Mathematical synesthesias

Some types of synesthesia, mainly the number-related types, can be considered mathematical synesthesias. However, it appears that “pure” mathematical synesthesia – the highly detailed visualisation of actual mathematical concepts – is extremely rare and tends to be either an acquired synesthesia, as a result of brain injury for example, or one of the types normally only found in savants (people with a prodigious talent who are usually on the autism spectrum).

Here are some types of synesthesia connected to mathematics:

1. The type of synesthesia where each number has its own unique shape, colour, spatial position, size and/or personality, not only one- or two-digit numbers but hundreds or thousands of individual numbers. The writer and high-functioning autistic savant Daniel Tammet has this very rare type. He has extraordinary mathematical capabilities: his synesthesia helped him achieve a record recital of Pi to 22,000 decimal places, as he can use the visual forms of his numbers to create highly memorable mental landscapes.

The number 1, for example, is a brilliant and bright white, like someone shining a flashlight into my eyes. Five is a clap of thunder or the sound of waves crashing against rocks. Thirty-seven is lumpy like porridge, while 89 reminds me of falling snow.”

2. Visualisation of mathematical operations

Possibly, many people – synesthetes and non-synesthetes alike – have some kind of mind's-eye visualisation on performing mental arithmetic. What they perceive and how they perceive it varies from person to person and might involve shapes, basic structures and spatial positioning. The more straightforward systems used by non-synesthetes as a habit to aid calculation, even if they automatically appear in the mind's-eye, wouldn't be considered synesthesia. However, for some synesthetes the numerical values have shape, colour, texture, spatial positioning and movement and a more complex automatic process takes place on solving maths problems. This could be considered a product of their synesthesia, or perhaps a way in which their synesthesia is useful, even if not considered a type of synesthesia in itself.

These examples would probably not be considered synesthetic processes:

"I simply see the numbers "stack" like tetris blocks."

"What's your thought process when solving 9+7? (...) 1 from the 7 gets tucked underneath it, that 1 gets slid under the table to 9, 9 uses it to build a 10, the remaining 6 jumps on top of the (10) table, & there we get 16."

(Source: this group conversation in the Facebook group Synesthesia2021.)

"For example, when I add together 4 and 8, I see a 8 as a shape that is missing two pieces or entities to make 10. Therefore, I see 8 scooping up half of 4's shape and then a 10 piece and a 2 piece being left over. I visualize this way of adding kind of like putting together pieces in Tetris. (…)

I physically see shapes combining with each other to form a stable shape like a square or rectangle. Also, sometimes when I add together numbers such as 7 and 8, I see the shape of the 7 hooking on to one of the holes of the 8. I was never taught to do addition this way, but I always understood that this was how it was done. (…) It happens involuntarily. (…) I don't see shapes when I look at individual numbers. It's only when I begin to add these numbers together that I begin to see shapes.

(Source: This post on the subReddit IAma: Ask Me Anything. 2011.)

This account by a number synesthete involving shape, texture and colour suggests that the process has connections with their synesthesia:

 “This is how it feels like to add single digit numbers for me personally.

The numbers on their own feel like different shapes, but if I am to visualise, let's say "3+7" in my head, this is what it looks like.

All numbers are sticks that get longer the higher they are, but they also have a distinctive shape at the top, that matches with another number. And those two numbers always add up to 10.

This association is a lot stronger for odd numbers, the texture on the even numbers is a lot less distinct.

1 and 9 have a very pointy triangle.
2 and 8 are kind of like a Lego brick.
3 and 7 have a semicircle.
4 and 6 have half a hexagon sticking out.
And 5 is a diagonal that matches with itself.

I don't necessarily see the colours when adding numbers, but I decided to make them the colours those numbers have for me on their own.

Personally, I would describe the material these pillars are made out of as solid, but when two non-matching odd numbers combine, they kind of turn into a kinetic-sand-like material than then shifts and morphs until they fit?

I really struggled with going beyond 10 as a kid, since that is where this model breaks down. (…) Well not just beyond 10, but when an addition goes across a 10-boundary, you know? Like 27 + 6 is so weird to me. I have to visualise it as 27 + 3 + 3 so my brain can actually do anything with that.”

(Source: This post/comments on the Synesthesia subReddit. 2022.)

Complex mathematical visualisations

“Seeing mathematics”, that is, involuntarily and physically visualising highly complex mathematical operations with a tremendous amount of detail, is however a rare phenomenon. It might occur as a natural (developmental) type of synesthesia, although I am personally not aware of any cases. It appears to be more typical of cases of acquired synesthesia (or similar phenomenon) resulting from brain injury. This acquired type is the case of Jason Padgettthe man who “became a maths genius overnight” after being hit on the head during a violent mugging. (In this video, Jason Padgett gives an engaging, interesting and very recommendable TED Talk).

3. Moving on to more common types, number-form synesthesia consists of perceiving numbers as having specific and consistent spatial positions. The people who have this type often benefit from it as they can mentally manipulate the numbers they can “see” or even “touch”, finding them very easy to remember.

Go to the page on number-form synesthesia

4. Synesthetes who have grapheme-colour synesthesia with numbers have a particularly significant relationship with their digits when each one has its own colour, and this can affect their mathematical abilities. Although some of them have natural skills and great enthusiasm in this area thanks to their synesthesia, for a lot of them it actually seems to create problems with maths ability, giving rise to awkward situations such as “3 is yellow and 5 is blue, but if you add them together they make 8, which should be green. But it’s purple! I don’t understand, it’s wrong and it’s impossible for me to memorise it!”

