The Game of Life

Last month the English mathematician John Conway passed away at the age of 80. His name may not be familiar to many of you, but he was the inventor of the Game of Life, in 1970. For several years I have been interested in this game. A suitable time to write a blog about it.

When Conway was still an undergraduate at Cambridge University in the sixties, he became interested in “recreational” mathematics and got in contact with Martin Gardner, who had a popular column “Mathematical Games” in the Scientific American. In October 1970 Gardner published in this column The fantastic combinations of John Conway’s new solitaire game “life” Read here and here interviews with Conway about how he invented the game and that for a long time he actually hated it.

The game is played on a grid of adjoining cells, which can be either alive (black) or dead (white). Each cell is surrounded by eight neighbouring cells and what happens to a cell depends on how many neighbours are alive. These are the rules:

1. When a living cell has 0 or 1 living neighbours, it will die.
2. A living cell with 2 or 3 living neighbours will stay alive.
3. A living cell with 4 or more living neighbours will die.
4. A dead cell with exactly 3 living neighbours will become alive.

With zero or one neighbours you will die from loneliness, with four or more neighbours you die from overcrowding. With two or three neighbours you survive and when there are three parents around, a new baby is born. It looks a bit like life 😉

Of course other rules are also possible and it took Conway considerable time to find rules that gave interesting results. As there were hardly any computers in those days, they used a go board to follow the development of (simple) patterns!

Here are a few simple examples to show how the rules work. I have indicated the number of living neighbours in each cell. Living cells that will survive have a yellow number, those that will die have a red one. An empty cells with three neighbours gets a blue number. After the start pattern I have only marked the cells with get a colored number.

Two patterns have died, two became stable (one of them oscillating).

Here is a pattern of 5 cells. Again I count the number of neighbours, using the same colours, blue, yellow and red for birth, life and death. After 4 generations the original pattern appears again, but diagonally shifted one cell! This pattern has got a name, it is called a glider, it will continue moving diagonally

This pattern of 5 cells is very similar, with only the leftmost cell shifted to the right, but the behaviour is very different! It “explodes” rather chaotically, grows to a maximum of more than 200 cells and finally becomes stable after 1103 generations with 116 living cells, including 6 gliders (notice that three of them are escaping at t = 150). This configuration is called the R-pentonimo .

Of course you can not follow the development of such a pattern with pen and paper or a go-board. You need a computer. In those days they were huge and expensive machines. Here is a PDP-7 computer similar to the one used by Conway. The right picture shows the display screen running a life pattern. Click here for a video. The computers were still so slow that Conway was only able to follow the development of the R-pentomino until t = 460, when the article was published.

Here are a few more interesting patterns. This one is called a Heavyweight spaceship. It moves, like the glider, but orthogonally, two cells in 4 steps.

And here are three oscillators. From left to right Figure Eight (period 8), Pulsar (period 3) and Fumarole (period 5)

The publication in the Scientific American aroused a frenzy of interest among professional mathematicians and amateurs alike. Conway thought himself that no pattern could grow indefinitely and offered a prize of 50 dollar to the first person who could prove or disprove this conjecture before the end of 1970.

It took only a couple of weeks before an MIT group constructed a pattern that generates a glider every 30 moves. therewith proving that patterns can grow indefinitely. It is called the Gosper Glider Gun. and one of the many guns that have been found since then. Keep in mind that all this is the result of 4 simple rules 😉 .

Fifty years have passed since Conway’s invention of the Game of Life, and there is still considerable interest, leading to new interesting patterns every year. There exists a comprehensive Life Wiki, similar to Wikipedia, containing at the moment more than 2100 articles. Here is the main page of the wiki.

Notice at the right a list of pattern categories. The Wiki contains at the moment more than 1350 pattern pages. Each page gives a description of the pattern and an option to watch the development of the pattern, by launching the so-called Lifeviewer at the right side of each page. I have linked the patterns described above to the corresponding Wiki page.

