solving the vitruvian man
DESCRIPTION
The geometries of the human body.TRANSCRIPT
SolvinPatrick M
The probl
almost 20
familiar n
human in
anthropo
geometric
replicatin
“…
ba
fin
th
fo
an
as
D
Albeit Vit
body fit i
geometric
problem i
relationsh
In looking
Proportio
g the Vit. Dey & Dami
lem of solving
000 years. Som
umber phi (ϕ
n the square
metrics, whe
c process is r
g geometry. V
… In the huma
ack, with his
ngers and toes
herefrom. And
ound from it. F
nd then apply
s the height, as
e Architectura
ruvius does n
nto both a c
cians, such a
n that we can
hips and prop
g at Leonard
ns, we can be
truvian Mian ‘Pi’ Lannin
g the Vitruvia
me probably
ϕ), a number
e and circle
ther the hum
related to oth
Vitruvius desc
n body the ce
hands and fee
s of his two h
just as the hu
For if we meas
that measure t
s in the case of
, Book III, 1 : 3
not present t
circle and a s
s Agrippa, Ce
n map the pro
portions. But w
Da Vinci’s Vit
egin to under
Man ngham
The Problem
an Man seem
never knew t
that seems t
is created,
man is in moti
her geometri
cribes the geo
ntral point is n
et extended, a
ands and feet
uman body yie
sure the distan
to the outstret
f planes and su
the question
square? This
esariano, Dür
oportions and
we must ask t
truvian Man,
rstand some g
m and the Ge
ms to have elu
there was pro
o be the mag
and how t
on or not. Fu
es, such as t
ometrical pro
naturally the n
and a pair of
will touch the
elds a circular
nce from the so
tched arms, th
urfaces which a
at hand, the
problem has
rer, and Da V
d geometries
then : how is
the most no
general probl
eometry
uded mathem
oblem. It use
gician of all n
he same ge
urthermore w
he pentagram
oportions of t
navel. For if a m
compasses ce
e circumferenc
outline, so too
oles of the fee
he breadth will
are perfectly sq
question sti
s been searc
Vinci. The pro
s of the huma
this geometr
otable interpr
ems :
maticians and
es that myste
umbers. We
eometries ca
we will illumin
m, and how
the human bo
man be placed
entered at his
ce of the circle
o a square fig
et to the top o
l be found to b
quare.”
ll exists : how
ched and wo
oblem is som
an body and d
ry created in
retation of Vi
geometricia
rious, and ye
will show ho
n account f
nate how the
it is a natura
ody as such :
d flat on his
s navel, the
e described
ure may be
of the head,
be the same
w does the h
rked out by
mewhat of a
discover num
the first place
itruvius’s Can
ns for
t very
w the
or all
same
l self‐
uman
many
latent
erous
e?
non of
The first i
do not m
error by r
either the
geometric
humanist
beautiful)
In order t
very ques
the fathe
collaborat
AFNOR on
“T
m
th
w
Logically t
inherent
square an
the oppo
adjacent
greater th
Figure
The produ
couple of
rectangle
the oppos
base of t
(Figure E)
90° (Figur
1 Le Corbus2004. P. 37
ndication tha
eet the boun
roughly 1.5%
e human body
cally perfect
ideals of the
) geometricia
to tackle this
stion was con
ers of Moder
tor, who is o
n standardizin
Take a man‐wi
meters each, su
hird square sh
where to put th
this process s
in human pr
nd then produ
osing side of
squares (Figu
han 90° (Figur
e A
uct of the roo
f years later
is constructe
sing mid‐poin
he geometry
, but still in e
re F) :
sier. The Modu7.
at there is a g
ndary of the c
of the circle’
y is not perfe
means of co
e body being
n, we have to
problem a b
nsidered and
rnist architec
only known as
ng products, g
ith‐arm‐uprais
perimposed o
ould give you
is third square
should incorp
roportions. T
uce golden re
the square
ure B), theref
re C) :
ot‐two solutio
r, meeting w
ed from a squ
nt of the initi
y at point c (
error by 0.63%
ulor. Trans. Pet
geometrical p
circle, but rat
s radius. Con
ect and natur
onstructing th
g perfect, as
o argue for th
it of history a
attempted b
cture. Betwee
s Hanning, as
goods, and bu
ed, 2 ∙ 20 m. i
n each other; p
u a solution. T
e.”1
porate and be
he first prop
ectangle a. Ne
from the go
fore the angle
Figure B
on rectangle i
with Mlle Elis
are. Next a li
ial square, po
Figure D). Th
% and produc
ter de Francia a
problem is th
ther overlap
nsidering this
e is not a per
he Vitruvian
well as the b
e latter. But o
and other att
by the archite
en 1943 and
s well as Mlle
uildings prese
n height; put
put a third squ
The place of th
e developed f
posal given b
ext, from the
olden rectang
e (ϴ) of the “
is in error by
sa Maillard a
ne is constru
oint b. A 90°
he resulting r
ces “regulatin
and Anna Bost
e fact that th
it. This is no
error one of
rfect geomet
Canon. Since
belief that na
once again : h
tempts to sol
ect Charles ‘L
d 1945 Le Co
e Elisa Mailla
ented a propo
him inside two
uare astride the
he right angle
from the gold
by Hanning (F
e initial square
gle. The resu
“regulating”
1.6%, and th
another solu
cted from po
is drawn fro
rectangle is c
ng” lines that
tock. Basel, Sw
he upper vert
mild error, a
f two things m
trician, or the
e the former
ature is the
how?
lve it should
Le Corbusier’
orbusier wor
ard. Le Corbu
ortioning pro
o squares 1 ∙ 1
ese first two sq
e should help
den mean, sin
Figure A) wa
e draw root‐t
ulting rectan
lines of the o
Figure C
erefore not a
ution was pro
oint a of the g
m line ab to
closer to two
have an ang
witzerland : Birk
tices of the s
as the overlap
must be accep
ere is a natura
r is antithetic
most perfect
be looked at
Jeanneret, o
ked with a y
usier, working
oblem to Hann
10 by 1 ∙ 10
quares. This
you decide
nce this prope
as to begin w
two rectangle
gle is almos
overall rectan
a viable solut
oposed. A g
golden rectan
intersect wit
o adjacent sq
le (ϴ) greater
khäuser Publis
quare
p is in
pted :
al and
cal to
t (and
. That
one of
young
g with
ning :
erty is
with a
e b on
t two
ngle is
ion. A
golden
gle to
th the
quares
r than
hers.
