Drawing 2025: A chromatic approximation of the number π

A digital drawing of mine, Untitled (A chromatic approximation of the number π, the first 768 hexadecimal digits), has been on view the last few months at the online exhibition DRAWING 2025 of John B. Aird Gallery, which was juried and curated by Kathleen Vaughan.  Please feel free to check it, along with great drawings by other amazing artists, out before the show concludes.

WEIHHNACHTEN (1+2+3+4+5+6+7+8+9)^2

TIME be on your side always.  Merry Christmas!  Frohe Weihnachten!

2025

The number 2025 is a very interesting number, which makes the year 2025 a very interesting year ...

The number 2025 = 45² is a perfect square.  The last time we had a perfect square year was the year 1936 = 44², which was 89 years ago.  And the next time we'll have a perfect square year will be the year 2116 = 46², which is 91 years into the future.  So for most of us, the year 2025 is the only perfect square year we experience in our whole life.

Also, perfect squares have this property:

k² = 1 + 3 + 5 +  ...  + (2k − 1)

So, we can also write 2025 as a sum of the first 89 odd integers:

2025 = 45² = 1 + 3 + 5 +  ...  + 89.

The number 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 is a triangular number.  So we have 

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)².

The last time we had a squared triangular number year was the year 1296 = ((1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)², and the next time we'll have a squared triangular number year will be the year 3025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)².  So we are really lucky to experience that in our short life.

The square of a triangular number has this very special property:

(1 + 2 + 3 +  ...  + n)² = 1³ + 2³ + 3³ +  ...  + n³  

So we have:

2025 = (1 + 2 + 3 +  ...  + 9)² = 1³ + 2³ + 3³ +  ...  + 9³,

which is the sum of the first nine cubes!

Aperiodic Ein-Stein

David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss published in March 2023 an infinite family of monotiles that tessellate the plane aperiodically (with the mirror image allowed).  Congratulations on solving the Einstein Problem of tessellation!  Here's an animation of the aperiodic monotiles:

Video from Craig Kaplan's youtube channel

Mural at Dundas West

Since it’s π-day today ...

This is a virtual mural I created last year for Dundas West subway station, Toronto.  The number is actually related to π but I'm not going to disclose the relation.  I will be impressed if someone knows what it is.  Let me know if you think you do.

Happy π-day!

Chromatic approximations of the number π

LUNA is an annual one-evening-only event at Toronto's Crescent School celebrating the arts.  This year's iteration, LUNA Nova, is the first time in three years that this event returns in-person on-site.  Chromatic approximations of the number π is my project for LUNA this year.

π is known to be an irrational number, which means that it is a non-terminating non-recurring decimal.  As it is not possible to exactly represent π with all its digits, we can only approximate it.  My project explores approximations of π using colours.  Here are some snapshots of my project from this past Thursday ...

The "low resolution" paintings:

One of the "high resolution" digital drawings:

Rational approximations of π and the distribution of primes

I found recently this very interesting video that reveals an unlikely connection between rational approximations of π and the distribution of prime numbers.  Enjoy!

Video from 3Blue1Brown's youtube channel

Bailey-Borwein-Plouffe

Here's a special edition for π-day today ...

I created a virtual mural last year using the Bailey-Borwein-Plouffe formula for π on a wall of Whitechapel Gallery, London.  The interior space was part of Elmgreen & Dragset's installation The Whitechapel Pool on the occasion of their survey exhibition This Is How We Bite Our Tongues in 2018:

The Bailey-Borwein-Plouffe formula for π is useful in quickly computing the nth hexadecimal digit of π without computing its preceding digits.

Happy π-day!

Probability for two randomly chosen positive integers to have no common factors

We have previously looked at the evaluation of the Riemann zeta function at 2:

This result can be used to calculate the probability for two randomly chosen positive integers to have no common factors.  It is easy to see that the probability for two randomly chosen positive integers to be both multiples of a prime p is 1/p².  So the probability that they do not share a common factor of p is 1 − 1/p².  Therefore the probability that they do not have ANY common factors is:

Very neat result!

A sequence of prime numbers (from the powers of π)

It's π-day today.  I am going to construct a sequence of prime numbers by applying the floor function to positive integer (n) powers of π,

⌊π ⁿ⌋.

