the picture. Look at a dark-brown shell that looks like a curvy rectangle. This shape has markings. Now look at a specific point R and its four closest brown shells. You can see that going around this point R, the brown shells alternate their orientation: the darker side of these shells either faces towards point R or away.
The big secret of this artwork is that it contains TWO symmetry groups: a group and a subgroup. If we ignore the markings on the brown shells and consider them one solid color, then point R is indeed a 4-fold symmetry point. In addition, the center of the brown shape is a 2-fold symmetry point. Thus, the symmetry group of this simplified drawing is 442 in orbifold notation.
If we take the markings of the brown shells into account, then point R is not a 4-fold rotation, it is a 2-fold rotation. Point G keeps the property of being a 4-fold rotation. If you know your symmetry groups, you can conclude that there should be another 4-fold rotation. But where is it?
I will spill my answer. The symmetry point G is not ONE symmetry point anymore. There are two different points where greenish shells meet. The dark side of the brown shells faces one of them and looks away from the other.
The dark secret of this drawing is that it demonstrates two symmetry groups: group 442 and its subgroup 442, with different fundamental regions. To see the secret, you must look closely at a dark brown shell and find its darker side.Share:
What is base 3/2? One of the ways to define such a base is to think of it in terms of exploding dots. What the heck are exploding dots? They are explained and popularized by James Tanton in his YouTube videos.
Essentially, “exploding dots” is a machine made of a row of boxes with rules describing how the dots loaded into the machine explode. As an example, let me describe the 1←2 machine, which corresponds to base 2. We load N dots into the rightmost box. Whenever there are 2 dots in one box, they explode into 1 dot in the box to the left.
For example, to write 5 in base 2, we would first load 5 dots in the rightmost box, as in the figure above. Then each group of 2 dots in the rightmost box would explode, and for each group, 1 dot would appear in the box to the left. Finally, the 2 dots in the second box would explode into 1 dot into the next box to the left. By reading the number of dots from left to right, we get 101, which is 5 in base 2.
The interesting thing here is that there is no reason this model should be exclusive to integer bases. Suppose our rule is that 3 dots explode into 2 dots in the box to the left. Such a rule is called the 2←3 machine, and it corresponds to base 3/2. To represent 5 in this base, we load 5 dots into the rightmost box, then we use the exploding rule shown in the figure below. Using this machine, 5 is represented in base 3/2 as 22.
The figures were made by my junior PRIMES STEP group, in the 2017-2018 academic year, for our paper, Variants of Base 3 Over 2.
But, in this post, I want to discuss a different paper from the same academic year. With my senior PRIMES STEP group, we wrote a paper On Base 3/2 and its Sequences. A shorter version which includes a tribute to John Conway, appeared in The Mathematical Intelligencer.
Speaking of John Conway, he liked inventing new sequences, especially ones with unusual behaviors. One of his hobbies was tweaking the Fibonacci rule to create new sequences, which he called Fibs. For example, the sorted Fibs sequence starts the same as the Fibonacci sequence with 0 and 1. To calculate the next term, we add the two previous terms and sort the digits in non-decreasing order. In base 10, this sequence is A069638: 0, 1, 1, 2, 3, 5, 8, 13, 12, 25, 37, 26, …. It is known that this sequence is periodic with a maximum value of 667.
With my senior PRIMES STEP group, we studied analogs of this sequence in base 3/2. We begin with the sorted Fibs sequence fn with the same two initial values that start the Fibonacci sequence: f0 = 0 and f1 = 1. To calculate fn+1, we add fn-1 and fn in base 3/2 and sort the digits in non-decreasing order. It follows that the numbers in the sequence are written as several ones followed by several twos. Unlike base 10, the sequence is not periodic and grows indefinitely: 0, 1, 1, 2, 2, 12, 12, 112, 112, 1112, 1112, 11112, …. In recognition of the constant growth of this Fibs sequence, we call it the Pinocchio sequence.
Obviously, you can start the sorted Fibs sequence with any two numbers. But we proved an interesting theorem which stated that any sorted Fibs sequence eventually turns into either the tail of the Pinocchio sequence or the 3-cycle 112, 1122, 1122.
However, we didn’t stop there. There are two natural ways to sort the digits of a number, in increasing or decreasing order. Naturally, there is another type of sequences worth considering, in which the digits are sorted in non-increasing order. We called such sequences the reverse sorted Fibs.