Figur
The differ
Although
Modulor
proportio
architectu
false and
tangentia
compasse
draftsman
was flawe
squares; o
being eco
negligible
suspect th
open and
of a fatefu
We, on th
happens t
paradigm
claim that
2 Ibid., p. 63 Ibid., p. 2
re D
rence in the e
this system
System, whic
nately grow (
ure. We mus
probably forc
l to its circle
es, and penci
n’s behalf. M
ed! In 1948
one of their s
onomical con
e quantity… it
hat these six
shut, it is no
ul equality wh
he other han
to be dead w
s on urbanism
t if there is an
64, Fig. 21. 234.
errors of the a
is in error,
ch is a syste
(or diminish)
t admit that
ced (for insta
e2). But given
ls, it seems lo
ore surprising
a Monsieur
sides is larger
nsiders : “In
is not seen w
x thousandths
t sealed; ther
hich is not exa
nd, argue tha
wrong (which
m do not nee
n error, no ma
Figure E
above solutio
Le Corbusier
m of measur
from the init
the geometr
ance a tangen
n that these g
ogical that a
gly, though, i
r R. Taton inf
r by six thous
everyday pra
with the eye.”
s of a value
re is a chink t
actly, not stri
at there is no
is certainly
ed to be acco
atter how triv
ns can be see
r used this g
rements built
tial geometric
ries are appr
ntial line prod
geometries w
geometrical
s the fact tha
forms Le Cor
andths of the
actice, six th
” But Le Corb
have an infin
to let in the a
ctly equal… …
o error in the
not the first
ounted here)
vial and small
en here :
geometric str
t on the prin
c construct. T
roximated, an
duced from th
were constru
error would
at Le Corbusi
rbusier : “yo
e other [gross
ousandths o
busier also be
nitely preciou
air; life is ther
… And that is
e initial probl
time he was
). As mathem
l, it is an erro
Figure F
ucture to de
nciple of ant
This was his m
nd at certain
he regulating
cted with T‐
be considere
er did know t
our two initia
s miscalculati
f a value are
eing a mystic
us importance
re, awakened
what creates
lem and that
s dead wrong
maticians and
or, and therefo
evise his infa
hropometrics
means to hum
times comp
lines, which
squares, tria
ed an error o
that this geom
al squares ar
ion].” Le Corb
e what is ca
further adds
e: the thing
by the recur
s movement.”
t Le Corbusie
g, but 20th Ce
geometrician
ore incorrect
amous
s that
manize
pletely
is not
ngles,
on the
metry
re not
busier
lled a
: “… I
is not
rrence
” 3
er just
entury
ns we
.
Furtherm
show that
rectangle
from its in
distance b
(Figure H)
vertical h
into accou
account fo
Figure G
Looking a
geometrie
“reason”
their likin
construct
simply co
more con
1) C
2) T
V
3) T
m
4) It
5) A
In conside
with a squ
through s
critical di
human bo
ore, we argue
t it is not eleg
created by t
nitial side of
between the
). The final pr
uman measu
unt horizonta
or shortly.
t Le Corbusie
es do not dis
and “logic”, c
ng. On the o
the whole.
nstructs harm
straints :
Construct a ge
The geometry
Vitruvian Can
The geometry
measurement
t is devised fr
And, finally, th
ering this pro
uare and cons
some other fo
viding lines c
ody, such as t
e that the sol
gant. For his g
the regulating
the square to
top of the h
roblem with
urements, tha
al measureme
er’s Modulor w
sassemble an
construct geo
other hand, t
Man disasse
mony that is
eometry that
y constructs th
on of Proport
y accounts no
ts and propor
rom the golde
he constructio
oblem we fin
structing a go
orm of geom
created by a
the height of t
lution has to
geometry to b
g lines (estab
o be placed n
ead to the ti
Le Corbusier’
at is, it only
ents, such the
we find that i
d then reass
ometries and
the geometri
embles and r
beautiful; nat
continuously
he human ins
tions
t only for the
rtions
en mean
on of the geo
nd that it is j
olden rectang
etrizing. Also
an overlappin
the head.
be elegant. A
be in accorda
blished at poi
next to the g
p of the fing
’s solution is
attempts to
e proportions
it is not only
semble a con
reassemble t
es of nature
eassembles h
ture just is. N
y builds upon
side both a ci
e vertical anth
ometry has an
ustifiable tha
gle that this co
o, we assume
ng third squa
Another look
nce with the
nt c) shown
golden rectan
ers when the
that it only a
solve the he
s of the arm
Figure H
incorrect, but
struction. Hu
them arbitra
e are built u
harmony to
Now we retur
itself
rcle and squa
hropometrics
n elegant solu
at Le Corbusi
ould inevitab
e he must also
are must be
at Le Corbus
human body
in Figure D h
gle (Figure G
e arm is raise
accounts for t
ight. The Mo
span to the w
(Drawing
from T
t it is also not
umans are th
rily in what e
p from an in
create “beau
rn the initial
are in accorda
, but also for
ution.
ier would ass
bly produce tw
o be right in
found in acc
ier’s geometr
y the portion o
has to be rem
G). This create
ed above the
the proportio
odulor never
whole. This w
g by Le Corbus
The Modulor)
t elegant. Na
he ones, with
every manner
nitial geomet
uty”, while n
question, but
ance with the
the horizont
sume that st
wo perfect sq
assuming tha
cordance wit
ry will
of the
moved
es the
head
ons of
takes
we will
ier,
ture’s
their
r suits
try to
nature
t with
e
al
arting
quares
at the
th the
When we
segments
present w
divides th
for the w
opfe are
squares, a
3).
Although
elegant, it
therefore
created i
anthropo
The easie
two golde
equal to
midpoint
yd is equ
segmente
create the
Square
Area = φ2 ‐
Perimeter
4 In this pa“squaring”
e began appr
s of line ab ar
with a square,
he square abc
whole geomet
equal. Finally
as well as giv
Figure 1
this does so
t simply is no
natural geo
nitially from
metrics. So th
st remedy to
en rectangles
(φ ‐ ϕ) creat
of line bc. Th
ual to line yh
ed into two li
e square beih
‐ 2φϕ + ϕ2
= 4φ ‐ 4φ
per we will use” refers to mak
roaching this
re a golden m
, but not nec
cd into a gold
try thus far (F
y, if the initia
ving the posit
olve the prob
ot elegant eno
metries, do
m another go
he constructio
o justifying th
s on perpend
e a golden re
hen construct
h, in which p
ines, instead
h. The new sq
Golden Rect
Area = φ2 ‐ φ
Perimeter =
e terms in drafking a square fr
problem we
mean (Figure 1
cessarily form
den mean is
Figure 2). The
al square is s
ion of the th
Figur
blem of the t
ough. It is not
not typically
olden geome
on of this geo
e golden me
dicular axes.
ectangle cdfe
another gold
point y is the
of just one,
uare’s area is
tangle
φϕ
4φ ‐ 2ϕ
fting with T‐sqrom a given lin
e segmented
1). We did thi
ming a golden
squared4, th
erefore, Figu
quared, this
ird square ov
re 2
wo squares
t elegant eno
segment a s
etry. Addition
ometry must
an segmenta
Essentially, s
e so that line
den rectangle
e midpoint o
so that the s
s φ2.