Here are the first few:

The next one ⌊π ³¹⁷⌋ is a 158-digit number.

21148363*

During the months of continual lockdowns, I discovered the livestreams by Romford guitarist Steve Reynolds (1985 - 2021), which were dedicated to health care workers.  His music was in the classic style of the Shadows.  Like so many among his growing circle of followers around the world, I anticipated and watched his shows every week, which have become a much welcomed consolation during this very difficult time.  Although most of us have never met him in person, we all feel fondly like we were in the household of this kind soul.  When I heard the news of his sudden death three weeks ago, it was so hard to believe.  I felt both shocked and deeply saddened.  He has been an inspiration and a moral support to many who were experiencing uncertainty and loss in their lives.  He and his music will be sorely missed.

Besides the Shadows, the music of Billy Fury (1940 - 1983) was often on Reynolds's playlist.  I have never heard of Billy Fury before.  One tragic fact linking Reynolds and Fury is that they both died young, both in January, abruptly.  In 1982 just before he died, Fury recorded his version of Bobby Rydell's 1963 hit Forget Him.  Reynolds's guitar interpretation of the song on video in 2014 was his tribute to Fury.  His own video now is my tribute to him.  RIP, my friend ...

Video from Steve Reynolds's youtube channel

Steve Reynolds ... your kindness shall I never forget.

*21148363 is the 1338771st prime number.

Fröhliche Weihnachten 2^2*5*101

For those of you who have been following my blog regularly, you must have noticed that I have hugely reduced my blog posting since October; and I owe you an explanation.  Because of COVID, my job suddenly became a lot more demanding, in terms of time commitment, since September.  My days, even weekends, have been mostly occupied.  The accumulated strain inevitably led to the reality of me falling sick in early October.  Not getting enough time to rest and recuperate, I continued to be sick on and off for the month that followed.  I was forced to slash my computer screen time, and hence blogging.  Now with the Christmas and New Year holidays, I finally get the chance to publish more posts again, at least for now.

It has been a tough year for many of us.  May reflection be our strength as we head into a new year full of hope and opportunities.  Here's my Christmas e-card this year:

A sequence of prime numbers (with the digits of π)

A sequence of prime numbers, each consisting of the leading digits of π:

3
31
314159
31415926535897932384626433832795028841

The next number in the sequence is 16208 digits long.

Frohe Weihnachten 3*{12^2+23^2}

... my Christmas e-card for 2019

Zeta 2 LUNA Rising

My project for LUNA Rising at Toronto's Crescent School this year is a video projection/performance Untitled (Zeta 2), which is inspired by the evaluation of the Riemann zeta function at 2:

I performed the calculation of the sum by placing adhesive vinyl squares successively on to the floor during the event evening while at the same time a projector screen and 24 computer monitors were displaying animated colours generated from the digits of π.  The installation was enlivened by mathematician/composer Tim Doyle's music Goya's Light.  The annual LUNA event was for one evening only.

Merry Christmas 2(15^2+28^2)

May your days be ...

... merry and bright.

Pythagorean primes

Fermat stated that an odd prime p can be expressed as a sum of two squares if and only if p leaves a remainder of 1 when divided by 4:

p = a² + b² ⇔ p ≡ 1 (mod 4),

where a and b are some positive integers.  Such primes are called Pythagorean primes.

If you are interested, the proof can be found in any elementary number theory textbooks.

A Continuous Becoming

It was π day when I arrived in London yesterday.  I didn't see π but I did see another interesting irrational number, φ, which is the golden ratio  1.618033988749894848204586834... .  Unlike π, which is a transcendental number, though, φ is an algebraic number since it is the positive root of the equation x² – x – 1 = 0.  φ appeared repeatedly at the exhibition A Continuous Becoming.

Giorgio Griffa from his solo exhibition currently at Camden Art Centre, London:

331313234313323332

I found this by Koo Jeong A on Serpentine Galleries' publication Reflections On Zaha Hadid.  Just wondering if anyone could help me decode it:

Snowy Christmas 9^2 + 44^2

Wish you all a hopeful Christmas

And a brave new year