▲ 2026 π DAY | Art for the 5%. Digits of `\pi` hidden in plain view. (explore the art)

▲ 2026 π DAY | Art for the 5%. Digits of `\pi` hidden in plain view. (explore the art)

π Day 2026 Art Posters - Art for the 5%

On March 14th celebrate `\pi` Day. Hug `\pi`—find a way to do it.

For those who favour `\tau=2\pi` will have to postpone celebrations until July 26th. That's what you get for thinking that `\pi` is wrong. I sympathize with this position and have `\tau` day art too!

If you're not into details, you may opt to party on July 22nd, which is `\pi` approximation day (`\pi` ≈ 22/7). It's 20% more accurate that the official `\pi` day!

Finally, if you believe that `\pi = 3`, you should read why `\pi` is not equal to 3.

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Most of the art is available for purchase as framed prints and, yes, even pillows. Sleep's never been more important — I take custom requests.

Welcome to this year's celebration of `\pi` and mathematics.

The theme this year is a homage to Shinobu Ishihara (1879–1963) and embodies the Japanese saying "To show something, hide something.".

This year, the digits of `\pi` are discernable only by people with colorblindness. It's the only test where a negative is a positive!

This year's `\pi` poem is Nothing Gold Can Stay by Robert Frost.

This year's `\pi` day song is Colors by Laleh.

▲ ART FOR THE 5% | The digits are much more obvious to someone with the most common type of colour blindness (deuteranopia).

Here I walk you through my method of creating patterns that are only visible to people with colour blindness. The approach is based on so-called “hidden” plates from Ishihara's Tests for Colour Deficiency.

About 5% of the male population have fewer-than-normal number of M-cones (deuteranomaly) and about 1.5% are missing them altogether (deuteranopia).

Putting it another way, in a group of 8 men, chances are 50% that at least one has some degree of colour blindness.

Other forms of colour deficiency are much rarer. About 2% of males have fewer (or missing) L-cones (protanomaly or protanopia) and only a tiny fraction (<0.01%) have fewer (or missing) S-cones (tritanomaly or tritanopia).

Females are very rarely affected because colour blindess is recessive and linked to the X chromosome and females have two X chromosomes and, therefore, very likely a working copy of the neceesary gene.

In colour blindness, one or more types of colour photoreceptors in the retina are reduced in number or missing entirely.

There are three types of cones in the retina. They're labeled with letters S, M, or L to mean that their sensitivity to light is highest for short (blue), medium (green), or long (red) wavelengths.

▲ HOW COLOURBLIND PEOPLE SEE | If you have a cone deficiency, you see the world in a different way than those with normal vision.

People with colour blindness see fewer colours. They also cannot (or only with difficulty) distinguish certain colour pairs.

It's useful to think of normal colour vision as 3-dimensional and colour blindness as 2-dimensional. In other words, normal vision requires three colour primaries whereas colour blindness requires only two.

In colour blindness, one of the primaries will be “invisible” invisible. The nature of the primary depends on the type of colour blindness.

The consequence of an invisible primary is that we can add as much of it as we want to any colour and this will not change how the colour is perceived by someone with colour blindness.

Mixing a visible with an invisible primary creates indistinguishable colours that are called (for obvious reasons) “confusion colours” In color space, they line along “lines of confusion”.

▲ The lines of confusion in `xy` chromaticity space for protanopes, deuteranopes and tritanopes. Colors along these lines are indistinguishable to those with the corresponding colorblindness. The colored triangle is the sRGB gamut — the range of colors for most monitors. The horseshoe shape is the 1931 CIE color space and represents the range of human color perception. The hollow point in the center is the D65 illuminant white point (midday light).

▲ The lines of confusion in `xy` chromaticity space for protanopia. The colored triangle shows the sRGB gamut transformed to as they would appear to a protanope. Two specific lines are also shown: the line of confusion through the D65 illuminant (long dash) and the line between the two colorblind primaries (short dash).