Two Golden R
Line be = bh =
uares, compase segment, an
a square ab
is assuming t
rectangle. If
is naturally c
re 1 = Figure
naturally cre
verlapping th
F
created from
ough simply b
square with
nally, it doe
be reconside
ation of the sq
starting with
e xd equal to
e adgh on the
of line ab. W
segments are
Rectangles
= φ
sses, and triangd not be confu
bcd with a li
that the golde
f the length o
creates a gold
e 2, and the s
eates two per
e two adjace
Figure 3
m a golden m
because sacre
a golden me
es not accou
red.
quare is to fo
square abcd
o line xe, in t
e perpendicul
We now have
e golden mea
Squar
Area =
Perim
gles. The term used with x2
ne op so tha
en mean had
of segment op
den rectangle
squares abcd
rfect and adj
ent squares (F
mean and is r
ed geometries
ean without
unt for horiz
orm a square
d whose edg
that point x
lar axis so tha
e a square t
ans. We ther
re
= φ2
meter = 4φ
“squared” or
at the
to be
p that
e bcef
d and
jacent
Figure
rather
s, and
being
zontal
e from
ges all
is the
at line
hat is
refore
Thus, the
a golden r
cg is equa
equal to l
that lines
and elmi
equally se
2φ2.
Golden Rec
Area = 2φ2
Perimeter
To reitera
1) The
2)The a
3) Squa
4) From
5) From
with
6) The
7) Thus
same proces
rectangle ejki
al to line cj. T
line zj. As a r
ie and el are
are all equa
egmented int
ctangle 2 ‐ φϕ
= 6φ ‐ 2ϕ
ate the steps a
initial square
area of the go
aring off the g
m this new sq
m the square
h an area of φ
rhythmic pat
s the two adj
ss of creating
i by squaring
This is the sa
result squares
e equal. The
al, thus locate
o a golden m
and the math
e with an area
olden rectang
golden rectan
quare create a
created in st
φ2, so that the
ttern of the p
acent square
Figures 1 – 3
line segment
me result as
s beih and cjk
result is two
es the placem
ean as the in
Proof of
Squares
hematics :
a of φ2 ‐ 2φφ
gle equals φ2
ngle creates a
another golde
ep 2 square o
e overall geom
erimeter of t
s are equally
can be reiter
t cg, that is, t
if the midpo
kg are equal
perfect and
ment of the
itial square. T
f Third Square
beih = cjki = φ
‐ ϕ2 creates a
– φϕ.
a new square
en rectangle w
one of its leng
metry has an
his geometry
segmented i
rated as follo
the golden m
oint of line be
. Subsequent
adjacent squ
third square
The overall ge
φ2
a golden mea
with an area
with an area
gths to produ
area of 2φ2.
y is : φ ‐ ϕ, ϕ,
nto golden m
ows. With squ
ean of square
e was rotated
tly take line e
ares, in that
e. The newly
eometry thus
Two S
Area =
Perim
an.
a of φ2.
of 2φ2 – φϕ.
uce two equa
φ ‐ ϕ, ϕ, φ ‐
means.
uare beih con
e beih, so tha
d, so that lin
ei and square
squares beih
created squ
s far has an a
Squares
= 2φ2
meter = 6φ
l squares eac
ϕ, ϕ, φ ‐ ϕ, ϕ
struct
at line
e zi is
e it so
h, cjki,
are is
rea of
h
ϕ… …
To empha
Placement
Here it m
problem
account f
you decid
If we wer
as such :
First cons
square ab
regulating
perpendic
square th
Next draw
two perfe
the vertic
point o. I
square ab
Thus the
Although
regulating
asize that this
t of Third Squar
ay be argued
as he posed
or the regula
de where to p
e to correct L
struct golden
bcd into a go
g lines, then t
cular to each
e line will inte
w a line from
ect and adjac
al anthropom
f a line pq is
bcd are bisect
above essen
this does sol
g lines to exp
s is correct th
re
d that we me
it. If we we
ting lines and
ut this third s
Le Corbusier’s
n rectangle a
olden mean b
the easiest wa
h other. So if
ersect at poin
point y that
ent squares,
metrics it also
s drawn to in
ted into golde
ntially create
ve Le Corbus
ress the rest
e following d
Creation of
erely establish
ere to solve t
d the 90° ang
square.” At th
s problem as
aefd. Then sq
by the line s
ay to create t
we draw a 4
nt y.
is perpendicu
so that squa
can create th
ntersect at p
en means.
es a reiteratio
sier’s problem
of the creatio
rawings will i
f Second Squar
hed our own
the problem
gle. As he said
his point the
he posed it t
quare line ef
segment xy. B
two squares i
45° (ϴ) from
ular to line ey
res abcd = xe
he horizontal
oint o as we
on of the sa
m as he posed
on of the Vitru
illustrate the
re
geometry, b
as Le Corbu
d : “The place
placement of
the results wi
f back towar
But if this co
s to draw two
point e tow
y so that it cr
efy = gxyh. T
measuremen
ell, then the
ame geometr
d it, it is not n
uvian Man.
squares and
Two Perfect A
but did not so
usier posed w
e of the right
f the right ang
ill still be the
rds the circle
onstruction is
o successive 4
ards the opp
reates point g
This construct
nts. Lines bc a
golden recta
ry we have
necessary to c
their placem
Adjacent Squar
olve Le Corbu
we would ha
t angle should
gle is rather t
same, and fo
e, which segm
s supposed t
45° angles th
posing edge o
g. This create
tion not only
and ey inters
angle befc an
already prop
continue usin
ents :
res
usier’s
ave to
d help
trivial.
ollows
ments
o use
at are
of the
es the
gives
sect at
nd the
posed.
ng the
So far the
not only
measurem
assume t
squaring l
is better t
that this p
grows int
off the lar
the first c
the way i
i.e. a gold
So, if the
This is do
Vitruvian
complete
proportio
S
L
From this
geometrie
equal as w
equally ve
midpoint
were draw
point u. T
e very structu
accurately
ments that ar
hat line jk is
line bj, based
to understand
process starts
o a golden m
rgest golden
constraint est
n which a na
en mean.
process cont
ne so that lin
Man rests.
