C

J.

Schmidhuber

LOW-COMPLEXITY

ART

JURGEN

SCHMIDHUBER

LOW-COMPLEXITY

ART

Report

-LRB-

revised

-RRB-

Jurgen

Schmidhuber

Fakultat

fur

Informatik

Technische

Universitat

Munchen

Munchen

Germany

schmidhu@informatik.tu-muenchen.de

Revised

January

Abstract

Many

artists

try

to

depict

the

essence

of

objects

to

be

represented

In

an

attempt

to

formalize

certain

aspects

of

the

the

essence

I

propose

an

art

form

called

low-complexity

art

Its

goals

are

based

on

concepts

from

algorithmic

information

theory

Suppose

the

task

is

to

draw

a

given

object

Usually

there

are

many

ways

of

doing

so

The

goal

of

low-complexity

art

is

to

draw

the

object

such

that

the

drawing

can

be

specified

by

a

computer

algorithm

and

two

properties

hold

-LRB-

-RRB-

The

drawing

should

look

right

-LRB-

-RRB-

The

Kolmogorov

complexity

of

the

drawing

should

be

small

-LRB-

the

algorithm

should

be

short

-RRB-

and

a

typical

observer

should

be

able

to

see

this

Examples

of

low-complexity

art

are

given

in

form

of

algorithmically

simple

cartoons

of

various

objects

Relations

to

previous

work

are

established

Attempts

are

made

to

relate

the

formalism

of

the

theory

of

minimum

description

length

to

informal

notions

like

good

artistic

style

and

beauty

Disclaimer

Despite

being

published

in

a

tech

report

series

this

is

not

a

scientific

paper

Although

attempts

are

made

to

formalize

certain

aspects

of

informal

notions

like

art

and

beauty

much

of

what

is

said

remains

informal

and

speculative

Note

The

title

of

the

original

version

of

this

paper

-LRB-

from

September

-RRB-

was

Algorithmic

Art

There

also

is

a

long

version

-LRB-

Schmidhuber

-RRB-

It

has

pages

contains

figures

and

is

written

in

German

c

J.

Schmidhuber

All

rights

reserved

Introduction

In

their

introduction

to

Kolmogorov

complexity

Li

and

Vit?anyi

-LRB-

-RRB-

write

We

are

to

admit

no

more

causes

of

natural

things

-LRB-

as

we

are

told

by

Newton

-RRB-

than

such

as

are

both

true

and

sufficient

to

explain

their

appearances

This

central

theme

is

basic

to

the

pursuit

of

science

and

goes

back

to

the

principle

known

as

Occam

's

razor

if

presented

with

a

choice

between

indifferent

alternatives

then

one

ought

to

select

the

simplest

one

Unconsciously

or

explicitly

informal

application

of

this

principle

in

science

and

mathematics

abound

This

paper

argues

that

the

principle

of

Occam

's

razor

is

not

only

relevant

to

science

and

mathematics

but

to

fine

arts

as

well

Some

artists

consciously

prefer

simple

art

by

claiming

art

is

the

art

of

omission

Furthermore

many

famous

works

of

art

were

either

consciously

or

unconsciously

designed

to

exhibit

regularities

that

intuitively

simplify

them

For

instance

every

stylistic

repetition

and

every

symmetry

in

a

painting

allows

for

describing

one

part

of

the

painting

in

terms

of

descriptions

of

other

parts

Intuitively

redundancy

of

this

kind

helps

to

shorten

the

length

of

the

description

of

the

whole

painting

thus

making

it

simple

in

a

certain

sense

It

is

possible

to

formalize

the

intuitive

notions

of

simplicity

and

complexity

The

appropriate

mathematical

tools

are

provided

by

the

theory

of

Kolmogorov

complexity

or

algorithmic

complexity

-LRB-

Kolmogorov

Chaitin

Solomonoff

-RRB-

The

Kolmogorov

complexity

of

some

computable

object

is

essentially

the

length

of

the

shortest

algorithm

computing

it

The

simpler

the

object

the

shorter

the

algorithm

See

-LRB-

Li

and

Vit?anyi

-RRB-

for

the

best

overview

See

-LRB-

Schmidhuber

-RRB-

for

a

machine

learning

application

In

this

paper

basic

concepts

from

the

theory

of

algorithmic

complexity

serve

as

ingredients

for

a

novel

form

of

simple

art

called

low-complexity

art

Although

the

focus

will

be

on

black-and-white

cartoons

the

basic

ideas

are

not

limited

to

them

Outline

Section

introduces

basic

concepts

of

the

theory

Section

specifies

the

nature

of

lowcomplexity

art

Section

focuses

on

a

particular

instance

of

low-complexity

art

based

on

a

fractal

way

of

specifying

drawings

The

bulk

of

this

paper

is

devoted

to

examples

of

low-complexity

art

to

be

presented

in

section

Section

attempts

to

relate

the

formalism

of

the

theory

of

minimum

description

length

-LRB-

MDL

-RRB-

to

informal

notions

like

good

artistic

style

and

beauty

Section

establishes

relations

to

previous

work

and

concludes

with

an

outlook

on

the

possible

future

of

low-complexity

art

Basic

Concepts

Each

Turing

machine

-LRB-

TM

-RRB-

C

-LRB-

mapping

bitstrings

to

bitstrings

without

loss

of

generality

-RRB-

computes

a

partial

function

fC

-LRB-

fC

is

undefined

where

C

does

not

halt

-RRB-

Compiler

theorem

It

is

well

known

that

there

is

a

universal

TM

U

with

the

following

property

for

every

TM

C

there

exists

a

constant

prefix

C

-LRB-

a

bitstring

-RRB-

such

that

fC

-LRB-

p

-RRB-

fU

-LRB-

Cp

-RRB-

for

all

bitstrings

p.

