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, spatial positioning and sometimes colour. Mentally visualising this kind of forms to aid basic maths wouldn't normally be considered synesthesia.

"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 conversation in the Facebook group Synesthesia. 2021.)

Here is another interesting description:

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 online debate platform Reddit/IAma: Ask Me Anything. 2011.)

**A number synesthete explains and illustrates their system**

Here is an interesting account with a great illustration, from a synesthete who automatically uses a visual system for adding
single digits to make 10.

“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.

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 online debate platform Reddit/Synesthesia.
2022.)

**Complex mathematical visualisations**

However, “seeing mathematics”, that is, involuntarily and physically visualising highly complex mathematical operations with a tremendous amount of detail, is 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 Padgett, the 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. 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: 27 November 2022**

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