d to create b
ns and measu
Squaring the He
ine bj = bo
geometry we
es. For instan
well. Given th
ertically, and
and point of
wn, for instan
hus it can be
ure of the ou
giving the h
re the result
the top of t
on Vitruvius’
d the logical c
s with a squa
mean. Thus it
mean thus fa
ablished abo
utilus shell c
tinues as befo
nes bj and bo
Then the w
lpo, a grid of
urements, an
eight
e find several
ce lengths jn
hese equalitie
line vw segm
intersection o
nce a line bp o
established t
r proposed g
human heigh
of squaring t
the human h
’s description
construction o
re that grows
can be logica
ar. This is not
ve states tha
ontinuously b
ore, the large
are equal, th
whole array o
squares and
nd henceforth
l aspects that
and jb are eq
s it can there
ments the cons
of lines eq an
or lo, then the
that point u is
geometric con
ht in the sq
the golden m
head, so we
n. But assumi
of the human
s into anothe
ally concluded
t a justificatio
t the geomet
builds upon i
est golden rec
hus creating t
of constructio
golden recta
h will be refer
t are related t
qual, and the
efore be estab
struct equally
nd vw. Likewis
eir midpoints
s the center o
nstruct incor
quare and ci
mean. It seem
can complete
ng is rather in
n geometry. L
er square by a
d that the ne
on but a comp
tries must bu
tself with the
ctangle bjkh
he square bjn
on lines and
angles. This is
rred to at the
Completin
to Vitruvius’s
refore a squa
blished that li
y horizontally
se, if a diagon
s and point of
of the constru
porates the g
ircle, but als
ms rather logic
e the rest of
nappropriate
Logically, thou
a golden mea
ext step is to
pletely ration
ild upon them
e same propo
is squared es
no, being the
d geometries
s the primary
“geometric c
ng the Grid
description o
are. Lengths e
ine eq segme
y. Therefore p
nal line bisect
f intersection
uct, or, in a se
golden mean
so has horiz
cal at this po
f the geomet
here, but rat
ugh, it can be
an, and this s
repeat and s
nal next step,
mselves, muc
ortional geom
stablishing po
e square with
s are orthogo
grid of the h
construct”.
of the human
eb, ei, and el a
ents the const
point u is the
ting the const
would still b
ense it is the
, thus
zontal
oint to
try by
ther it
e seen
quare
quare
since
ch like
metry,
oint o.
in the
onally
uman
n
are
truct
truct
e
center of
well as th
V
A
P
Not only
anthropo
the shou
Proportio
construct
square, th
Logically f
be able to
more pro
anthropo
the hands
establishe
they shou
Le Corbus
derived fr
or produc
squaring
provide th
One mino
positions
inertia. Acco
e center of th
Vitruvian Cano
Area = 4φ2 ‐ 2φ
Perimeter = 4φ
is this soluti
metrics, as th
lders. As opp
ns was slight
another squa
hen square th
from this bas
o be construc
oper way of
metrics of th
s and digits) a
ed above. Sin
uld be able to
sier’s Modulo
rom anthropo
ced from the
or producing
he most accu
or flaw on Le
of body parts
rding to Vitru
he circle that
on of Proportion
φϕ
‐ ϕ
on elegant a
he initial squ
posed to Le
tly irregular,
are from the
he previous go
ic geometrica
cted from the
f saying it,
e human bod
and bodily loc
nce these pro
be construct
or is a propor
ometrics that
e golden mea
g a golden m
rate Modulor
Corbusier’s
s. For instanc
uvius it can be
encompasses
ns
and builds up
are, golden m
Corbusier’s
here the sol
initial (or pre
olden mean,
Creating a
al construct t
e overall geom
correcting L
dy. Furthermo
cations (i.e. lo
oportions can
ted inward, o
rtionately gro
t grow or dim
an. Since the
mean follows
r System.
part is that h
ce the positio
e logically con
s the human
pon its own
mean, and th
geometry, w
ution provide
evious) golden
et cetera.
ll Anthropom
the rest of the
metry. In esse
Le Corbusier
ore, we shou
ocation of the
n be built out
r fractalize.
owing (or dim
minish by eith
solution pro
s this exact
his geometric
on of the third
ncluded that t
figure.
Vitruvian
geometries,
he reflected s
whose definin
es a regular
n mean, then
metrics
e measureme
ence, we hav
r’s Modulor
ld be able to
e elbow) by b
tward, i.e. gr
minishing) sys
her squaring (
oposed involv
same patter
c construct o
d square in th
this is the loc
Man
it also accou
square establ
ng geometry
grid that can
n square the i
ents of the h
ve to account
System and
o account for
building off of
row upon the
tem of meas
(or doubling)
ves the same
rn, therefore
nly accounts
he Modulor p
cation of nave
unts for horiz
ish the bread
y of the Can
n be repeate
nitial (or prev
uman body s
t for creating,
d achieve al
all body part
f the same pr
emselves, lik
urements tha
the measure
e process of e
our solution
for a few pr
provides the c
el, as
zontal
dth of
on of
d, i.e.
vious)
hould
, or, a
ll the
ts (i.e.
rocess
kewise
at are
ement
either
n can
rimary
crown
of the he
course, t
positions
various nu
H above).
Thus, our
positions
construct
solution is
Since rest
must star
square) is
larger squ
geometric
through a
above the
and there
the arms
they fall i
body. So t
Construct
golden re
measurem
the nipple
elbow.
ad, and the “
his only acco
on the body
umbers, and
solution’s “M
of body par
ing the over
s as follows :
t of the bodi
rt with a squa
s to be chose
uares, say sq
c construct, a
a golden mea
e head) to cr
efore is not a
are held perp
nto the great
the groin squ
t golden recta
ectangle and
ments produc
es in males, b
“common” ed
ounts for th
are accounte
therefore an
Modulor syste
rts. The solut
rall geometri
ly positions a
are. The squa
en, since it is
uare abcd is
and therefore
n). If the squa
reate golden
prime square
pendicular to
t circle and th
are appears t
angles on all f
continue the
ced account f
but center of
dge of the tw
eir vertical p
ed for second
rather arbitr
em”, so to sa
tion to defin
c construct,
and measure
are surroundi
the smallest
s chosen, the
e produce no
are above the
means, then
e to choose. T
the body or
he navel beco
to be the mos
four sides of
em to the ed
for the vertic
the breasts in
wo squares p
positions and
arily by eithe
rary means of
ay, has to be
ning the less
as it follows
ments must
ng the groin
t square that
en the constr
new results.
e head is cho
the geomet
The groin so
lower. If the
omes the cen
st logical choi
the groin squ
dges of the g
al height of t
n adult femal
rovides the p
d not their
er squaring or
f accounting
able to provi
er anthropom
s an identica
be defined b
(henceforth
t lies within t
ruction proce
(Since square
sen (in the re
tries will lie o
happens to b
arms are hig
nter, thus pro
ice.
uare. Then ta
great square,
the knees and
les), as well a
position of th
horizontal po
r producing th
for these pos
de both hori
metrics are j
al means of
by a golden m
will be referr
the human b
ess will only
e abcd create
egion created
outside the st
be the center
her than the
oducing geom
ke the edges
therefore cr
d the center
as the horizon
he belly butto
ositions. All
he golden me
sitions. (See F
zontal and ve
just as elega
construction
mean, the so
red to as the
body. If one o
repeat the o
es the groin s
d by the arm r
tatic human
r of the body
shoulder line
metries outsid
produced by
reating a grid
of the breast
ntal position o
on. Of
other
ean of
Figure
ertical
ant as
n. The
lution
groin
of the
overall
quare
raised
body,
when
e then
de the
y each
d. The
ts (i.e.