C

is

the

compiler

that

compiles

programs

for

C

into

equivalent

programs

for

U

Less

formally

any

program

for

a

given

machine

can

be

compiled

into

an

equivalent

program

for

a

given

universal

machine

by

a

compiler

program

of

constant

size

Kolmogorov

complexity

The

Kolmogorov

complexity

KU

-LRB-

s

-RRB-

of

a

finite

string

s

is

the

length

of

the

shortest

program

p

that

computes

s

on

a

universal

Turing

machine

U

and

halts

where

the

set

of

possible

halting

programs

forms

a

prefix

code

-LRB-

no

halting

program

can

be

the

prefix

of

another

one

-RRB-

KU

-LRB-

s

-RRB-

min

p

fj

p

j

fU

-LRB-

p

-RRB-

sg

where

j

p

j

denotes

the

length

of

p

In

general

KU

-LRB-

s

-RRB-

is

noncomputable

-LRB-

Kolmogorov

Solomonoff

Chaitin

-RRB-

Invariance

theorem

Due

to

the

compiler

theorem

-LRB-

s

-RRB-

-LRB-

s

-RRB-

O

-LRB-

-RRB-

for

two

universal

machines

and

Therefore

we

may

choose

one

particular

universal

machine

U

and

henceforth

write

K

-LRB-

s

-RRB-

KU

-LRB-

s

-RRB-

Essentially

the

invariance

theorem

says

Kolmogorov

complexity

is

objective

in

the

sense

that

it

is

independent

of

the

particular

computer

used

leaving

aside

an

additive

machine

specific

constant

Low-Complexity

Art

Suppose

an

artist

's

task

is

to

produce

a

drawing

which

obeys

a

set

of

-LRB-

possibly

informal

-RRB-

specifications

given

in

advance

The

goal

of

low-complexity

art

is

to

represent

the

essence

by

achieving

two

conflicting

goals

simultaneously

Goal

Given

the

specifications

the

drawing

should

look

right

Goal

-LRB-

A

-RRB-

The

Kolmogorov

complexity

of

the

final

design

should

be

provably

small

In

other

words

there

should

be

a

short

algorithm

computing

the

drawing

-LRB-

by

generating

appropriate

instructions

for

a

printer

say

-RRB-

-LRB-

B

-RRB-

It

should

be

easy

for

the

observer

to

perceive

the

algorithmic

simplicity

of

the

drawing

He

ought

to

see

the

essence

extracted

by

the

low-complexity

artist

It

is

predicted

that

drawings

that

do

a

good

job

on

both

conflicting

goals

will

be

appreciated

by

the

observer

The

next

subsection

addresses

the

extent

to

which

both

goals

are

subjective

How

Subjective

is

Low-Complexity

Art

Goal

is

clearly

subjective

in

the

sense

that

it

strongly

depends

on

a

given

observer

and

on

the

way

he

interprets

the

-LRB-

possibly

informal

-RRB-

specifications

What

looks

right

to

an

observer

from

one

-LRB-

sub

-RRB-

culture

may

look

wrong

to

an

observer

from

another

-LRB-

sub

-RRB-

culture

-LRB-

or

another

time

-RRB-

Goal

-LRB-

A

-RRB-

depends

on

the

nature

of

the

computer

running

the

algorithm

In

what

follows

this

dependency

will

be

ignored

In

the

limit

this

is

justified

by

the

above-mentioned

invariance

theorem

Like

Goal

Goal

-LRB-

B

-RRB-

depends

on

the

observer

But

in

a

sense

Goal

-LRB-

B

-RRB-

is

less

subjective

than

Goal

This

is

because

intelligent

human

observers

in

principle

can

learn

to

compute

anything

a

digital

computer

can

compute

-LRB-

only

the

converse

is

a

matter

of

controversy

-RRB-

In

particular

a

short

algorithm

running

on

a

conventional

digital

machine

can

be

quickly

taught

to

an

intelligent

human

being

Note

that

if

the

human

observer

was

just

another

universal

computer

then

we

could

immediately

apply

the

invariance

theorem

thus

-LRB-

in

the

limit

-RRB-

removing

subjectiveness

from

Goal

-LRB-

B

-RRB-

Then

the

only

remaining

subjective

aspect

of

low-complexity

art

would

be

given

by

Goal

Low-Complexity

Art

is

Hard

This

paper

describes

the

goals

of

low-complexity

art

not

the

way

to

achieve

the

goals

The

latter

requires

intuition

and

like

with

any

other

form

of

art

a

sometimes

rewarding

but

often

frustrating

struggle

for

capturing

the

essence

In

the

beginning

of

his

attempts

to

create

a

work

of

low-complexity

art

the

low-complexity

artist

will

usually

not

be

able

to

predict

the

details

of

the

final

result

The

noncomputability

of

Kolmogorov

complexity

implies

that

there

is

no

general

method

for

finding

the

shortest

description

of

some

piece

of

data

-LRB-

Kolmogorov

Solomonoff

Chaitin

-RRB-

This

seems

to

indicate

that

the

formal

art

form

proposed

above

will

always

represent

a

big

challenge

to

any

artist

willing

to

pursue

it

Low-Complexity

Design

To

have

regard

for

the

difference

between

art

and

design

-LRB-

this

difference

is

important

to

many

artists

-RRB-

I

would

like

to

use

the

expression

low-complexity

design

instead

of

low-complexity

art

in

cases

where

no

artistic

purpose

is

pursued

by

a

designer

trying

to

achieve

Goal

and

Goal

The

next

section

first

defines

a

fractal

-LRB-

Mandelbrot

-RRB-

coding

scheme

for

encoding

drawings

in

algorithmic

form

The

coding

scheme

is

easily

learned

and

understood

by

most

humans

Section

presents

a

set

of

cartoons

Each

cartoon

satisfies

some

informal

specifications

provided

in

advance

Furthermore

each

cartoon

is

algorithmically

simple

-LCB-

its

description

-LRB-

based

on

the

coding

scheme

implemented

on

a

conventional

digital

computer

-RRB-

does

not

require

many

bits

of

information

To

achieve

Goal

-LRB-

B

-RRB-

the

algorithmic

simplicity

of

each

cartoon

is

made

obvious

with

the

help

of

text

and

additional

drawings

Fractal

Circles

for

Coding

Drawings

This

section

introduces

a

fractal

coding

scheme

for

encoding

drawings

in

algorithmic

form

The

scheme

is

general

enough

to

design

arbitrary

drawings

As

will

be

seen

in

section

it

is

sophisticated

enough

to

allow

for

specification

of

non-trivial

drawings

with

a

very

limited

amount

of

information

Finally

it

is

simple

enough

to

be

implemented

by

a

short

algorithm

and

to

be

quickly

taught

to

typical

human

observers

The

ancient

Greeks

considered

the

circle

as

the

ideal

two-dimensional

geometric

form

Without

necessarily

agreeing

with

the

Greeks

in

what

follows

I

will

use

circles

as

the

basis

for

designing

drawings

One

motivation

is

that

a

circle

can

be

drawn

by

a

very

short

algorithm

Another

motivation

is

that

circles

are

something

most

humans

can

relate

to

most

people

know

something

about

circles

and

their

properties

This

will

make

it

easy

to

teach

the

algorithmic

simplicity

of

the

drawings

from

section

to

a

typical

observer

-LRB-

to

achieve

Goal

-LRB-

B

-RRB-

from

section

-RRB-

Sizes

and

relative

positions

of

legal

circles

will

be

greatly

limited

by

the

following

fractal

rules

Rules

for

Making

Legal

Circles

Initialization

Arbitrarily

define

the

first

circle

Arbitrarily

select

a

second

circle

with

equal

radius

whose

center

is

on

the

first

circle

The

first

two

circles

are

defined

to

be

legal

circles

The

rules

for

generating

additional

legal

circles

are

as

follows

Rule

I.