▲ The lines of confusion in `xy` chromaticity space for deuteranopia. The colored triangle shows the sRGB gamut transformed to as they would appear to a deuteranope. Two specific lines are also shown: the line of confusion through the D65 illuminant (long dash) and the line between the two colorblind primaries (short dash).

▲ The lines of confusion in `xy` chromaticity space for tritanopia. The colored triangle shows the sRGB gamut transformed to as they would appear to a tritanope. Two specific lines are also shown: the line of confusion through the D65 illuminant (long dash) and the line between the two colorblind primaries (short dash).

Let's pick three confusion colour pairs for deuteranopia.

Pink/cyan colours map onto grey, rose/blue map onto blue, and red/green colours map onto yellow,

▲ EFFECT OF DEUTERANOPIA | If your M-cones are missing, you have deuteranopia. Here are three colour pairs that you cannot distinguish.

At this point we can make an observation that will be the basis of how this year's `\pi` Day art was created: yellow and grey are quite different but the red/pink and green/cyan pairs are relatively similar to someone with normal vision.

By using one colour from each pair we can create an image that's discernable to someone with deuteranopia but vexing to look at for someone with normal colour vision.

▲ CONCEPT BEHIND THE ART. | Red and pink (or green and cyan) are similar to someone with normal vision and map onto yellow and grey.

For the rest of this page, when I mention colour blindness assume deuteranopia, unless otherwise specified.

Now that we've seen what confusion color pairs look like next to each other, we can now start creating patterns that will be hidden to people with normal vision but discernable by those with colour blindness.

The effect is subtle. I've spent a lot of time figuring out how to tune this process to create as much confusion for the normal eye and as little confusion to the colour blind eye.

I'm not colour blind, but I can use colour blindness simulations to show me what the art will look like to someone with colour blindness.

YMMV, depending on your committment to seeing a pattern or the severity of your colour deficiency.

Let's start with a reference image in which the digits are visible to everyone.

▲ REFERENCE PLATE | Everyone can see the digits.

The first row of plates is how they look to someone with normal colour vision. The second row simulates deuteranopia.

If you can't see a digit, then you have achromatopsia, a very rare condition in which the eye has no colour perception whatsoever.

I've previously photographed and analyzed the size distribution of circles on Ishihara test plates.

All the art reflects this distribution and plates are sized to reflect Ishihara test plates, which are 9 cm in diameter.

If we use the green and green-cyan pair for our test plates, the digits take their first steps towards disappearing for someone with normal colour vision.

▲ GREEN/CYAN CONFUSION COLORS | Green and cyan will frustrate someone with normal vision.

The same thing happens when we use the red and pink pair.

▲ CONFUSION COLORS | Red and pink will frustrate someone with normal vision

Since (red,green) and (pink,cyan) are confusion colour pairs, we can use one pair for the background of the plate and another for the digit of the plate and the deuteranope will not know the difference.

▲ RANDOMLY SAMPLING CONFUSION COLOR PAIRS | By randomly sampling from confusion colors we can make the image look busier to someone with normal vision without affecting how someone with colour blindness sees it.

We're taking our first real steps towards hiding the digits from normal eyes.

In the above example, the colours were sampled randomly within each pair. In other words, a given dot in the background (or digit) had the same chance to be red or green (pink or cyan).

For his hidden plates, Ishihara grouped colours into stripes — this helped confuse the normal eye.

▲ ISHIHARA'S TEST PLATE 18 | Colors in Ishihara's hidden plates flow across the plate like a stream. These large-scale patterns further confuse the normal eye.

I've taken this idea and implemented it by using a 5-tone texture generated from a reaction-diffusion pattern. to assign each dot one of 5 tones.

▲ TEXTURE PATTERN | A 5-tone texture pattern created by a reaction-diffusion system will be used to control how colours are sampled from confusion pairs.

Each dot now has two properties: group (digit, background) and texture level (0–4).