of the
The next s
the groin
just the s
account f
the femu
offset fro
Again, the
for the ve
sternum),
The same
centerline
palm. Doi
meet the
smallest squa
square – is th
same manne
or the vertica
r) and the po
m the nose (i
e next smalle
ertical positio
, and the hori
e constructio
e of the arms
ing this once
palm.
are – that is, o
hen used to p
r as in the p
al height of th
osition of the
.e. the outer
st square the
on of the mo
izontal positio
n repeated f
s and the heig
more will ac
one of the sm
produce four
previous step
he groin (i.e.
collar bone,
edge of the e
en produces f
outh and the
on of the wris
for the next
ght of the eye
ccount for th
maller squares
golden mean
p illustrated
the height of
as well as th
eye).
four golden m
e trough of t
st.
smallest squ
es, as well as
he horizontal
s produced by
ns on each sid
above. The
f the pubic tr
he horizontal
means. The m
he breastplat
are accounts
the horizont
location of t
y squaring th
de of the squa
measuremen
iangle and th
position in w
measurements
te’s center (
s for the vert
tal position o
the moment
e golden mea
are. This is do
nts produced
he fulcrum po
which the eye
s created acc
i.e. bottom o
tical height o
f the center o
where the fi
ans of
one in
here
oint of
es are
counts
of the
of the
of the
ingers
This is, of
the golde
iterative
measurem
accounts
earlobe. T
vertical an
the Modu
construct
Now, Le C
measurem
product d
5 Ibid., p. 8
course, an it
en means the
process, any
ment or posit
for every hum
The asymptot
nd horizontal
ular to accoun
.
Corbusier wit
ments for the
design, and an
85 ‐ 87, Figures
terative const
e asymptote
y square on
tion. Specula
man measure
tic grid forme
Modulor me
nt for horizon
th him Modu
e human bod
nything with
33 – 35.
truction with
approaches
any side of
ative, though
ement from t
ed here echoe
easurements
ntal bodily m
ulor System, a
dy for applic
ergonomic ap
no end. Logic
the opposing
any square
most probab
the length of
es a similar g
of the human
measurements
and even Alb
cation in the
pplications. S
cally, because
g vertex of t
can be cho
bly, this asym
f the small int
grid Le Corbus
n. 5 Albeit, his
s, and it is no
brecht Dürer
visual arts,
So as a final n
e of the cont
the square. A
osen to crea
mptotic grid o
testines to th
sier construc
s is ad hoc an
ot constructe
before him,
furniture de
note to Le Co
inuous squar
And, since it
te another b
of anthropom
he thickness o
ted to devise
nd is trying to
d from the o
devised num
sign, archite
rbusier’s Mod
ring of
is an
bodily
metrics
of the
e both
force
overall
merical
cture,
dulor,
we do not wish to correct his Modulor values. The reasons for not doing so have nothing to do with the
math and tedious calculations. Rather we object to his value system altogether for democratic reasons.
The values derived from the Modulor only correspond to the ideal male, who is six feet in height. It must
be stressed that the ideal male height is different from the average male height. Since Le Corbusier is
French, the average male height is 5 foot 9 ½ inches. He got the idea to use the six foot tall man from
English detective novels, in which the good looking man is always six feet tall. But in England the average
man is 5 foot 9 inches tall.6 In the States the average Caucasian man is 5 foot 9 ½ inches tall.7 Clearly his
anthropometrics are ideal, rather than average. Additionally, his anthropometric values are not in
harmony for someone that is neither average nor ideal in height. Obviously, a house built for an average
man is not in anthropometric accordance with the Modular, and therefore out of harmony. Finally, his
system could be deemed sexist, as it does not account for women, whose proportions and
anthropometrics are slightly different from a man (for instance, the hips and breasts). The very fact that
Vitruvius’s description has always been referred to as the “Vitruvian Man” has been played upon with
Vitruvian Woman in the Feminist movement; most notable is the Vitruvian Woman by Susan Dorothea
White, as well as one by Nat Krate.
One can see the can of worms that would be opened here in trying to accurately and equally account for
the vast amount of variations of human proportions and measurements in all cultures, races, age
ranges, and the sexes. We will leave that for the ergonomists.
But then again, when Le Corbusier devised his Modulor System he rounded off the numbers so that they
may be feasibly implemented for practical purposes. No contractor would ever try to make a concrete
wall exactly 1.61803399… meters long. In this sense, we were being misleading when we said we would
“correct” Le Corbusier’s Modulor. Really, we are just providing the most accurate means for creating the
Modulor. There really is nothing wrong with the values of Modulor, except that they are only for a six‐
foot tall man. And in reality Le Corbusier did not devise his Modulor values from his anthropo‐geometry,
but rather from some basic numbers, such as the height of the head, the navel, and the arm raised
above the head. He then took these numbers and either squared them, or produced a golden mean
from them. In short, his values would actually be identical to the numbers of our geometry.
Geometry and Metrics of the Great Circle
Another significant problem with the Vitruvian Man is the great circle8. We have been assuming thus far
that it is simply there because Vitruvius said so. Nowhere in did the geometric construct say to us :
“Hereupon thou shalt place thy circle, and it shall be ye radius.” (Point in fact the construct says that wu
6 National Center for Social Research. Health Survey for England 2008. United Kingdom : National Center for Social Research. 2009. 7 National Center for Health Statistics. Anthropometric Reference Data for Children and Adults : United States 2003 – 2006, Number 10, October 22, 2008. Hayattsville, Maryland : National Center for Health Statics. 2008. 8 The term “great circle” will be used to refer to the circle created from the arms raised above the head and whose center is the navel. The same will apply for the “great square”, in which the width is created from the horizontal span of the arms (negating the vertical height) and the vertical position of the feet (negating the horizontal position). It is these specific geometries and their specific relation to the human body that is important, for, as we will see in the discussion on dynamic anthropometrics, that the great square and circle will change sizes, but still maintain a certain relation to the human body.
will be th
geometry
much is t
has alread
construct
circumfer
upraised.
these que
nature de
kingdom.