Wherever

two

legal

circles

of

equal

radius

touch

or

intersect

another

legal

circle

of

equal

radius

may

be

drawn

with

this

point

as

its

center

Rule

II

Every

legal

circle

with

center

point

p

and

radius

r

may

have

within

it

a

legal

circle

whose

center

point

is

also

p

but

whose

radius

is

r

Figure

shows

the

result

of

an

iterative

application

of

the

rules

above

Rules

for

Making

Legal

Drawings

A

legal

drawing

is

defined

by

-LRB-

a

-RRB-

legal

lines

-LRB-

segments

of

legal

circles

-RRB-

or

-LRB-

b

-RRB-

legal

areas

-LRB-

defined

by

legal

circles

that

touch

or

intersect

-RRB-

The

rules

for

legal

lines

and

areas

are

Rule

III

Each

legal

line

must

be

a

segment

of

a

legal

circle

Rule

IV

At

both

endpoints

of

a

legal

line

some

legal

circles

must

touch

or

intersect

Rule

V

The

line

width

of

a

legal

line

must

be

equal

to

the

radius

of

some

legal

circle

Rule

VI

A

legal

area

is

an

area

whose

border

is

a

closed

chain

of

legal

lines

Legal

areas

may

be

shaded

using

a

small

set

of

grey

levels

Comments

-LRB-

-RRB-

On

each

legal

circle

you

can

find

the

centers

of

legal

circles

with

the

same

radius

-LRB-

-RRB-

Define

the

radius

of

the

initial

circle

as

The

radius

of

any

legal

circle

can

be

written

as

n

where

n

is

a

non-negative

integer

The

same

holds

for

legal

line

widths

-LRB-

-RRB-

On

a

given

area

there

are

about

times

as

many

circles

with

radius

n

as

there

are

circles

with

radius

n.