Let's make a very simple mapping between texture and color that alternates the colors between adjacent levels of texture tone.

▲ SAMPLING CONFUSION COLORS BASED ON TEXTURE | Large-scale patterns in color distribution will confuse the normal eye. A deuteranope will not see this variation.

Here's what our digit test plates look like once we've gone from random colour selection to selection based on the texture tone.

▲ SAMPLING CONFUSION COLORS BASED ON TEXTURE | Large-scale patterns in color distribution will confuse the normal eye. A deuteranope will not see this variation.

The texture pattern does not repeat across the artwork — each panel will have a different arrangement of colours.

To confuse the normal eye even more, let's map one of the texture tones onto its colourblind equivalent.

We'll use the grey mapped to by pink/cyan. This will take the colors that previously appeared only within the digit and spill them onto the rest of the plate.

▲ ADDING A COMMON COLOR | Using a common color across the entire plate further breaks up the pattern for the normal.

The grey now does a good job in breaking up the shape of the digit. The normal eye is going to be more confused by this than the deuteranope, who will see a grey digit on a background of yellow and with grey stripes.

▲ ADDING A COMMON COLOR. | Using a common color across the entire plate further breaks up the pattern for the normal.

In this example, there's quite a lot of grey dots outside of the digit. In the final art I apply a probability to this effect. For example, dots outside the digit that correspond to texture level 4 have a 33% of being grey.

We can work even harder to confuse the normal eye by continuing to add colours (and patterns) seen (only) by the normal eye.

One way to do this is to vary the luminance (perceived brightness) and chroma (saturation) for each dot. This breaks up the lines of the digit even more.

▲ VARYING LUMINANCE AND CHROMA | By varying the luminance and chroma of each dot, we add more burden for the normal eye.

Finally, it would be very useful to have is a single tuning parameter that hides the digits from everyone, but more so from the normal eye.

To do this, I've generated another set of dots that use the digit confusion pair (pink/cyan), with mild luminance and chroma differences.

These dots act like a filter. If they are placed on top of the test plate with 100% opacity, the digits completely disappear.

▲ GLOBAL FILTER | Overlaying pink and cyan dots at 100% opacity completely hides the pattern.

If the filter is applied at (for example) 35% opacity, we will be able to see the underlying pattern, though with more difficulty. This has the effect of adding in some browns and purples for the normal eye.

▲ 35% GLOBAL FILTER | Overlaying pink and cyan dots at 35% opacity slightly obscures the pattern.

In the above example the filter was combined with the dot colour in a simple way. Regardless what the dot colour underneath was, the filter's dot was placed on top with 35% opacity.

Below, I adjust this scheme so that the resulting colour depends on the luminance of the dot. The filter strength is maximum for dots with `L = 60` and the strength linearly decreases to 0 outside the interval `L = [50,70]`.

▲ GLOBAL FILTER (35%) | Filter strength is maximum at luminance 60 and falls to zero outside `L=[50,70]`.

The art starts as a bitmap with three regions: digit, plate, and background. The output is designed to be printed at a specific size so that the panels and dot size distribution mimicks actual Ishihara test plates.

▲ TEMPLATE | The art is created from a template bitmap like this one. The image is split into regions for digit, plate, and background. The colors here are the ones used for Ishihara's demonstration plate, but are arbitrary since they are only used for classification.

▲ TEMPLATE (DEUTERANOPIA) | Regions are filled by dots of various size. The color of the dot reflects which area it came from: digit, panel, or background. The dot size distribution reflects the sizes of dots on actual Ishihara panels.

Let's color some of the panel and background dots with the colors used for the digit. This is done by randomly changing the classification of a dot from panel to digit (or background to digit) based on a probability that increases up to some maximum (e.g., 50%) with distance to the nearest digit dot.

With this scheme, plate dots that are near digit dots will not have their color flipped, which would make the digit shape hard to discern. I played around with various probabilities and distance cutoffs.