The term
which wil
will refer
ground in
from the
accordanc
In order t
Vitruvius,
Agrippa, C
these ind
originally
First let u
Occult Ph
geometrie
drawing s
that Di Vi
are alchem
with som
geometrie
Plate 1
9 Agrippa,
he radius). B
y? Where doe
hat extra len
dy been acco
(that is, so
rence is tange
In this sense
estions we wi
esigns in dyn
“static huma
l satisfy vertic
to the static
n order to cre
static positi
ce with the gr
to further ad
Da Vinci, an
Cesare Cesar
dividuals we
assumed (we
us look at the
hilosophy Ag
es.9 It seems
style does not
nci’s Vitruvia
mal symbols
e science be
es. Here are t
Heinrich Corne
But where do
es the extra w
gth? These m
ounted for. Ha
to say, the
ential to the
e the circle h
ill demonstra
namics. The f
an” would be
cal anthropom
c human as a
eate the squa
ion, or the l
reat circle. Th
ddress this p
nd Le Corbusi
iano, and Alb
will find tha
e will reveal D
e former, the
rippa produc
easy for us to
t look propor
n Man is only
inscribed abo
hind it. But a
the six plates
elius. De Occul
oes it come
width on eith
may seem like
asn’t it? We e
“center of
bottom of t
has been acco
te that the g
fact that natu
e assumed to
metrics, but n
a person with
are. Therefor
egs are swu
his may alread
roblem it is
er had to say
brecht Dürer
at there is m
Da Vinci’s secr
e alchemist an
ced six plate
o dismiss his d
rtionate or as
y a sketch). O
out the huma
as we will pr
he produced
Plate 2
ta Philosophia
from? How
her flank of th
e trivial quest
established th
inertia”), an
the feet and
ounted for v
eometries of
ure designs i
o mean a per
not horizonta
h their arms
e any instanc
ng outwards
dy be obvious
important to
y on the mat
. In examinin
more informa
rets as we pro
nd philosoph
es that diag
drawings of t
s well drawn
Or we might w
an figure, and
rove Agrippa’
:
Libri Tres. Boo
is it created
he square fro
tions. It may
hat the nave
nd therefore
the tips of t
ertically and
f nature are n
in dynamics
rson standing
al. Although t
held outrigh
ce in which t
s new geome
s with Da Vin
o consider w
tter. Of partic
ng the Vitruv
ation embed
ogress).
her Cornelius
grammatically
he human ge
at Di Vinci’s s
want to dismi
d we know al
’s diagrams a
ok II, Chapter X
d from the re
om the circle
be argued th
l is the cente
we can dra
the fingers w
not horizont
not designed
is rather crit
g with their a
his is comple
ht and the fe
the arms are
etries are cr
ci’s sketch.
hat other int
cular importa
ian Canon of
dded in Di V
Agrippa. In h
y describe th
eometries bec
sketch (and it
iss these for t
lchemy to be
are in accord
Plate 3
XXVII. 1533.
est of the h
e come from?
hat the great
er of the geom
aw a circle w
when the arm
tally. In answ
in statics, bu
tical to the a
arms at their
tely legitimat
eet are flat o
raised or low
reated that a
tellectuals be
ance are Cor
f Proportions
Vinci’s sketch
his Three Boo
he overall h
cause his Med
t must be str
the fact that
a lot of myst
dance with h
uman
? How
circle
metric
whose
ms are
wering
t that
animal
r side,
te, we
on the
wered
are in
esides
nelius
from
than
oks of
uman
dieval
ressed
there
ticism
uman
Plate 4
Plates 1 ‐
that the n
stance the
represent
inscribed
arm span
illustrates
circle, wh
the cente
new featu
the vertic
still the c
navel lies
the same
geometric
Plates 5 a
in accorda
smaller th
rather tha
Let us see
geometry
1521, twe
Vitruvian
that at so
square ca
circle :
‐3 illustrate t
navel is the c
e body is insc
ts the full hei
in square (th
n is equal to
s that when t
ich, in this ill
er of the body
ure : when th
cal centerline
enter of the
on the cente
e geometry i
c construct of
and 6 illustrat
ance with ano
han the great
an an actual h
e how Plate 5
y is proposed
elve years bef
Canon of Pr
ome point a h
an be placed
three aspects
center of the
cribed in a cir
ght of the ar
he great squa
the human
he arms are
ustration, the
y. Plate 4 dem
e arms are ra
of the body,
body. There
erline of the
n his sketch
f the human b
te something
other square
square; as w
human geome
5 works first.
d by Cesare C
fore Agrippa’
oportions the
uman can rai
within the g
Plate 5
s of the hum
body when t
rcle (the grea
rms raised ab
are) when th
height; in th
raised to the
e circle can b
monstrates t
aise above th
, the human
fore, Plate 4
legs, i.e. the
of the Vitru
body we have
that seems r
formed by th
well as the pe
etry.
. Agrippa is n
Cesariano in
s De Occulta.
e following a
ise the arms
reat circle, an
an geometry
the arms are
t circle to be
bove the head
e arms are h
his stance the
ir full height
be inscribed in
he same geo
e shoulders a
is still inscrib
illustrates th
legs appear
uvian Man. T
e proposed ab
rather forced,
he arms and l
ntagram. The
not the only o
his Italian tr
. Of the two w
appears to ha
and legs to so
nd the squar
y we already
e raised above
exact). The c
d. Plate 2 illu
held perpend
e groin is th
above the he
nside of a squ
ometric conse
and the feet a
bed in the sam
hat when the
to hinge upo
Therefore al
bove.
, as if the hum
egs as diagon
e latter may s
one to propo
ranslation of
woodcuts Ces
ave some geo
ome specific
re’s corners l
Plate 6
understand.
e the should
circle above t
strates that t
icular to the
he center of
ead the body
uare; in this s
equences as P
are swung ou
me great circ
e legs are sw
on the navel.
l these plate
man body co
nals, which m
seem more m
ose this geom
f Vitruvius’s D
sariano illustr
ometric logic
and respectiv
ie on the circ
Plate 1 illust
er line, and i
the head in P
the human bo
body, so tha
the body. Pl
y fits into the
stance the na
Plate 1, but w
utwards away
le and the na
wung outward
Di Vinci illust
es logically f
uld not actua
means this squ
mystically cont
metry, as the
De Architectu
rates depictin
c within it, na
ve height so t
cumference o
trates
n this
Plate 1
ody is
at the
late 3
great
avel is
with a
y from
avel is
ds the
trates
it the
ally be
uare is
trived
same
ura in
ng the
amely
that a
of the
Ceasiano’
circle we
therefore
the navel
still be the
Logically,
square in
illustrate
in relation
If the pro
only the g
great circ
Actually,
great squ
generated
applies. T
complete
on the gr
are the sh
orientatio
respective
Let’s see h
and when
generated
s woodcut e
are concerne
, the distance
to the cente
e center of th
if the legs ar
accordance w
how the grea
n to the great
ocess starts w
great square
le is generate
any moment
are no long
d. But certain
This occurs ro
ly upright the
eat circle’s c
houlder’s bre
on) cannot be
ely, if they do
how this wor
n they are r
d.
stablishes th
ed with. When
e from the na
r of the feet.
he body due t
re swung out
with the legs.
at square cha
t circle, and v
with a static h
is generated
ed and square
t in which th
applies. It is
nly there com
oughly when
en this is the
ircumference
adth apart. In
e determined
o not share th
ks. So when t
aised to the
e navel as th
n the legs sw
avel to the tip
So, even tho
to the limbs’
twards within
. This can be d
anges position
ice versa.