We defined the reverse sorted Fibs sequence rn in base 3/2 as follows. To calculate rn+1, we add rn-1 and rn in base 3/2 and sort the digits in non-increasing order, ignoring zeros. It follows that after the initial terms, the terms of such a sequence are represented with several twos followed by several ones. We call the reverse sorted Fibs that start in a similar way to the Fibonacci sequence with r0 = 0 and r1 = 1, the proper reverse sorted Fibs. Here are several terms of the proper reverse sorted Fibs: 0, 1, 1, 2, 2, 21, 21, 221, 2211, 221, 221, 2211, 221, 221, 2211, …. This sequence becomes cyclic, starting from r7.
We also found one reverse sorted Fibs growing indefinitely: 2211, 2211, 22211, 22211, 222211, 222211, and so on. We proved that any reverse sorted Fibs sequence eventually turns into either this sequence or a 3-cycle sequence: 221, 221, 2211. The similarity between the sorted Fibs and the reverse sorted Fibs surprised us. Up to the initial terms, they both have exactly one sequence which grows indefinitely and one 3-cycle. To emphasize this similarity, we reversed the word Pinocchio and named this growing reverse Fibs sequence the Oihcconip sequence.
Now I need to figure out how to pronounce the name of this new sequence.Share:
My friend, John Conway, had a trick to help him with tricky situations. Whenever he needed to make a non-trivial decision, he would ask himself, “What would John Conway do?” As he explained to me, he had in mind the public image he himself created. He liked the image and thought this mental trick helped him be a better, more productive, and not-to-forget, flashier person.
From time to time, I catch myself in need of a decision and ask myself, “What would John Conway do?” And he gave me the answer: I should change the question and ask myself, “What would Tanya Khovanova do?”Share:
Yesterday I stumbled upon a picture of John Conway I completely forgot I had: it was saved in a wrong folder. The photo was taken at the MOVES conference in 2015. There is a third person in the original shot, but I do not remember her. I decided to cut her out as I didn’t know how to contact her for permission to post this photo.
(I wrote this piece for La Recherche. It was translated into French by Philippe PAJOT. You can find this piece and pieces by others at John Horton Conway: a magician of maths disappears.)
Unlike many other mathematicians I know, John Conway cared a lot about the way he presented things. For example, in the puzzle he invented—known as Conway’s Wizards—the wizards had to be riding on a bus. Why was the bus so important? You see, the numbers in the puzzle were related to the age of one of the wizards, the number of the bus, and the number of the wizard’s children. It was important to John that the readers be able to use a convenient notation a, b and c for these numbers and remember which number is which.
When I give my lecture on integers and sequences, I show my students a list of different famous sequences. The first question from the audience is almost always: “What are the Evil Numbers?” As you can guess the name for this sequence was invented by John Conway. This name was invented together with the name of another sequence which is called Odious Numbers. These two sequences are complementary in the same sense as even and odd numbers are complementary: every natural number is either evil or odious. The names are good, not only because they attract, but also because they help remember what the sequences are. Evil numbers are numbers with an even number of ones in their binary representation. I assume that you can interpolate what the odious numbers are.
When he was lecturing, John used all sorts of tricks to emphasize important points: From time to time I saw him shouting or throwing his shoes. Once I remember him staring at his statement written on the blackboard for a really long time. My neighbor in the lecture hall got uncomfortable. He assumed that John, who was at that time way over 70, was blanking out and had forgotten what he wanted to say. I calmed my neighbor down. It was my fourth time listening to the same lecture, including the same pause. John Conway didn’t forget.Share:
Once John Conway showed me a cute way to enumerate Latin squares of size 4, up to the movements of the plane. It was a joint result with Alex Ryba, which is now written in a paper Kenning KenKen.
For starters, I want to remind you that a Latin square of size n is an n by n table filled with integers 1 through n, so that every row and column doesn’t have repeated integers. KenKen is a game that John Conway likes, where you need to recover a Latin square given some information about it.
Let me start by describing a particular shape of four cells that one digit can occupy in a Latin square of size 4. There are only seven different shapes. To get to the beautiful result, we need to number these seven shapes in a particular order starting from zero. The shapes are shown below.
There are 12 different Latin squares up to movements of the square and relabeling the digits. Here is how Conway and Ryba matches shapes and squares. For each Latin square, take the shapes of all four digits, remove the duplicate shape numbers and sum the leftover shape numbers. You will get a unique number from 1 to 12 that represents a particular Latin square. For example, consider the square on the picture below.
