The art of Pi (`\pi`), Phi (`\phi`) and `e`
This section contains various art work based on `\pi`, `\phi` and `e` that I created over the years.
Some of the numerical art reveals interesting and unexpected observations. For example, the sequence 999999 in π at digit 762 called the Feynman Point. Or that if you calculate π to 13,099,586 digits you will find love.
`\pi` day art and `\pi` approximation day art is kept separate.
The accidental similarity number (ASN) is a kind of overlap between numbers. I came up with this concept after creating typographical art about the `i`-ness of `\pi`.
To construct the accidental similarity number (ASN) for three numbers `\pi`, `\phi` and `e`, we first align these numbers and then identify positions for which the numbers have the same digit.
π φ e
3.1415926535897932 … 21170679821 … 10270193852 … 1.6180339887498948 … 93911374847 … 08659593958 … 2.7182818284590452 … 51664274274 … 32862794349 …
These digits are then used to create the accidental similarity number. In this case,
ASN(π, φ, e) = 0.97911 48920 55221 …
By definition, the decimal is held in place.
The posters of `asn(pi,phi,e)` show the accidental similarity number created from the first 1,000,000 digits of each number. The numbers have the same digit at 9,997 positions.
The poster shows 9,996 ASN digits (last one is omitted) because I use the distance between the index of the digits that make up the ASN for the color mapping.
The distribution of distances follows a Poisson distribution with an average of 100, with about 1-1/`e` values being smaller than 100.
The font is Neutraface Slab Display Medium.
properties of the accidental similarity number
Any properties are accidental, but curiously ASN(`\pi`,`\phi`,`e`) ≈ 1.
If you find other curiously accidental properties, let me know.
data files
Download the first 9,997 digits of the accidental similarity number. This file provides the ASN digit index, `i`, the digit, `ASN_i` and the position from which it is sampled, `\text{index}(ASN_i)`.
i ASN_i index(ASN_i) 0 9 13 1 7 100 2 9 170 3 1 396 # e.g. 4th ASN digit is 1, sampled from digit index 396 4 1 500 5 4 596 6 8 607 7 9 694 8 2 825 9 0 828 10 5 841 11 5 941 12 2 1283 ...
Nasa to send our human genome discs to the Moon
We'd like to say a ‘cosmic hello’: mathematics, culture, palaeontology, art and science, and ... human genomes.
Comparing classifier performance with baselines
All animals are equal, but some animals are more equal than others. —George Orwell
This month, we will illustrate the importance of establishing a baseline performance level.
Baselines are typically generated independently for each dataset using very simple models. Their role is to set the minimum level of acceptable performance and help with comparing relative improvements in performance of other models.
Unfortunately, baselines are often overlooked and, in the presence of a class imbalance5, must be established with care.
Megahed, F.M, Chen, Y-J., Jones-Farmer, A., Rigdon, S.E., Krzywinski, M. & Altman, N. (2024) Points of significance: Comparing classifier performance with baselines. Nat. Methods 20.
Happy 2024 π Day—
sunflowers ho!
Celebrate π Day (March 14th) and dig into the digit garden. Let's grow something.
How Analyzing Cosmic Nothing Might Explain Everything
Huge empty areas of the universe called voids could help solve the greatest mysteries in the cosmos.
My graphic accompanying How Analyzing Cosmic Nothing Might Explain Everything in the January 2024 issue of Scientific American depicts the entire Universe in a two-page spread — full of nothing.
The graphic uses the latest data from SDSS 12 and is an update to my Superclusters and Voids poster.
Michael Lemonick (editor) explains on the graphic:
“Regions of relatively empty space called cosmic voids are everywhere in the universe, and scientists believe studying their size, shape and spread across the cosmos could help them understand dark matter, dark energy and other big mysteries.
To use voids in this way, astronomers must map these regions in detail—a project that is just beginning.
Shown here are voids discovered by the Sloan Digital Sky Survey (SDSS), along with a selection of 16 previously named voids. Scientists expect voids to be evenly distributed throughout space—the lack of voids in some regions on the globe simply reflects SDSS’s sky coverage.”
voids
Sofia Contarini, Alice Pisani, Nico Hamaus, Federico Marulli Lauro Moscardini & Marco Baldi (2023) Cosmological Constraints from the BOSS DR12 Void Size Function Astrophysical Journal 953:46.
Nico Hamaus, Alice Pisani, Jin-Ah Choi, Guilhem Lavaux, Benjamin D. Wandelt & Jochen Weller (2020) Journal of Cosmology and Astroparticle Physics 2020:023.
Sloan Digital Sky Survey Data Release 12
Alan MacRobert (Sky & Telescope), Paulina Rowicka/Martin Krzywinski (revisions & Microscopium)
Hoffleit & Warren Jr. (1991) The Bright Star Catalog, 5th Revised Edition (Preliminary Version).
H0 = 67.4 km/(Mpc·s), Ωm = 0.315, Ωv = 0.685. Planck collaboration Planck 2018 results. VI. Cosmological parameters (2018).
constellation figures
stars
cosmology