human, i.e. a
. But if only
e no longer ap
e arms are r
when the ar
mes a point w
the arms ar
height of the
e because the
n other word
d by points R
he same posit
the arms are
height of th
he center, the
ing outwards
ps of the finge
ough the nave
relationship t
n the great ci
demonstrate
n and size wh
human with
the arms are
pplies.
aised more t
rms are at he
hen the arms
re raised alm
e crest of the
e crest of a c
s, the tangen
R and L, whic
ion on the cir
perpendicula
he head the
erefore the c
s they always
ers will alway
el will not be
to the circle (
rcle, then the
d through a s
hen the arms
its arms rais
e raised to th
than exactly
ead height or
s are raised s
most straight
e great circle.
circle cannot
ntial crest of c
ch could be d
rcumference
ar to the body
square no a
circle present
lay within th
ys be equal to
the center of
(i.e. their equ
e arms can b
series of itera
and legs are
sed perpendi
he height of t
perpendicula
r higher that
o high that th
up. Once th
Here the fin
lie in two dif
circle (assum
denoted by r
of the circle.
y the great sq
applies and t
t here is the
e great circle
o the distance
f the square,
ual length / ra
be raised to fo
ative diagram
e raised or low
icular to the
the head, the
ar to the bod
the great cir
he circle no l
e arms are r
nger tips cann
fferent point
ing some arb
right and left
quare is gener
the great cir
great
e, and,
e from
it will
adius).
orm a
s that
wered
body,
en the
dy the
rcle is
onger
raised
not be
s that
bitrary
arms
rated,
rcle is
If the arm
regenerat
the huma
new squa
If the arm
the point
the diago
with abso
sketch is
legs is vie
here to re
motion as
ms remain at h
ted. This is pr
an with the ar
re is smaller t
ms and legs ar
in which all f
nals of the sq
olute certainty
only the resu
ewed in reve
ealize that na
s well at a sta
head‐height a
recisely what
rms raised to
than the grea
re raised a lit
for corners o
quare are equ
y that the pe
ult of dynami
rse then the
ture does des
tic stance.
and the legs a
Da Vinci is de
head‐height
at square of t
ttle bit more
of the square
ual to the dia
erceived error
c human geo
great square
sign in dynam
are swung ou
epicting in his
and the legs
he static hum
the square w
will lay on th
ameter of the
r between th
ometries. If th
e naturally an
mics, and that
utwards to a c
s sketch : the
swung out. I
man.
will become e
he circumfere
e circle. It can
e great squa
his iterative p
nd logically f
t nature wou
certain point
static human
It is clear in t
even smaller.
ence of the g
n therefore b
re and great
process of rai
falls into plac
ld account fo
then the squ
n superimpos
his stance tha
There then c
great circle, so
e established
circle in Da V
ising the arm
ce. It is quite
or its geometr
uare is
ed on
at the
comes
o that
d here
Vinci’s
ms and
clear
ries in
It is also
Additiona
Dürer’s ill
Al
Thus the
held perp
center of
the cente
the great
can now b
not come
In order t
not if the
geometrie
head to t
circle. So
10 Dürer, A
clear that Ag
ally this is co
ustration is W
lbrecht Dürer’s
square is at i
pendicular to
the body. Lik
er of the squa
circle and sq
be understoo
from a 0.6%
to avoid conf
ey are natur
es of the hum
he great squ
how can this
Albrect. Vier Bü
grippa’s huma
rroborated b
William Blake’
s “Proportion o
its maximum
the body an
kewise, the sq
are shares th
uare did not
od the reason
error by Le C
fusion it mus
al or comfor
man body on
are is not eq
geometry be
ücher von Mens
an geometry
by Albrecht D
’s watercolor
of Man”
size when th
d the legs ar
quare is at its
e same posit
overlap in th
n for this ove
Corbusier).
t be noted h
rtable to per
ne thing that
qual to the di
e possible? W
schlicher Propo
shown in Pla
Dürer’s interp
Glad Day (17
he human ass
e not swung
minimum siz
tion as the ce
e first place t
rlap, as natu
ere that thes
rform is ano
may be not
stance from
Would not the
ortion. Nüremb
ate 5 and Ce
pretation of t
794).
William Bl
sumes the sta
outwards; as
ze when the a
enter of the c
that this prob
re designs in
se geometrie
other matter.
iced is that t
the humeral
e arm arc out
berg. 1528.
esariano’s wo
the Vitruvian
lake’s “Glad Da
atic stance, w
s well as whe
arms are legs
circle (the na
blem wouldn’
motion (and
es are possibl
. If we are t
the distance
head to the
tside of the gr
oodcut are co
Man.10 Simi
ay”
when the arm
en the groin
s are raised so
vel). It seems
’t exist at all.
d that motion
le, but wheth
trying to ma
from the hu
crest of the
reat circle at
orrect.
lar to
ms are
is the
o that
s as if
But it
n does
her or
p the
meral
great
some
point? Alt
the shoul
head of th
is higher,
then a hu
Primates
This imag
depiction
represent
the heigh
with arms
referred t
and there
orthogona
position i
Saint Pete
fashion th
much the
Cr
Ra
Before m
nature of
establishe
fit into th
from the
with what
fails to do
sketch : n
though this is
der is not a f
he humerus h
and when th
uman (and all
would certain
ge of the ar
s of the Vitr
tation during
ht of the hea
s at head‐heig
to as a “salti
efore the legs
ally; or the b
n which the
er, Jesus, Sain
he Vitruvian
same canon,
rucifixion
affaello Sanzio
oving forwar
f the extra
ed that the ci
e human geo
asymptotic g
t we have be
o so. Here we
notice that D
s true, we mu
fixed point th
has flexibility
e arm is dow
l primates) w
nly have had
m upraised
uvian Man, t
the Middle A
d. The Talism
ght. Saint And
re”. Depictio
s are parted i
beams of the
square is ful
nt Andrew, O
Canon. Imag
, albeit not al
o (1503 CE)
ds and analy
lengths flank
rcle is the res
ometry and ca
rid, or a varia
en working w
e will look at a
Da Vinci plac
ust stress tha
hat the arm s
in its positio
wn the head o
would not be
a hard time c
above the s
the same sta
Ages and the
man of Orphe
drew was cru
ns of his cru
in accordance
cross do inte
ly inscribed w
Orpheus / Bac
ges and statu
l these cultur
Talisman
3rd Centu
zing Agrippa’
king outside
sult of dynam
an be account
ation from it)
with so far the
another little
ces lines at s
at nature des
swings around
n. So when th
of the humeru
able to lift th
climbing arou
houlder line
ance is in eve
Proto‐Renais
eus from the
cified on crux
cifixions assu
e with the an
ersect orthog
within the gr
cchus, Mithra
ues of Hindu
res were fami
n of Orpheus
ury CE
’s human and
of the body
mic anthropom
ted for in som
. If these dim
en there wou
key to the h
several key b
signs in motio
d about. Bec
he arm is upr
us is lower. If
he arms abov
nd in the tree
is really no
ery crucifixio
ssance depicts
e 3rd Century
x decussata, o
ume either th
ngle of the cr
gonally (at a 9
reat circle. Bu
a, Krishna, et
deities with
iliar with Vitr
C
1
d the pentacl
y created by
metrics. But t
me manner by
mensions of th
uld be no prob
uman geome
bodily mome
on and not st
ause of the s
raised the he
f the shoulder
ve the head (
es.
othing new.
on ever seen.
s Christ with
BCE depicts
or X‐shaped c
he arms raise
ross, which w
90° angle) an
ut whether th
cetera, they
multiple arm
uvius.