Go to the page on grapheme-colour synesthesia

5. For some grapheme-colour synesthetes, not only numbers and/or letters have colour but also mathematical symbols:

“Math symbols have colors, too, like the root sign [√] is red.”

(Source: the book Synesthetes by Sean Day (2016), p31. -You can get a pdf copy here.)

6. Consistently and automatically associating particular colours not just with mathematical symbols but with other concepts in this domain – such as theorems, objects, disciplines or perhaps the mathematical operations themselves – would be considered a type of coloured sequence synesthesia.

“I have a math PhD with synesthesia for mathematical concepts. I don’t think it’s highly detailed, but I instinctively associate certain theorems, objects, and entire disciplines with colors. Complex analysis is tan, topology is green, and differentiable manifolds are orange.”

Source: a comment on this page of the Synesthesia Tree, 2023.

Go to the page on coloured sequence synesthesia

7. For some people each number has a well-defined personality, so they also have a very special relationship with numbers, seeing them almost as if they were people. This is ordinal linguistic personification synesthesia. When they do maths they sometimes have to deal with internal struggles between the characters that affect them emotionally, although they also get the chance to enjoy exhilarating stories of adventure, love, betrayal, vengeance or solidarity, all produced by the dynamics of the digits and their interaction. Obviously this can have an important effect on both their maths skills and their concentration.

Go to the page on number personification

Go to the page on ordinal linguistic personification and personification in general

This page last updated: 22 April 2023


  1. One thing I do experience is the idea that numbers have their own personality. However, it's not in a very anthropomorphic sense. Instead, a number like 17 is just 17. It's this prime number that has its own way of existing. I don't automatically know what a number's personality is; I have to get familiar with it. The 10 base system we use has an effect on how I see a number; specifically, it allows me to view it as an addition of powers of ten (duh). But as I get to know the number better, I think of its prime factors and stuff and that becomes central to its identity. I don't have a perfect memory; it's just a matter of how familiar I am with a number. I'm never going to forget that 111 is 3 times 37.

    All that being said, I consider numbers to have an intrinsic "personality" that I can become familiar with and this seems like a logical thing to do. That's probably not synesthesia, but are there any other opinions on this?

    Numbers also have shape to them, but these shapes are not necessarily fixed, just like the personality. They are basically just polygons, so larger numbers aren't really that distinct, aside from the modular arithmetic you get, but I don't have all that memorized. I created this tool partly because I wanted to be able to look up the shapes of various interactions:

    1. For clarification, I don't only view numbers as polygons. The number 8, for example is easy to think of as the Earth cut into octants, as it is the simplest proper perfect cube (not the unit 1). 8 also makes me think of spiders :).

    2. Sometimes. Not all the time. My brain doesn't have time to think about spiders while doing math. 8 is too simple a number. But if I see 57, I'll often be like "Ah, 57. so common, yet so mysterious." but in a faster, less conscious way.

    3. Very interesting comments. I love your modular visualiser, I will be having a go at that and see what it turns out! Maths synesthesia isn't my speciality at all, unfortunately, so I can't really point you in any useful direction. Numbers having personality is certainly considered synesthesia, but you're saying your ordinal linguistic personfication is of higher numbers too, any and all numbers, is that right? so it's different from the vast majority (where only single-digit numbers have personality and the bigger numbers are just combinations of those). But it seems quite logical to me that some people could have this, and it's perfectly logical that you would have to "get to know" the number before you realise what its personality is. It makes me think of Daniel Tammet's hundreds or thousands of numbers, each one with their own specific presence, shape, colour and so on, I can't remember if his numbers had personality exactly, but I believe they certainly had attitude! Anyhow you're probably familiar with the literature about him. I can't remember ever having come across or read about a case of anyone with separate personalities for all multiple digit numbers, but if I do I will reply to your comment here. I was thinking that Shereshevsky might have had separate personalities for all numbers but reading what he said I think he didn't: he just had them for numbers up to 9 and the two-digit ones would just be something like those two people meeting each other.
      When you mentioned 111 thought of Oliver Sacks and the savant twins in the book The Man who Mistook his Wife for a Hat who had that instant concept of "one hundred and eleven-ness".
      It would be interesting to know more about the personalities of your numbers. But I imagine they're a long way from the rather topical characters that ordinal linguistic personification seems to produce sometimes, and probably more about "attitude" and uniqueness.

  2. I’m have a math PhD with synesthesia for mathematical concepts. I don’t think it’s highly detailed, but I instinctively associate certain theorems, objects, and entire disciplines with colors. Complex analysis is tan, topology is green, and differentiable manifolds are orange.

    1. Very interesting! I’ve included your comment as a new section on this page (no. 6) as this would fit into the Coloured sequence type of synesthesia and I only had sections on people who have colours for numbers or mathematical symbols, not other concepts, so it’s different. I’ve also added this category to the list on the page on Coloured sequence synesthesia.

      Thank you for commenting!

  3. I view multiples of numbers in shades of a color and with a physical location. For example, multiples of 3 are green and in a column above my head and to the right, extending up into space. And number colors get influenced by the numbers that can be multiplied to equal them... For example, my multiples of 4 are shades of blue, so 12 is a green-blue. Prime numbers tend to be yellow because 1 is yellow to me. My colors also have an even or oddness relationship that then gets applied to other things. For example: yellow, green, orange, and red are odd while blue, pink, and purple are even, and, because I associate letters with colors, H is odd, while B is even. I then associate people with colors (and corresponding even or oddness) based on the spelling of their names.