-LRB-

-RRB-

Low-complexity

art

is

in

no

way

limited

to

Rules

I-VI

For

instance

a

drawing

that

only

partly

obeys

Rules

I-VI

may

be

a

work

of

low-complexity

art

as

long

as

the

deviations

can

be

uniquely

determined

by

a

short

algorithm

Coding

Drawings

by

Circle

Numbers

There

are

many

straight-forward

schemes

for

encoding

drawings

generated

by

rules

I-VI

This

section

gives

an

example

Have

another

look

at

Figure

Define

the

radius

of

the

initial

circle

-LRB-

the

frame

-RRB-

as

A

visible

circle

is

a

circle

partly

covered

by

the

initial

circle

Starting

with

the

initial

circle

we

generate

all

visible

circles

Each

gets

a

number

The

initial

circle

gets

the

number

There

are

visible

circles

with

radius

intersecting

the

initial

circle

They

are

numbered

-LRB-

in

some

deterministic

clockwise

fashion

say

-RRB-

There

are

visible

circles

with

radius

-LRB-

partly

-RRB-

covered

by

the

initial

circle

They

are

numbered

And

so

on

Obviously

there

are

few

big

circles

with

small

numbers

There

are

many

small

circles

with

high

numbers

In

general

the

smaller

a

circle

the

more

bits

are

needed

to

specify

its

number

An

unshaded

drawing

is

specified

by

a

set

of

legal

lines

For

each

legal

line

l

we

need

to

specify

the

number

of

the

corresponding

circle

cl

the

start

point

sl

the

end

point

el

and

the

line

width

wl

By

convention

lines

are

drawn

clockwise

from

sl

to

el

Once

we

know

cl

we

can

specify

sl

by

specifying

the

number

of

the

circle

touching

or

intersecting

cl

in

sl

In

general

an

extra

bit

is

necessary

to

disambiguate

between

two

possible

intersections

Similarily

for

el

Thus

all

pixels

of

a

legal

line

may

be

compactly

represented

by

a

triple

of

circle

numbers

two

bits

for

intersection

disambiguation

and

a

few

bits

for

the

line

width

Clearly

the

bigger

the

used

circles

the

fewer

bits

are

needed

to

specify

the

corresponding

legal

lines

and

the

simpler

-LRB-

in

general

-RRB-

the

drawing

By

using

very

many

very

small

circles

-LRB-

beyond

the

resolution

of

the

human

eye

-RRB-

anything

can

be

drawn

-LRB-

using

Rules

IV

and

V

-RRB-

such

that

it

looks

right

This

would

not

be

very

impressive

however

because

a

lot

of

information

would

be

required

to

specify

the

drawing

It

would

be

more

impressive

if

it

were

possible

to

draw

something

non-trivial

that

looks

right

from

legal

lines

defined

by

few

big

circles

In

a

way

this

would

be

related

to

capturing

the

essence

provided

one

agrees

that

the

essence

of

an

object

rests

in

the

shortest

algorithm

describing

the

object

Section

however

will

show

that

such

compact

representation

can

be

difficult

I

found

that

it

is

much

easier

to

come

up

with

acceptable

complex

drawings

than

with

acceptable

simple

drawings

of

given

objects

Often

the

low-complexity

artist

will

use

drawing-specific

symmetries

and

the

like

to

further

compress

the

description

of

a

drawing

The

next

section

will

present

examples

of

this

Examples

of

Cartoons

Based

on

Rules

I-IV

Throughout

the

centuries

most

artist

's

favorite

object

has

been

the

human

body

In

what

follows

I

will

for

the

most

part

adhere

to

this

tradition

Here

is

a

set

of

informal

specifications

of

the

cartoons

to

be

presented

Figure

The

-LRB-

informal

-RRB-

goal

is

to

draw

a

butterfly

approaching

a

vase

with

a

flower

Figure

The

goal

is

to

design

a

woman

's

profile

Figure

The

goal

is

to

design

a

logo

for

a

gym

based

on

a

weight

lifter

's

upper

body

Figure

The

goal

is

to

draw

a

caricature

of

a

pin-up

girl

Figures

and

are

examples

of

cartoons

designed

by

using

rules

I-VI

only

Instead

of

providing

each

drawing

's

somewhat

opaque

coding

sequence

-LRB-

a

list

of

small

numbers

generated

according

to

the

coding

scheme

from

section

-RRB-

Figures

and

provide

corresponding

graphical

explanations

In

conjunction

with

Figure

each

explanation

allows

the

algorithmic

simplicity

of

the

corresponding

cartoon

to

be

described

quickly

to

the

human

observer

The

figures

demonstrate

that

the

circle

scheme

is

quite

flexible

In

terms

of

bits

it

is

cheaper

to

encode

all

cartoons

-LRB-

Figures

-RRB-

simultaneously

than

to

encode

each

cartoon

separately

because

the

algorithm

for

generating

legal

circles

and

their

numbers

-LRB-

see

section

-RRB-

is

shared

by

all

four

cartoons

In

the

terminology

of

algorithmic

information

theory

the

cartoons

share

a

non-trivial

amount

of

mutual

algorithmic

information

The

circle

scheme

may

be

viewed

as

something

like

a

common

recognizeable

artistic

style

Many

additional

examples

of

algorithmically

simple

drawings

appear

elsewhere

-LRB-

Schmidhuber

-RRB-

On

Beauty

and

Minimum

Description

Length

Sometimes

an

artist

is

appreciated

for

his

distinctive

style

Sometimes

certain

works

of

art

are

perceived

as

beautiful

This

section

attempts

to

relate

the

formalism

of

the

theory

of

minimumdescription

length

-LRB-

MDL

see

-LRB-

Solomonoff

Kolmogorov

Chaitin

Levin

Wallace

and

Boulton

Rissanen

-RRB-

for

important

contributions

see

-LRB-

Li

and

Vit?anyi

-RRB-

for

an

overview

-RRB-

to

informal

notions

like

beauty

and

good

artistic

style

What

is

a

Beautiful

Drawing

What

is

beautiful

What

is

not

There

can

clearly

be

no

objective

answers

to

these

questions

What

is

considered

beautiful

by

one

observer

may

be

regarded

ugly

by

another

observer

Ideals

of

beauty

are

different

in

different

cultures

and

subcultures

they

have

changed

over

the

centuries

and

they

are

not

even

stable

with

respect

to

a

single

individual

Therefore

any

theory

of

beauty

has

to

take

the

observer

-LRB-

the

subject

-RRB-

into

account

Following

common

intuition

I

assume

that

a

typical

human

observer

tries

to

represent

input

data

in

terms

of

what

he

already

knows

To

take

care

of

the

observer

's

subjectiveness

I

assume

that

the

Church-Turing

is

true

and

postulate

the

following

setting

At

a

given

time

the

current

knowledge

of

a

human

observer

can

be

described

as

a

coding

algorithm

This

algorithm

maps

input

data

-LRB-

such

as

retinal

activation

caused

by

a

work

of

art

in

the

visual

field

-RRB-

onto

internal

representations

of

the

data

The

coding

algorithm

C

the

data

D

and

its

internal

representation

can

be

written

as

strings

of

symbols

from

a

finite

alphabet

If

conveys

all

the

information

about

D

but

the

length

of

is

less

than

the

length

of

D

then

D

is

compressible

or

redundant

with

respect

to

the

observer

's

knowledge

The

observer

already

knew

something

about

D

Similar

statements

can

be

made

in

cases

where

allows

only

for

partial

reconstruction

of

D

The

observer

's

subjectiveness

is

embodied

by

his

coding

algorithm

C

One

may

be

tempted

to

define

the

beauty

of

a

drawing

with

respect

to

C

In

the

following

preliminary

attempt

to

do

so

-LRB-

inspired

by

the

MDL

approach

-RRB-

I

assume

that

beauty

simply

corresponds

to

high

probability

with

respect

to

C

Given

C

the

best

way

of

selecting

a

drawing

s

from

a

set

or

class

S

of

possible

drawings

satisfying

certain

specifications

may

be

to

maximize

P

-LRB-

s

j

C

-RRB-

the

conditional

probability

of

s

given

C.

Bayes

formula

tells

us

P

-LRB-

s

j

C

-RRB-

P

-LRB-

C

j

s

-RRB-

P

-LRB-

s

-RRB-

P

-LRB-

C

-RRB-

-LRB-

-RRB-

or

equivalently

logP

-LRB-

s

j

C

-RRB-

logP

-LRB-

C

j

s

-RRB-

logP

-LRB-

C

-RRB-

logP

-LRB-

s

-RRB-

-LRB-

-RRB-

speaking

the

Church-Turing

thesis

claims

everything

that

can

be

computed

by

a

human

being

can

be

computed

by

an

appropriate

program

for

a

general

purpose

computer

-LRB-

like

a

Turing

machine

-RRB-

Let

us

interpret

this

Since

C

is

given

P

-LRB-

C

-RRB-

may

be

viewed

as

a

normalizing

constant

It

may

be

neglected

logP

-LRB-

C

j

s

-RRB-

can

be

interpreted

as

the

information

-LRB-

or

length

of

the

observer

's

shortest

algorithm

-RRB-

required

to

compute

C

from

s.

P

-LRB-

s

-RRB-

is

given

by

some

a

priori

distribution

on

the

drawings

For

simplicity

let

us

assume

that

this

prior

is

uniform

Then

given

C

s

S

is

optimal

-LRB-

most

likely

most

beautiful

-RRB-

if

the

information

required

to

compute

C

from

s

is

minimized

How

can

this

be

related

to

human

experience

The

following

example

attempts

to

establish

such

a

relationship

Beautiful

faces

Humans

appear

to

have

a

certain

coding

scheme

for

storing

faces

This

scheme

is

certainly

different

from

the

circle

scheme

from

section

It

is

probably

based

on

previous

experiences

with

many

different

faces

and

it

is

probably

adapted

to

code

many

faces

efficiently

-LRB-

see

also

section

-RRB-

One

way

of

doing

so

is

to

store

a

prototype

face

and

code

new

faces

by

coding

only

the

deviations

from

the

prototype

The

principle

of

minmal

description

length

suggests

that

the

ideal

-LRB-

most

likely

-RRB-

prototype

FP

maximizes

P

-LRB-

FP

j

F

-RRB-

thus

minimizing

logP

-LRB-

F

j

FP

-RRB-

logP

-LRB-

FP

-RRB-

where

F

is

a

given

set

of

all

faces

to

be

coded

In

words

The

optimal

prototype

minimizes

the

sum

of

the

description

lengths

of

all

faces

relative

to

the

prototype

and

the

description

length

of

the

prototype

itself

-LRB-

relative

to

the

remaining

observer

knowledge

-RRB-

Assume

that

all

faces

are

equally

likely

to

appear

in

the

visual

field

The

formalism

above

predicts

that

the

most

beautiful

face

is

the

one

that

makes

it

easiest

to

compute

the

coding

scheme

from

the

face

It

seems

reasonable

to

assume

that

the

information

required

to

specify

the

coding

scheme

is

dominated

by

the

information

required

to

specify

the

prototype

face

If

the

current

face

looks

like

the

prototype

face

then

there

is

hardly

anything

to

compute

This

would

imply

that

the

prototype

face

is

perceived

as

the

most

beautiful

one

Previous

work

on

attractive

faces

The

statement

above

seems

compatible

with

results

presented

in

-LRB-

Langlois

and

Roggman

-RRB-

The

authors

claim

that

the

average

face

-LRB-

computed

by

digital

blending

of

numerous

photos

of

real

faces

-RRB-

is

perceived

as

the

most

attractive

one

Perrett

May

and

Yoshikawa

-LRB-

-RRB-

partly

dispute

this

claim

however

Their

test

subjects

also

appreciate

average

faces

computed

by

blending

-LRB-

Benson

and

Perrett

-RRB-

but

prefer

attractive

average

faces

constructed

from

faces

perceived

as

attractive

Indeed

the

most

attractive

faces

are

caricatures

obtained

by

digitally

exaggerating

the

deviations

between

average

and

attractive

average

Problems

with

previous

work

The

studies

above

however

do

not

say

much

about

the

plausibility

of

the

algorithms

used

to

compute

average

faces

Let

us

assume

that

the

brain

does

indeed

support

face

processing

by

an

ideal

-LRB-

in

the

information

theoretic

sense

-RRB-

prototype

face

It

would

be

naive

to

assume

that

the

latter

equals

the

one

computed

by

blending

There

are

many

possible

plausible

algorithms

for

computing

prototypes

based

on

many

plausible

metrics

for

distances

between

faces

Therefore

the

studies

above

including

the

statement

that

the

average

face

is

not

the

most

attractive

one

have

to

be

judged

with

scepticism

The

presented

claims

depend

on

the

definition

of

the

average

and

the

corresponding

nature

of

the

blending

algorithms

which

need

not

be

very

much

related

to

a

hypothetical

method

the

brain

might

be

using

for

generating

the

optimal

prototype

FP

Unfortunately

at

the

present

time

it

seems

impossible

to

analyze

the

way

the

brain

stores

representations

of

objects

Therefore

it

also

seems

impossible

to

test

the

predictions

made

by

the

formalism

presented

above

Comments

Beauty

and

evolution

One

may

continue

to

speculate

as

follows

A

society

with

a

distribution

of

faces

corresponding

to

an

algorithmically

simple

prototype

face

may

have

an

evolutionary

advantage

This

is

because

face

recognition

-LRB-

based

on

the

given

hardware

the

brain

-RRB-

may

be

more

successful

or

efficient

in

such

a

society

Evolutionary

pressure

may

favor

beautiful

prototypes

where

beauty

is

defined

by

the

nature

of

the

computations

our

brain

is

good

at

On

the

other

hand

the

nature

of

these

computations

is

influenced

by

typical

face

recognition

tasks

to

be

solved

It

seems

to

be

hard

to

analyze

such

mutual

dependencies

Learning

coding

schemes

Most

certainly

the

fractal

circle

scheme

from

section

is

different

from

the

typical

human

coding

scheme

Therefore

the

most

beautiful

cartoons

relative

to

the

circle

scheme

will

be

different

from

the

most

beautiful

cartoons

relative

to

the

coding

scheme

of

most

humans

Unfortunately

I

can

not

extract

the

latter

-LRB-

although

many

artists

implicitly

try

to

guess

it

I

believe

-RRB-

However

humans

can

learn

new

coding

schemes

In

particular

it

is

not

hard

for

them

to

learn

the

circle

scheme

Therefore

I

hope

that

some

of

the

cartoons

in

this

paper

will

be

easily

accessible

by

some

readers

Something

beautiful

needs

not

be

interesting

Interestingness

has

to

do

with

unexpectedness

But

not

everything

that

is

unexpected

is

interesting

-LCB-

just

think

of

white

noise

One

reason

for

the

interestingness

-LRB-

relative

to

some

observers

-RRB-

of

some

of

the

pictures

from

section

may

be

that

they

exhibit

unexpected

structure

Certain

aspects

of

these

pictures

are

not

only

unexpected

-LRB-

relative

to

typical

observers

-RRB-

but

unexpected

in

a

regular

non-random

way

The

formalization

of

interestingness

requires

an

extension

of

the

formalism

above

This

however

is

beyond

the

scope

of

the

paper

What

is

Good

Artistic

Style

Above

I

made

use

of

the

fact

that

humans

can

learn

new

coding

algorithms

In

a

way

their

original

coding

algorithm

is

universal

enough

to

allow

for

implementing

new

coding

algorithms

One

important

subgoal

of

low-complexity

art

is

to

devise

good

coding

algorithms

What

does

this

mean

A

new

coding

scheme

may

be

considered

as

good

relative

to

a

given

observer

if

-LRB-

-RRB-

it

does

not

require

many

bits

to

be

specified

-LRB-

given

the

observer

's

previous

knowledge

-RRB-

and

-LRB-

-RRB-

many

different

drawings

-LRB-

satisfying

typical

specifications

-RRB-

can

be

encoded

efficiently

by

it

In

that

case

the

drawings

share

a

non-trivial

amount

of

algorithmic

information

and

the

coding

scheme

represents

something

like

a

common

style

A

given

coding

scheme

may

be

representative

for

a

given

artist

which

will

make

him

stylistically

recognizeable

More

formally

the

quality

of

an

artistic

style

or

coding

scheme

C

may

be

evaluated

as

follows

An

optimal

style

C

maximizes

P

-LRB-

C

j

S

-RRB-

the

conditional

probability

of

the

style

given

a

set

of

drawings

S

-LRB-

defined

by

a

set

of

specifications

-RRB-

Equivalently

an

optimal

style

C

minimizes

logP

-LRB-

C

j

S

-RRB-

logP

-LRB-

S

j

C

-RRB-

logP

-LRB-

S

-RRB-

logP

-LRB-

C

-RRB-

-LRB-

-RRB-

Since

S

is

given

P

-LRB-

S

-RRB-

may

be

viewed

as

a

normalizing

constant

and

it

may

be

ignored

The

term

logP

-LRB-

S

j

C

-RRB-

can

be

interpreted

as

the

information

required

to

compute

all

elements

in

S

from

C

The

term

P

-LRB-

C

-RRB-

is

given

by

some

a

priori

distribution

on

the

coding

schemes

and

depends

on

the

observer

logP

-LRB-

C

-RRB-

can

be

interpreted

as

the

information

necessary

to

specify

C

given

the

observer

's

knowledge

Thus

given

S

C

is

optimal

-LRB-

most

likely

-RRB-

if

the

sum

of

two

terms

is

minimized

-LRB-

-RRB-

The

information

required

to

compute

S

from

C

and

-LRB-

-RRB-

The

information

required

to

compute

C

from

the

observer

's

previous

coding

scheme

The

circle

scheme

from

section

is

easy

to

teach

to

typical

humans

which

may

be

another

way

of

saying

that

not

much

information

is

required

to

compute

it

from

typical

human

knowledge

In

this

case

the

second

term

appears

negligible

Therefore

given

the

drawings

presented

in

this

paper

the

circle

scheme

appears

to

correspond

to

a

good

-LRB-

although

probably

non-optimal

-RRB-

artistic

style

Final

Remarks

On

the

Difficulty

of

Creating

Low-Complexity

Art

This

paper

specifies

-LRB-

and

exemplifies

-RRB-

the

nature

of

low-complexity

art

without

providing

a

general

method

for

creating

low-complexity

art

Although

it

is

trivial

to

redraw

the

concrete

examples

from

section

they

appear

out

of

the

blue

They

do

not

tell

how

they

were

discovered

They

do

not

give

many

clues

about

how

to

draw

other

objects

No

universal

algorithm

for

generating

low-complexity

art

is

known

At

the

moment

a

human

artist

is

required

I

found

it

difficult

to

discover

acceptable

but

algorithmically

simple

cartoons

I

found

it

easier

to

come

up

with

acceptable

cartoons

that

appeared

to

be

algorithmically

very

complex

For

instance

I

had

to

do

more

than

sketches

all

of

them

seriously

flawed

in

one

way

or

another

before

finally

ending

up

with

the

algorithmically

simple

Figure

Relation

to

Previous

Work

This

paper

certainly

is

not

the

first

to

introduce

formal

rules

in

art

For

instance

Lyonel

Feininger

writes

Aber

die

Erkenntnis

ist

mein

dass

es

in

der

ganzen

Welt

in

allen

Welten

nichts

Ungesetzliches

nichts

Zufalliges

nichts

ohne

Form

und

Rhythmus

gibt

noch

geben

kann

Warum

dann

gerade

in

der

Kunst

Soll

diese

nicht

dann

indem

sie

des

Menschen

schopferischen

Willen

offenbart

gerade

voll

Form

voll

Gesetz

und

Geist

sein

English

But

the

insight

is

mine

that

in

the

whole

world

in

all

worlds

nothing

unlawful

nothing

random

nothing

without

form

and

rhythm

exists

nor

can

exist

Why

then

in

the

arts

Should

n't

the

arts

by

exhibiting

man

's

creative

will

be

filled

with

form

law

and

spirit

The

ancient

Greeks

Leonardo

da

Vinci

Albrecht

Durer

LeCorbusier

and

many

others

devised

formal

rules

to

draw

things

right

Most

rules

are

based

on

simple

proportions

Examples

are

The

distance

between

the

eyes

should

equal

the

eye-width

-LRB-

origin

unknown

-RRB-

Or

The

ratio

of

the

distance

from

toes

to

navel

and

the

distance

from

toes

to

top

of

the

head

should

be

the

harmonic

-LRB-

LeCorbusier

-RRB-

Artists

know

many

rules

like

that

Most

of

these

rules

however

are

very

informal

and

leave

almost

everything

to

intuition

In

contrast

to

previous

informal

approaches

this

paper

adopts

an

extreme

standpoint

It

advocates

the

position

that

the

simplicity

of

a

work

of

art

should

be

proven

by

demonstrating

that

it

can

be

computed

by

a

short

algorithm

Nothing

in

a

work

of

low-complexity

art

should

be

without

such

a

motivation

This

paper

does

not

claim

to

be

the

first

to

present

examples

of

low-complexity

art

however

Certain

representations

of

artistically

interesting

and

aesthetically

pleasing

representations

of

fractal

objects

may

be

regarded

as

works

of

low-complexity

art

such

as

hills

coast

lines

etc.

-LRB-

Mandelbrot

Peitgen

and

Richter

-RRB-

This

is

because

they

are

based

on

relatively

short

and

understandable

algorithmic

descriptions

Also

certain

data

compression

methods

can

be

used

to

generate

patterns

that

may

be

regarded

as

works

of

low-complexity

art

-LRB-

see

e.g.

-LRB-

Culik

and

Kari

-RRB-

and

the

references

to

be

found

therein

-RRB-

Similar

statements

could

be

made

about

certain

other

simplifying

computer

models

for

representing

objects

-LRB-

see

e.g.

-LRB-

Essa

et

al.

-RRB-

for

work

on

face

animation

-RRB-

See

also

-LRB-

Li

-RRB-

Also

many

graphical

designers

consciously

or

unconsciously

use

their

tools

to

come

up

with

algorithmically

simple

designs

Apparently

however

nobody

so

far

has

identified

low-complexity

art

and

low-complexity

design

as

such

in

its

most

general

form

A

contribution

of

this

paper

is

to

make

explicit

the

nature

of

lowcomplexity

art

and

low-complexity

design

-LCB-

the

creation

of

understandable

works

of

art

-LRB-

or

designs

-RRB-

with

low

Kolmogorov

complexity

This

also

provides

a

framework

for

the

categorization

of

previous

work

Finally

it

should

be

mentioned

that

there

have

been

attempts

to

use

classical

information

theory

-LRB-

Shannon

-RRB-

to

formalize

what

's

aesthetically

pleasing

e.g.

-LRB-

Nake

-RRB-

Another

contribution

of

this

paper

is

to

offer

an

alternative

approach

based

on

algorithmic

information

theory

translation

by

the

author

Subtleties

in

the

original

text

source

may

be

lost

harmonic

proportion

is

what

you

get

if

you

divide

a

straight

line

of

length

a

into

two

segments

of

lenghts

b

and

c

such

that

ab

bc

One

solution

is

b

Outlook

Artistic

expressibility

depends

on

available

technology

The

cave

artists

from

the

stone

age

did

not

have

the

technology

for

creating

colors

that

made

impressionism

possible

The

impressionists

did

not

have

today

's

computer

graphics

This

does

neither

imply

that

the

best

cave

drawings

are

less

perfect

than

the

best

impressionist

paintings

nor

that

the

best

impressionist

paintings

are

less

perfect

than

the

best

computer

graphics

Each

age

however

tends

to

have

its

preferred

means

of

artistic

expression

What

can

we

expect

with

regard

to

the

future

of

low-complexity

art

There

will

be

tools

that

will

simplify

the

creation

of

low-complexity

art

I

expect

significant

extensions

of

the

programs

used

to

speed

up

and

simplify

the

creation

of

the

drawings

from

section

In

particular

I

expect

virtual

ateliers

implemented

on

powerful

machines

A

virtual

atelier

will

be

accessible

via

stereoscopic

virtual

reality

interfaces

It

will

allow

complex

-LRB-

or

more

dimensional

-RRB-

objects

to

be

quickly

composed

from

simpler

ones

by

hand

movements

-LRB-

perceived

by

the

machine

via

data

gloves

or

similar

devices

-RRB-

For

instance

with

a

virtual

atelier

it

will

be

easy

to

extend

the

circle

scheme

from

section

to

an

analoguous

sphere

scheme

or

bubble

scheme

The

algorithmic

atelier

will

permit

the

artist

to

quickly

generate

sequences

of

sketches

of

sculptures

to

evaluate

them

with

respect

to

their

artistic

value

to

discard

them

or

refine

them

In

principle

the

technology

for

building

virtual

ateliers

is

available

Given

the

current

inflation

of

cheap

computing

power

we

may

expect

that

it

will

not

take

very

long

until

many

artists

will

have

access

to

acceptable

virtual

ateliers

Acknowledgements

Thanks

to

Ulrike

Krommer

Christian

Freksa

Mark

Ring

Christof

Schmidhuber

Kurt

Prelinger

Karolin

Schmidhuber

Gerhard

Weiss

Astrid

Kiehn

Rohan

Baxter

Margit

Kinder

Martin

Eldracher

Daniel

Prelinger

Sepp

Hochreiter

and

Gary

Greenfield

for

useful

comments

References

Benson

P.

J.

and

Perrett

D.

I.

-LRB-

-RRB-

Extracting

prototypical

facial

images

from

exemplars

Perception

-LCB-

Chaitin

G.

-LRB-

-RRB-

On

the

length

of

programs

for

computing

finite

binary

sequences

statistical

considerations

Journal

of

the

ACM

-LCB-

Culik

K.

and

Kari

J.

-LRB-

-RRB-

Parallel

pattern

generation

with

one-way

communications

In

Karhumaki

J.

Maurer

H.

and

Rozenberg

G.

editors

Results

and

trends

in

Theoretical

Computer

Science

pages

-LCB-

Springer

Essa

I.

A.

Darrell

T.

and

Pentland

A.

-LRB-

-RRB-

Modeling

and

interactive

animation

of

facial

expressions

using

vision

Technical

report

MIT

Media

Lab

Perceptual

Computing

Section

TR

Kolmogorov

A.

-LRB-

-RRB-

Three

approaches

to

the

quantitative

definition

of

information

Problems

of

Information

Transmission

-LCB-

Langlois

J.

H.

and

Roggman

L.

A.

-LRB-

-RRB-

Attractive

faces

are

only

average

Psychological

Science

-LCB-

Levin

L.

A.

-LRB-

-RRB-

Laws

of

information

-LRB-

nongrowth

-RRB-

and

aspects

of

the

foundation

of

probability

theory

Problems

of

Information

Transmission

-LRB-

-RRB-

-LCB-

Li

M.

-LRB-

-RRB-

Minimum

description

length

based

shape

description

In

IEEE

Int

Conf

on

Computer

Vision

pages

-LCB-

Li

M.

and

Vit?anyi

P.

M.

B.

-LRB-

-RRB-

An

Introduction

to

Kolmogorov

Complexity

and

its

Applications

Springer

Mandelbrot

B.

-LRB-

-RRB-

The

Fractal

Geometry

of

Nature

Freeman

and

Co.

San

Francisco

Nake

F.

-LRB-

-RRB-

Asthetik

als

Informationsverarbeitung

Springer

Peitgen

H.

O.

and

Richter

P.

H.

-LRB-

-RRB-

The

Beauty

of

Fractals

Images

of

Complex

Dynamical

Systems

Springer

Perrett

D.

I.

May

K.

A.

and

Yoshikawa

S.

-LRB-

-RRB-

Facial

shape

and

judgements

of

female

attractiveness

Nature

-LCB-

Rissanen

J.

-LRB-

-RRB-

Modeling

by

shortest

data

description

Automatica

-LCB-

Schmidhuber

J.

H.

-LRB-

-RRB-

Algorithmische

Kunst

Manuscript

Schmidhuber

J.

H.

-LRB-

-RRB-

Discovering

problem

solutions

with

low

Kolmogorov

complexity

and

high

generalization

capability

Technical

Report

Fakultat

fur

Informatik

Technische

Universitat

Munchen

Shannon

C.

E.

-LRB-

-RRB-

A

mathematical

theory

of

communication

-LRB-

parts

I

and

II

-RRB-

Bell

System

Technical

Journal

XXVII

-LCB-

Solomonoff

R.

-LRB-

-RRB-

A

formal

theory

of

inductive

inference

Part

I.

Information

and

Control

-LCB-

Wallace

C.

S.

and

Boulton

D.

M.

-LRB-

-RRB-

An

information

theoretic

measure

for

classification

Computer

Journal

-LRB-

-RRB-

-LCB-

FIGURES

Figure

Fractal

result

of

an

iterative

application

of

Rules

I

and

II

-LRB-

following

initialization

see

section

-RRB-

Recall

that

Rule

I

says

wherever

two

legal

circles

of

equal

radius

touch

or

intersect

define

the

center

of

another

legal

circle

with

equal

radius

Recall

that

Rule

II

says

the

center

of

any

legal

circle

with

a

given

radius

is

the

center

of

another

legal

circle

with

half

the

radius

The

self-similar

figure

builds

the

basis

of

all

figures

to

follow

On

each

legal

circle

there

are

the

centers

of

legal

circles

with

equal

radius

For

clarity

legal

circles

are

emphasized

in

proportion

to

the

logarithm

of

their

radius

Parts

of

legal

circles

located

outside

the

initial

circle

-LRB-

the

frame

-RRB-

are

not

shown

Also

circles

whose

radius

is

smaller

than

one

sixteenth

of

the

radius

of

the

frame

are

not

shown

C

J.

Schmidhuber

Figure

Butterfly

approaching

a

vase

with

a

flower

Only

Rules

I-VI

-LRB-

see

section

-RRB-

were

used

to

design

the

drawing

Its

low

Kolmogorov

complexity

-LRB-

or

its

algorithmic

simplicity

-RRB-

is

explained

by

Figure

C

J.

Schmidhuber

Figure

Explanation

of

the

low

Kolmogorov

complexity

-LRB-

or

algorithmic

simplicity

-RRB-

of

Figure

Take

a

little

time

to

analyze

the

structure

of

the

drawing

All

circles

shown

are

taken

from

Figure

Very

few

of

them

however

are

needed

to

specify

the

drawing

They

are

drawn

with

a

thicker

line

width

Many

of

them

are

big

and

do

not

require

many

bits

to

be

specified

Compare

the

coding

scheme

in

section

C

J.

Schmidhuber

Figure

Woman

's

profile

Again

only

Rules

I-VI

-LRB-

see

section

-RRB-

were

used

to

design

the

drawing

Its

low

Kolmogorov

complexity

-LRB-

or

its

algorithmic

simplicity

-RRB-

is

explained

by

Figure

C

J.

Schmidhuber

Figure

Explanation

of

the

low

Kolmogorov

complexity

-LRB-

or

algorithmic

simplicity

-RRB-

of

Figure

All

circles

shown

are

taken

from

Figure

Most

of

the

used

circles

-LRB-

drawn

with

a

thicker

line

width

-RRB-

are

comparatively

big

-LCB-

their

unique

specification

requires

only

few

bits

Compare

the

coding

scheme

in

section

Take

a

little

time

to

analyze

the

structure

of

the

drawing

For

instance

note

the

symmetries

in

nose

lips

and

chin

Many

contours

of

the

eyes

nose

mouth

and

chin

repeat

themselves

again

and

again

thus

fitting

each

other

in

a

certain

sense

C

J.

Schmidhuber

Figure

Cartoon

of

a

weight

lifter

's

upper

body

designed

for

the

logo

of

a

gym

Again

only

Rules

I-VI

were

used

to

specify

the

cartoon

Its

low

Kolmogorov

complexity

-LRB-

or

its

algorithmic

simplicity

-RRB-

is

explained

by

Figure

C

J.

Schmidhuber

Figure

Explanation

of

the

low

Kolmogorov

complexity

-LRB-

or

the

algorithmic

simplicity

-RRB-

of

Figure

All

circles

shown

are

legal

circles

taken

from

Figure

Compare

the

coding

scheme

from

section

Some

comments

Note

that

shoulders

and

biceps/triceps

are

shaped

by

circles

of

equal

sizes

This

may

be

viewed

as

an

idealization

of

what

can

be

observed

with

certain

human

weight

lifters

The

same

circle

size

is

used

for

many

additional

features

such

as

the

top

of

the

head

parts

of

the

chest

etc.

Mirror

symmetry

is

broken

only

for

the

abdominal

muscles

The

visible

part

of

the

dumb-bell

belongs

to

a

legal

circle

with

nearly

infinite

radius

-LRB-

drawn

with

a

huge

but

legal

line

width

-RRB-

C

J.

Schmidhuber

Figure

Caricature

of

a

pin-up

girl

Again

only

Rules

I-VI

-LRB-

see

section

-RRB-

were

used

to

design

the

cartoon

The

drawing

has

low

Kolmogorov

complexity

and

is

algorithmically

simple

in

the

sense

that

it

can

be

specified

by

a

short

sequence

of

small

integers

using

the

coding

scheme

from

section

The

algorithmic

simplicity

of

the

caricature

is

explained

by

Figure

-LRB-

face

details

-RRB-

and

Figure

C

J.

Schmidhuber

Figure

Explanation

of

the

low

Kolmogorov

complexity

-LRB-

or

algorithmic

simplicity

-RRB-

of

the

face

of

the

caricature

in

Figure

All

circles

shown

are

taken

from

Figure

Very

few

of

them

however

are

needed

to

specify

the

drawing

-LRB-

they

are

drawn

with

a

thicker

line

width

-RRB-

Many

of

them

are

big

and

do

not

require

many

bits

to

be

specified

-LRB-

compare

the

coding

scheme

in

section

-RRB-

Some

comments

Cheek

forehead

and

part

of

one

eyebrow

-LRB-

drawn

with

a

huge

legal

line

width

-RRB-

are

shaped

from

equally-sized

legal

circles

The

bridge

of

the

nose

is

made

from

a

legal

circle

twice

this

size

and

so

is

part

of

the

chin

The

basic

shapes

of

the

eyes

are

constructed

by

two

circles

each

They

happen

to

fit

an

ancient

rule

The

distance

between

the

eyes

equals

one

eye-width

Many

contours

of

the

eyes

nose

mouth

and

chin

repeat

themselves

again

and

again

thus

fitting

each

other

in

a

certain

sense

C

J.

Schmidhuber

Figure

Explanation

of

the

low

Kolmogorov

complexity

-LRB-

or

the

algorithmic

simplicity

-RRB-

of

Figure

Most

legal

circles

shown

are

not

necessary

to

specify

the

drawing

-LCB-

the

dotted

ones

are

shown

only

to

provide

a

frame

of

reference

-LRB-

compare

Figure

-RRB-

Circles

actually

used

are

drawn

with

a

thicker

line

width

Take

a

little

time

to

study

them

Some

comments

The

initial

circle

is

the

one

defining

the

shin

of

the

bent

leg

Its

radius

is

twice

the

radius

of

the

circle

used

for

the

frame

Note

the

symmetry

of

calf

bottom

and

waist

They

are

shaped

by

a

row

of

three

touching

equally-sized

legal

circles

Shoulders

cheek

and

forehead

and

most

parts

of

the

chest

are

all

defined

by

circles

of

the

same

size

thus

creating

numerous

mirror

and

rotation

symmetries

Observe

that

the

leg

with

the

boot

is

shaped

by

legal

waves

of

decreasing

amplitude

Note

that

all

circles

used

for

the

face

-LRB-

see

again

Figure

for

details

-RRB-

correspond

to

legal

circles

of

the

more

complex

Figure

C

J.

Schmidhuber

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O 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