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