There is a yearly competition for the Pattern of the Year . Here are a few winners :

  • David Hilbert (2019) , 122 cells an oscillator with period 23
  • Sir Robin (2018) , 282 cells, a spaceship moving in an oblique direction
  • Lobster (2011), 83 cells, another spaceship, diagonally moving

Not always use the creators fancy names. Here is the p416 60P5H2V0 gun (left image) It has 26342 cells and fires gliders from four directions which collide in such a way that every 416 generations a 60P5H2V0 spaceship (right image) is produced. The center image shows a just completed spaceship, while gliders are already approaching to form a new one. Fascinating. When you click on the image, you can watch a YouTube video of the process.

You can play with the Lifeviewer , by clicking the image below. You start with a blank grid, where you can draw any pattern you like. The Lifeviewer is versatile and powerful, it may take some time to get to know all the options, just give it a try!

Let me finish this post with a few general remarks.

  • The Game of Life is deterministic but unpredictable. Simple rules lead to complex behavior. All my life that has been a topic of great interest to me.
  • When I got my first desktop computer, in the eighties, I wrote my own Game of Life program, in Pascal and partly in Assembler. I took part in a competition. No prize but a honourable mention that my program was very fast. Nowadays a lot of software exists, powerful and of course much faster. Golly is the most popular one, you can download it here.
  • The Game of Life is much more than a collection of beautiful patterns. It has been shown by Conway himself that the Game of Life is a Universal Turing Machine, it can perform any calculation that a computer can do. The Pi-calculator calculates the decimal digits of π , the Primer uses the Sieve of Eratosthenes to determine the prime numbers. AND and OR gates can be constructed, etc.
  • The Game of life is an example of what is called a Cellular Automaton.
  • Martin Gardner has written three columns about the Game of Life. Here is the pdf-file with all three articles.

Lockdown!

On Monday night, 16 March 2020, the Prime Minister of Malaysia announced in a live telecast that the country would go on lockdown Wednesday 18 March, because of the increasing number of Covid-19 infections in the country. A very strict lockdown, schools and borders (even interstate) would be closed, people had to work from home, only essential shops (groceries, pharmacies, etc) would remain open. People had to stay at home, no outdoor exercise, no social visits allowed. It was called a Movement Control Order (MCO)

In this post I will give an impression about our life the past 2 months.

The next day there was a rush on supermarkets and groceries, to buy food. We were a bit late, many shelves were empty already.

That day was also my last chance to visit Bukit Kiara. This time I walked in the lower part of the park (green line). The prison fence in red.

The first day of the lockdown. Visiting a supermarket to buy food was still allowed, so I walked to Tesco, carpark almost empty. Not much useful stock left in Tesco, because many people had been panic buying. Other shops closed, also next door IKEA and Mcdonalds.

We managed to buy some canned food, crackers, maggi mee, just in case there would be a shortage.

People were advised to wear masks when going out. Although I was personally not convinced that it would help, we went with the flow.

But our supply of masks was very limited. Hin, a Kiara friend of mine, had ordered a few boxes, and I could buy one from him. A transaction without physical contact, I drove to his house where the box was waiting for me on a pillar next to his gate. Payment online 😉 .

We were not allowed to receive visitors in our condo. So, for many weeks I didn’t talk to anybody, except the occasional cashier in the supermarket. I don’t think I would have managed without Aric. Whatsapp also helped, I spent many hours a day chatting with family and friends. Physical distancing led to social bonding! But still I had a few days of depression.

Most of our shopping we did in the Jaya supermarket, about 500 m from our condo. After the first days of “hoarding”, stock was generally sufficient. When I went shopping I always walked, to have at least some exercise.

Supermarkets had introduced a door policy, limiting the number of customers, measuring their temperature and sometimes providing them with plastic gloves. It resulted in sometimes large queues, but as a senior citizen I didn’t have to queue!