Digit 1 is shape 4, digits 2 and 4 form shape 2, and digit 3 forms shape 6. Shape 2 is used twice, and we ignore multiplicities. So we have shapes 2, 4, and 6 used. The resulting Latin square is number 2 + 4 + 6, that is 12. It is a fun exercise to try to find all the squares. For example, square 1 can only use shapes 0 and 1. But shape 1 uses exactly one corner. So the first square should use each of the digits in shape 1.
John likes finding interesting ways to remember which shape is which. You can find his and Alex’s suggestions in the paper which Alex submitted to the arxiv.
John Conway came to the 2017 MOVES conference and told me that he wanted to talk to me about subprime fibs. The subprime Fibonacci sequence was invented by John Conway, and I wrote a paper about it. The paper, Conway’s Subprime Fibonacci Sequences, wasn’t written with John, but rather with Richard Guy and Julian Salazar, and is published in Mathematics Magazine.
I wanted to visit my friend Julia, who lives in Princeton, and this was a good opportunity to discuss the mysteries of subprime fibs with John. On my second day in Princeton, I came to the math department around 3:00 pm carrying some apples. John never goes out for lunch, as he has trouble walking, so he is always hungry by the end of his work day. Thus, each time I go visit him, I come with food. We have very different tastes in apples: unlike me, he likes his apples unwashed.
Anyway, by the time I arrived to the department, John had already left. This was somewhat unusual, so I called him. He sounded weird and not very coherent, as if he wasn’t feeling well. Considering also that he had left early, I started to worry. Unfortunately, there was a lot of background noise during our conversation and I only understood that he was at a pizza place. John walks very slowly, so he couldn’t have gone too far away from campus. I found him in the second pizza place I checked. It was Tiger’s Pizza. He told me that he felt very sleepy and tired. However, I was gratified to see how much having an interested listener gave him energy. He started telling me stories of his trip to Germany a long time ago. He had already eaten, but decided to have some more fries. As a perfect gentleman he offered me some, but I didn’t want any.
At some point he dropped a couple of fries on the floor. He tried to reach them and I jumped to help. That was a mistake. I actually know that he likes proving to me and to himself that he can do stuff independently. He accepts my help when I am subtle about it, or when it is unavoidable. Anyway, he looked at me angrily and I backed off. He picked up his fries from the floor and ate them.
I liked his T-shirt and tried to take a picture of it. As you can see, I am no photographer. The T-shirt shows a test question: Name the triangles. Then it features three triangles: an equilateral, isosceles, and right. It also provides someone’s answers to this naming test: Geoffrey, Frederick, Eugene.
John asked me if I am more scared of Donald Trump or Kim Jong Un. We agreed that Trump is scarier. At this time he seemed his usual self.
I offered John a ride home, as I do whenever I visit him. He was very glad as he felt very tired. He started to get up. This time, I remembered not to try to help. He couldn’t get up, I waited. He tried to push his weight off the table top, but the table was wobbly. I leaned on the table, as if to rest. We often play this sort of game in which he welcomes my help as long as we both pretend that I’m not helping.
My car was a block away and he wanted to walk the block. But after making two steps out of the pizzeria he changed his mind and asked me to bring the car to him. This was the first time ever. This visit he was so much worse than ever before.
On the drive to his place, he gave me a puzzle:
John’s puzzle. Given a Mebius strip with a hole, how do you embed it in 3-D so that the two circular borders of the surface are equivalent?
I dropped him off at his house and offered to walk him to the door. He refused. I sat in my car and watched him walking very slowly along his path. I had this sinking feeling in my gut that I was seeing John for the last time. I drove away, once he disappeared behind his door.
On my way back to Boston I visited my friend Vitaly in East Brunswick, and the next day my high school friend Olga in Edison. In Edison, my car started beeping and I panicked. I was far away from home, and didn’t want to be stuck in NJ. I started to look for the source of the sound. It was John’s phone. As always, my gut feeling deceived me: I had to go back to Princeton.
I drove back to John’s apartment. His door was unlocked and I entered. He was resting in bed. He was greatly annoyed at being disturbed. I explained the reason, and gave him his phone. He took the phone and said, “Off you go.” I had this sinking feeling in my gut that these words would be the last words that I would hear from John.Share: