Art inspired by mathematics

i’m an artist

and i’m inspired by mathematics

now some people think of mathematics as

algorithms and

boring formulas that have very little to

do with life

or being alive but with my art i want to

show that mathematics is all about life

and beauty and patterns and connections

and that these can be some of the most

inspiring things of all

i’m going to show you how one

mathematical pattern led me to create

art and music and poetry and literature

and

a lot of other cool stuff and this story

begins about 20 years ago when i was

playing with a puzzle

called the towers of hanoi it usually

looks like this with a it’s a stack of

discs

on some pins and the idea is to take the

stack

and move it to one of two other

locations

just by moving one piece at a time and

you can never place a larger piece on

top of a smaller piece

it’s a fun puzzle it’s easy to make

mistakes

and it’s full of some really cool

patterns and when you get into the

patterns

you can solve it very quickly and easily

like this

one two one three one two

one four one two

one three one two

one five one two

one three one two one

four one two one three

one two one

okay here’s the list of moves that i

just did and you can maybe see some

structure here already

there’s interesting groupings that

repeat themselves

on larger and larger scales and one of

the first things that i did with this

was i represented these numbers as

lengths that doubled from one to two to

three

and we can see some shapes already

here’s some some triangles it looks kind

of like mountains maybe we could turn

this into

a landscape hey that could be cool let’s

see what other patterns we have in here

this looked familiar to me this is the

pattern on an english

ruler that i grew up using and here’s

the solution to that puzzle

in the space of two inches

if we rotate this 90 degrees then you

might recognize this shape as

a playoff tree

we rotate it again and we get well

like an actual tree and if we

even out those angles a little bit we

get something that’s

that’s very organic looking in fact this

is a shape that we find

all over nature

another place we find this in nature is

in a family tree

right we each come from two parents and

each of our parents come from two

parents and

each of them from two parents so in a

sense

we are right in the middle of one of

these trees

that inspired me to make this picture

where instead of using

line segments now i used actual people

and this shows how we’re all part of

these trees

that connect backwards in time through

countless generations

if we want to get a little bit

metaphysical we could also think of this

tree

as all of the paths that are available

in life

and imagine you’re going along and you

need to make a decision at that point

the universe splits into two nearly

identical copies

but depending on what decision we make

we could end up in two entirely

different places i thought this

structure could

make a neat poem so i wrote this one

it’s actually 16

poems in one it’s called decision tree

i’ll turn it sideways so we can read it

a little bit better

and so you start here at the trunk of

the tree is the question

should i do it should i not then there’s

your decision

i shouldn’t do it so i thought or i went

and did it anyway

next is your reaction to that decision

and then in the fourth stage

is like how that affects you in the

future

let’s take a look at a couple of these

poems there’s

should i do it should i not

i went and did it anyways but now i’m

full of guilty

thoughts it wasn’t worth it i would say

well that’s kind of depressing right

let’s try a different one okay

should i do it should i not i shouldn’t

do it so i thought

now the chance has slipped away i guess

i’m doomed to live this way

that one’s kind of depressing too

actually a lot of these are kind of

depressing

and i was a little surprised by that but

i think what’s happening is i put some

extra structure in here

i had this idea that one direction of

the poem

would be like how a good person would

react

and the other direction would be like

how a bad person

would react and so

there’s somebody here who actually feels

good about themselves and is happy with

who they are

it’s this one right here should i do it

should i not

i shouldn’t do it so i thought i’ll do

what i know is right

and keep my conscience clear and bright

but there’s another person here who i

think feels equally

good about themselves can you see that

one

yeah it’s uh should i do it should i not

i went and did it anyway

i just hope i don’t get caught they

didn’t catch me yesterday

so i’m not sure this says something

about morality here but i think it says

something about happiness

that in life as in mathematics

we’re free to choose our own rules

we just need to make sure that our

actions are consistent with our rules

or we’re going to risk being unhappy

it’s kind of a cool thing we can learn

from math huh

so all of these ideas here so far have

just come from changing these numbers

into shapes and we could try turning

them into all sorts of other

shapes and we could connect this to lots

of ideas in mathematics

here’s a bunch of shapes that all have

the same

pattern in them and these are cool and

interesting ideas they’re all worth

exploring and they all can be great

source

for inspiration before we go too much

further

let’s give this pattern a name instead

of using ones

twos and threes we’re going to replace

the ones with a’s

the twos with b’s the threes with c’s

the fours with d’s

et cetera and we get kind of this

interesting word pattern

we’re going to call this the abba cabba

pattern

and if you want to try writing this down

uh it’s it’s pretty fun to do

the first step is you just write the

letter a

okay then you double it and put the next

letter of the alphabet in the middle

so we get abba

double that and put the next letter in

the middle and we get abba cabba

double it again and put the next letter

in the middle and we come to abba

kabba dabba kabba it’s kind of fun to

say

would you mind saying it with me can we

say it together okay

ready abba cabadabakaba

okay right on okay next step then okay

double it and add the next letter in the

middle now this has a little diphthong a

little double vowel in the middle so

it’s got kind of an

yabba cab in the middle there okay you

ready for this one

ready abba cabba dabba cabba

yabba kabba dabba kabba all right you

guys are awesome all right let’s uh

let’s go to the next one

okay let’s say it together here we go

abba cabba dabba

cabba yabba kabba dabba kabba

fabba kabba dabba

all right yeah

[Applause]

okay that was the sixth step we only

have 20 more to go

okay next one uh you know you can do the

you can do this one uh you can practice

this uh by yourself later

because this is g if we double it this

is what it looks like to h

double it again it’s and

this gets ridiculously big

fast okay and if we take this all the

way out to z

it has over 67 million letters in it

and it would take us over three months

non-stop to say that name

imagine if you were to try to write it

down

well i did that actually

i published this it was a it took a set

of four volumes

each was uh over 400 pages

and it was written in four point font

so i think this i think this sets two

world records right here

it sets the world’s record for the

longest published word

and it simultaneously sets the world

record for the

world’s most boring book

i think you’ll agree one page is

probably

enough right but i still really like

this idea here’s a

magical sounding word that’s impossibly

long to say i thought this would make a

cool

children’s story so i wrote this story

called maggie and the abba cabbage

genies

about a young girl who finds a genie

bottle and she has to say these genie

names to call it more and more powerful

genies

until she comes to the genie who lives

at the letter z

so these ideas now have come from

changing that number pattern

to letters and also when i see letters

when i see a’s b’s and c’s

i think of musical notes

what does abikabha sound like

let’s give it a try here’s a we’d begin

with a

[Music]

and then we take the next letter next

note b

and repeat everything that came before a

and we go to c and repeat everything

that came before

now we go to d and repeat

and it gets longer and longer until we

use up the whole piano i’m going to add

a little bass line to this and

turn it into a little song

do

[Applause]

okay thanks that was fun so that’s a

short version of a

little longer song and i’ve got this

sheet music available for you

if you want to try playing this at home

what i thought was uh

what i thought was neat about this song

was that it sounds nice

even though it’s so structured and i

thought well

if this is so structured could we make a

machine

that plays this song so i built this

this is the abba kaba music machine

[Applause]

you can hear us playing the abba kaba

pattern by dropping balls onto his

xylophone

and the balls go through this simple

series of dates that send

every other ball to the first note a

every fourth ball to a b

every eighth to a c and so on

and what’s kind of cool is the gates

down the side

they’re actually counting in binary from

0 to 127.

now normally when we write numbers we

write them in decimal which is base 10.

so we have a ones place a tens place and

a hundreds place

and when we write like 375 we

really mean three hundreds seven

tens and five ones and with binary

instead of powers of ten

we use powers of two so we have a ones

place a twos place a fours place an

eighths place and so on

so if we write one zero one one that

means one eight

zero fours one two and one one which is

how we would write the number thirteen

so with just zeros and ones we can write

all of the numbers

from uh here’s the start of the list

right here from

0 to 16

and binary is full

of abikab patterns for example if we

just look at how many zeros there are at

the end of each number

there’s an abba caba pattern

lots of other places too uh how about if

we color them

color in the digits i’ll leave the zeros

white and we’ll color in

the ones black we get this kind of weird

looking shape if you turn your head

sideways you might be able to see

there’s an abba cabba tree right in

there in the binary numbers

i thought this looked a little lopsided

so i made two copies and stuck it

together so it’d be a little more

symmetrical

and got some interesting little shapes

do you see what i see

in there yeah there’s fish

it’s full of fish so

this became uh this became a picture uh

here’s

two solid fish and two ghost fish

swimming in the seaweed made out of the

numbers from

zero to fifteen and i call this one

pisces

after the zodiac sign with with two fish

i wrote the name a little bit funny

because the picture has a nice uh

upside down symmetry to it so i wanted

the name to also

read the same if it was going upside

down

if we go back to this idea with uh

lengths

if we write this uh binary numbers

using these blocks and doubling lengths

we can begin to count

one two three four five six seven and we

begin to see some

neat shapes coming up we get a bunch of

squares

it looks like steps it maybe looks a

little bit three-dimensional

and if we think of the smallest squares

as a’s and the next bigger ones as b’s

and the next bigger ones is c’s then we

get this

abacaba pattern as we walk up the

staircase

with the steps that we step on

so i extended that pattern out to 255

and i took

several copies of it and put them

together and got something that really

is looking

3d this became a piece of like op

art and it also started me thinking

about three-dimensional what would this

look like in 3d

so i tried some paper clipping and then

here’s a pop-up card that i made

i have some instructions available for

this too if you want to make one at home

it makes a

great art project and i really wanted to

walk up these steps i built

models out of wood and i began making

computer models

here’s one that is completely symmetric

in 3d and i flooded this with water and

i got like this awesome looking planet

i really wanted to be on this planet so

i began designing

landscapes and i began to live in this

world

in my head and this became the basis for

a fantasy novel that i’ve just written

it’s called abba qabex

and it takes place on this planet where

the middle block

is this letter x in the pattern

and i filled this world with people and

monsters and magic and mathematics

and lots of connections to the abba kaba

pattern

and to me there is nothing closer to

real magic

than mathematics so

all of these ideas that we’ve seen have

come from this one

abba caba pattern and there’s so much

more i’ve made a lot of resources

available on abacabba.org

if you want to see some more art or

listen to some music

or download some activities or games

so just think if one pattern

can inspire art and music

and poetry and sculpture and landscape

and architecture and

literature what might other

patterns do so

i hope you go out there play with some

ideas and some patterns and put them

together in

different ways and see what you can find

you might be surprised at

all the beautiful things that you could

create

thank you