To have more exercise I sometimes walked to Tesco about 2.5 km one way. The first time I was a bit worried about the police, as they were sometimes overreacting. One senior citizen had been arrested because he was walking to the grocery, 300 meter from his home, wearing sport shoes. Read the report here. But nothing happened to me (I was wearing sandals haha)

Regarding food, before the lockdown it was our usual routine to go out for dinner a few times a week, or order food to be delivered, and only prepare food ourselves one or two times a week. But now all restaurants were closed and Aric was reluctant to have food delivered by Grab or Panda, because of the hypothetical risk of infection. So from 18 March until last week, we have been preparing dinner ourselves every day!

Here is a selection of dishes prepared by Aric. Mostly Chinese cuisine, from the nice composition you can see that he is a designer 🙂

Here is a selection of my “creations” , Italian food and traditional Dutch fare.

With so much nice food and without my usual hiking in Bukit Kiara, it was no wonder that I gained some weight during the past period.

Did I do anything else beside eating and chatting? Yes, I watched a lot of movies. I am a fan of Pier Paolo Pasolini and many of his films can be downloaded from the Internet. I watched Edipo Re (1967), Teorema (1968), Porcile (1969), Arabian Nights (1974) and a few more.

Another favourite of mine is the Taiwanese film director Tsai Ming Liang . I watched Rebels of the Neon God (1992), Vive l’Amour (1994), The River (1997) and The Hole (1998). I am now watching I don’t want to sleep alone (2007), shot in Malaysia and originally banned here because it showed the country “in a bad light”.

Two more films I watched and (only) one book I read. Monty Python’s Life of Brian (1979) is a religious satire about Brian, a Jewish boy, who is born on the same day as Jesus and is mistaken for the Messiah. It was so controversial that it was banned in several countries and I had never watched it, although I was a big fan of Monty Python. Hilarious movie.

Not hilarious at all, actually quite scary, is the movie Contagion (2011) . It describes quite accurately a virus outbreak similar to the Covid-19 pandemic. Contact tracing, fomites, the frantic efforts to develop a vaccine.

I was planning to read more, but I only finished one book. 21 Lessons for the 21st Century by Yuval Noah Harari, the author of two bestsellers, Sapiens and Homo Deus. The book was published in 2018, before the Covid-19 pandemic, otherwise he would probably have added a 22nd lesson, read for example this article written by him: In the Battle Against Coronavirus, Humanity Lacks Leadership.

I also tried to blog, but I was not always in the mood. I only published one post about an unknown opera from Domenico Scarlatti. So unknown, that it took me a long time to Google for information. Here is the result, click on the image to go to the blog.

Of course I followed the news about Covid-19, especially in Malaysia and the Netherlands. Here are two graphs (from the Worldometers site). It is interesting to compare them. First of all, the vertical scales are different! Many more people have been infected in the Netherlands than in Malaysia. When you take into account the difference in population (Malaysia has almost twice as many people), the difference becomes even more dramatic. Per 1 million people Malaysia has ( as per 17 May) 213 infections and 3 deaths, compared to the Netherlands 2561 infections and 331 deaths.

Another conspicuous difference is the shape of the graph. You expect a bell-shaped curve (for many years already I am planning to write a blog about exponential growth and S-curves) , for the Netherlands that is roughly the case, but not for Malaysia, where there is a sudden start of infections around 17 March. An explanation can be found here.

On 17 April I celebrated my 76th birthday. A lockdown birthday, no visitors of course. But a few days before my birthday I received an email “16282 is out for delivery” from a company selling liquor. As I had not ordered anything, I thought it might be spam and didn’t pay attention to it. So I was very surprised that GDEX actually delivered a package with a nice bottle of whisky! Turned out to be a present from our UK friend Rodney! Very much appreciated 😉 . Aric surpassed himself by preparing a sublime meal and baking a delicious birthday cake.

The first MCO was for two weeks, but it was extended by the government several times. On 1 May the Prime Minister announced that from 4 May some of the regulations would be relaxed, the Conditional MCO . For me the most important relaxation was that we were again allowed to be outside , walk and exercise. Immediately I started walking around our condo. No jungle, but at least green and refreshing.

Unfortunately Bukit Kiara still remains closed, but it is possible to hike in North Kiara. Of course keeping social distance 🙂

The CMCO has been extended until 9 June. If any new developments happen, I will update this post

An opera and some history

Domenico Scarlatti (1685 – 1757) is nowadays mainly known for his 555 keyboard sonatas. Click here for a nice collection. His father Alessandro specialised in operas (114!) and that may have been a reason that young Domenico wrote one himself, in 1703 when he was 18 years old.

Here is the opera L’Ottavia restituita al Trono (Octavia returned to the throne).

Pleasing music, fresh and youthful. It is a concertante recording, seven singers, no stage, no choir.

I listened several times to the recording, trying to understand the “plot”with the help of the (French) subtitles.

The main characters are Nero, Octavia and Poppaea. Nero, Emperor of Rome is sung by a countertenor. Octavia, his divorced wife and Poppaea, his future bride , are both sung by sopranos.

Floro is a Prince of Epirus and Rosilda his wife/bride. They were shipwrecked and got separated. Both are sung by sopranos (Floro in travesti)

Two supporting roles. Dorillo is a young servant of Floro, sung by a sopranist (a male soprano). And Belisa, an elderly female, is sung by a tenor!

So the three male parts, Nero, Floro and Dorillo, are performed by a countertenor, a soprano and a sopranist, whereas the female role of Belisa is sung by a tenor. No wonder that at first hearing I was confused.

The first performance of the opera took place in November 1703 in Naples. Apparently the work received a favorable reception, but later it has fallen into complete oblivion until recently. The original libretto has been preserved, but the music was thought to be lost, until copies of almost all arias were found in the library of the Naples Conservatory. Based on this material, two Italian musicologists, Antonio Florio and Alessandro Ciccolini, have made a reconstruction of the opera. It was performed in 2007, with Antonio Florio himself conducting.

Before giving a summary of the opera , here is some historical background.

Nero, Octavia and Poppaea are historical characters during the Roman empire. Nero was born in 37 AD and married Claudia Octavia in 53 AD when he was 16 years old and she 14. One year later , at the age of 17, he became Emperor of the Roman Empire. Octavia became Empress and was widely respected by the Roman citizen body. But their marriage was unhappy and Nero soon got mistresses. One of them was Poppaea Sabina, seven years his senior, and an ambitious schemer. When she became pregnant in AD 62 he divorced Octavia and almost immediately married Poppaea. Octavia was banned to the island of Pandateria. The citizens of Rome protested so strongly that Nero almost remarried Octavia, but instead he decided, supported (urged?) by Poppaea, that she should be killed. She was 23 years old and, according to the historian Tacitus, she was decapitated. In this painting (Giovanni Muzzioli – 1876 ) Poppaea presents Octavia’s head to Nero.

From the title “Octavia returned to the throne” , you may guess already that the content of Scarlatti’s opera is not historically accurate. The opera consist of three acts, although in the recording there is no interval between them.

———————————————–
Act I

Octavia is back in Rome, disguised as a shepherdess and planning revenge against Poppaea. Nero is going to get married to Poppaea, but also charmed by every nice lady he meets. And Poppaea loves Nero, but she is concerned about the negative reaction of the Roman people and willing to give up her position as empress.
Floro tells Nero about his shipwreck and is appointed as captain of the army. Belisa introduces Rosilda to Nero who is charmed by her beauty, much to Poppaea’s concern. Floro recognises Rosilda and is worried too about Nero’s avances. He decides to hide his identity to Rosilda and “become” Clearque . The act ends with a hilarious duet between Belisa and Dorillo. Belisa fancies Dorillo, who is not interested. Click on the picture to hear the duet.

———————————————–
Act II

Nero and Clearque meet a shepherdess, not realising it is Octavia. She tells Nero her sad fate and that she is looking for revenge. Nero is charmed by her beauty and tells Clearque that it is no problem to love more than one. He tells the same to Poppaea, who is furious about his interest in Rosilda. When Nero meets Rosilda, he tries to flirt with her. She resists and asks Clearque to help her.
Octavia is ready to take her revenge on Poppaea, who is complaining and sad about what is happening. When she falls asleep, Octavia prepares to kill her, but Nero arrives in time to prevent it. Octavia flees and Poppaea blames Nero, who promises to find and punish the shepherdess.
Clearque finds Octavia and after he discovers that they both have a grudge against Nero, he decides to help her. This act ends with another funny duet between Dorillo and Belisa. Click on the picture for the duet.

———————————————–
Act III

Octavia tells Clearque who she is and tells him that he can help her by raising the anger of the people and tell Nero that the only solution is to take back Octavia. Later she meets Belisa, who is looking for Dorillo, and also tells her who she is, asking Belisa for help. Belisa advises Octavia to hide in a garden where Poppaea and Nero will meet.

In the garden Octavia attacks Poppaea, Poppaea defends herself, and again Nero arrives in time. Clearque tells Nero that the people are revolting and want his death unless he takes back Octavia back on the throne. Nero laments that, still loving Poppaea, he has to give her up. Poppaea says that she will give up her position.

Clearque is finally convinced of Rosilda’s faithfulness and reveals to her that he is actually Floro. Both are happy. Nero asks Floro to call Octavia, he apologises to her for his behaviour. She suggest that he go back to Poppaea, but Floro says again that the people want her back. Then she accepts to come back to Nero, for the love of the country , although he is not worth it. Poppaea is not jealous and Octavia forgives her. Finally all (including Poppaea !) sing about the triumph of the God of Love.

———————————————–

When you have read the above summary of the opera, I think you will agree with me that there is hardly anything in it that is historically correct. Yes, the fear of Nero for a revolt of the people. What I found surprising is that Poppaea, considered by most historians as an ambitious, ruthless schemer , is here portrayed as being considerate, genuinely in love with Nero, whereas it is Octavia who , by hook and by crook, wants to get back her position as empress. Nero comes across as a womanizer who is ultimately only interested in his own survival.

A few more comments

  1. The (infamous) Nero period of the Roman Empire must have fascinated composers. In 1643, Monteverdi wrote the opera L’incoronazione di Poppea (The Coronation of Poppaea), based more or less on the historical events. This opera is performed regularly, here is a beautiful recording by the Monteverdi Choir. And in 1704 Handel wrote the opera Nero, the music of which has been lost, only the libretto survived.
  2. Bach, Handel and Scarlatti were all three born in 1685. Bach never wrote an opera, Scarlatti only a few, Handel about fifty! In 1709 he wrote the opera Agrippina about Nero’s mother. Here is a brilliant modern recording by an American ensemble Ars Lyrica Houston
  3. Almost no information exists on the Internet about Scarlatti’s opera. Finally I found this useful link: OTTAVIA, 300 anni fa… It is written in Italian, I needed Google Translate 😉 As far as I know only one recording of the opera exists on YouTube, I found two arias, not related to this recording, a duet between Poppaea and Nero and an aria by Belisa That’s all.
  4. A bit more history. Nero murdered Agrippina in 59 AD. And in 65 AD, in a fit of rage, he trampled on the belly of pregnant Poppaea, killing her. Nero went in deep mourning and two years later married the boy Sporus, who resembled Poppaea, after having him castrated. Nero committed suicide in 68 AD , 31 year old.

It was a real pleasure, although time consuming, to find out more about this delightful “little” opera. Hopefully it will be performed more often.