▲ NON-DIGIT DOTS MAY USE DIGIT COLORS | Non-digit dots that are not near digit dots may be randomly reclassified as digit dots.

▲ PANEL DOTS | Dots that make up the panel. Some of these appear missing because they were randomly reclassified as digit dots.

▲ BACKGROUND DOTS | Dots that make up the background. Some of these appear missing because they were randomly reclassified as digit dots.

Here's the first take of the full piece that makes use of all the steps described above.

▲ WHAT CAN YOU SEE? | The first 9 digits of `\pi` are hidden from normal eyes and substantially more visible to people with colour blindness.

Now that I've walked you through the essentials of the method, I'll briefly go into some of the additional adjustments that are included in the creation of the final art pieces.

In the example above, I used color confusion pairs: pink/cyan and red/green. But, as I mentioned, confusion colors exist on lines of confusion and we are not limited to choosing a pair of confusion colors. We can pick as many as we want (with diminishing returns).

Below I show the equivalent colors for deuteranopia. All colors in a given row will be perceived in the same way by a deuteranope. You can think of the row as a kind of line of confusion, though real lines of confusion are defined in a specific color space.

▲ EQUIVALENT COLORS IN DEUTERANOPIA. | All colors in a given row will be perceived in the same way by a deuteranope.

These equivalent colors were generated useing the method by Viénot, Brettel & Mollon, 1999, which I describe in detail on my math of colour blindness pages.

In general, colour blindness simulations vary slightly, particularly in the luminance of the output.

The actual color selection for the art is a little more complicated than what I show above, though it follows the same principle.

Let `r_i` be the reference color (as seen by the colour blind person) and `e_i` be the set of corresponding confusion colors, where `i=0` for pink/cyan and `i=1` for red/green.

Next, divide up the line of confusion into `j=1..5` equal-sized bands. For example, the color set `e_{ij} = e_{01}` is all the equivalent colors from the first band (`j=1`) of the pink/cyan line of confusion (`i=0`).

Using this scheme, I select 3 colors from each `e_ij` set at luminance `L= 55,60,65,70,75`.

For each region on the canvas (digit or panel/background), I create a large set of colors composed of the smaller sets `e_ij` of equivalent (or reference) colors according to this scheme for each texture tone:

For example, digit dots for texture tone `0` are drawn from `r_0` (grey reference) and `e_12` (band 2 of red/green line of confusion) with a `4:1` probability ratio (this is why `r_0` appears 4 times in the list).

You can see that I sometimes use colors from the red/green confusion lines in the digit (`e_{1j}`) to subtly disturb the pattern of the digit.

Finally, I normalize all the colors to be within luminance `L \in [50,75]` and chroma `C \in [30,40]`.

There's nothing magical or specific about these particular settings. I tweaked the values until I found something that was hard to see.

So far, I talked about creating art based on deuteranopia. But what about protanopia or tritanopia?

Protanopia is fairly similar to deuteranopia since the absorption spectra of M-cones and L-cones largely overlap.

▲ EQUIVALENT COLORS IN PROTANOPIA | All colors in a given row will be perceived in the same way by a protanope.

▲ 2026 π DAY | Art for the 5%. Digits of `\pi` hidden in plain view. (explore the art)

Tritanopia is interesting because the the lines of confusion are quite different than for deuteranopia and protanopia. Recall that protanopia and deuteranopia lines of confusion are similar because the L-cones and M-cones absorption spectra largely overlap.

The S-cones, affected in tritanopia, on the other hand,

Recall that in deuteranopia the pink/cyan and red/green lines had similar colors, and that's why we were able to confuse the normal eye.

The tritanopia lines are purple/yellow, blue/cyan, and pink/red/yellow.

▲ GLOBAL FILTER (35%) | Filter strength is maximum at luminance 60 and falls to zero outside `L=[50,70]`.

▲ 2026 π DAY | Art for the 5%. Digits of `\pi` hidden in plain view. (explore the art)