Crucifixion of Sa
14th Century CE
le we need to
y the circle.
the extra leng
y our proport
he circle could
blem, but ou
etry that Da V
ents, and no
tatics. The he
shoulder blad
ad of the sho
r was a fixed
at least no e
Aside from
. The most t
the arms rais
Orpheus cru
cross, but com
ed to head‐h
would not inte
nd he assume
he crucifixion
all depict in
ms and legs d
aint Andrew
E
o fully addres
It has now
gth must som
tioning system
d be accounte
r proposed sy
Vinci gives us
tice the dist
ead of
de the
oulder
hinge
asily).
literal
ypical
sed to
ucified
mmon
eight,
ersect
es the
n is of
some
depict
ss the
been
mehow
m (i.e.
ed for
ystem
in his
ances
between
Da Vinci is
divide the
Da Vinci i
height), a
seems rig
noticed e
then agai
geometric
great circ
be illustra
the great
geometry
Breadt
Is this nec
their hypo
and respe
distance b
them. Althou
s illustrating i
e human in th
llustrates tha
and therefore
ght on par, it
arlier in the a
n his drawing
c construct is
le is half the
ated that leng
circle and th
y (that is of th
th of Shoulders
cessarily so?
otenuses hor
ectively, then
between the
ugh these line
is the proport
he square into
at this extra w
e its dimensi
is actually in
anthropomet
g is only sketc
revisited we
breadth of th
gth x is half t
hat the two a
e square insc
s = Extra Length
Indeed it is.
izontally divid
the vertical d
two the corre
es do not alw
tions of squa
o fourths and
width created
on is half th
error by 5.6%
tric grid that
ch and is mea
discover tha
he shoulders
he breadth o
are equal. As
cribing the gre
h of Circle
If two simila
de the great
distance betw
esponding 45
ays align with
res that prop
eighths.
d by great cir
e breadth of
% of being on
some of the
ant to explore
t Da Vinci is s
in the geome
of the shoulde
s a conseque
eat circle) is 4
Leng
r 45 ‐ 45 righ
square and t
ween the two
5° vertices. Th
h the anthrop
portionally en
rcle is one eig
f the shoulde
ne eighth of t
lines do not
e an idea, not
somewhat co
etric construc
ers, as well a
nce of this g
4ϕ2.
gth AB = x
ht triangles a
he square ins
o hypotenuses
his is true for
pometric‐asy
large from th
ghth of the a
er. Although
he arm span.
align with D
t to finalize it
orrect : the ex
ct proposed e
s the extra le
geometry, the
are constructe
scribing the g
s will be equa
r all similar sq
mptotic grid,
he head in ord
rms span (or
in his sketc
. It may have
a Vinci’s line
t. But if the o
xtra breadth o
earlier. Here
ength produc
e final area o
ed so that ea
great circle eq
al to the horiz
quares that sh
what
der to
r body
h this
been
s, but
overall
of the
it can
ced by
of this
ach of
qually
zontal
hare a
common
and proof
It can the
groin, and
motion. A
in size. Co
finally the
asymptot
So as the
centers u
point that
Agrippa :
degree th
featured
Technical
and the u
The quest
circle, a c
smaller ci
To consid
are const
pentagram
the penta
accordanc
must lay o
placemen
edge, and up
f is rather spe
en be establis
d the conseq
As the arms ri
onsequently
e two centers
ic grid can ac
arms raise hi
nite at the n
t the circle’s
that if the a
he human w
in Robert Flu
History of th
niverse as a m
tion at hand i
circle whose
rcle with the
er if this geo
tants. First, t
m (point o). S
agram, and t
ce with the g
on the circum
nt of the pent
pon that com
ecial.
shed here the
quences of th
se above the
as this happ
s share the sa
count for this
igher and the
navel. Can the
center lays u
arms are low
ill rest upon
udd’s alchem
he Two World
macrocosm o
Des
is : does this
center is at
groin as the c
ometry is eve
he crown of
Second, the l
herefore a co
geometries es
mference of th
agram will ap
mon edge a c
e relationship
he great squ
shoulder line
pens the cent
ame position.
s extra length
e legs swing m
e same be tr
upon the cen
ered to a ce
a pentagram
al and occult
ds, published
f man.
stiny by Robert
really work?
the navel. H
center?
n correct it m
the head is
egs can be sw
onstant. In o
stablished abo
he great circle
ppear as such
common mid
p between th
are in relatio
e and the legs
ter of the sq
We will acco
h created by t
more outward
rue for the c
nter of the g
rtain degree
m, with the
t philosophic
in 1617, dep
t Fludd (1617)
As the legs sw
ow, then, ca
must first be
constant, as
wung outwar
order for two
ove, these tw
e (points m a
:
dpoint. But in
e two bodily
on to the gre
s are swung o
uare approa
ount more ful
the great circ
d the square
ircle? Can th
roin? This is
and the legs
groin as the
cal treatise Th
picting man a
wing outward
an the legs sw
established w
s this is the v
rds in order t
o lower vertic
wo vertices (i.
and n). Given
the human g
y centers, i.e.
eat circle wh
outwards the
ches the cen
lly on how th
le shortly.
diminishes in
he circle dimi
what is depi
s swung outw
center. A s
The Metaphys
s a microcos
ds they alway
wing outwar
which points
vertex of the
to create two
ces of the pe
.e. the placem
these three p
geometry this
the navel an
hen the body
square dimin
nter of circle,
e anthropom
n size until th
nish in size t
icted in Plate
wards to a ce
imilar depict
sical, Physical
m of the univ
ys fall on the
d and lie wit
on the penta
e top point o
o lower vertic
entagram to
ment of each
points the siz
s case
nd the
y is in
nishes
, until
metric‐
e two
to the
e 6 by
ertain
ion is
l, and
verse,
great
thin a
agram
of the
ces of
be in
foot)
ze and
As can be
the great
x) of the p
in which t
0.087% o
remembe
this neglig
with the g
Corbusier
nature of
further, p
Although
pentagram
because t
the cente
the groin
without e
1/64 of th
in that cas
Of all geo
pentagram
considere
pentagram
pentagram
the exact
e seen, the tr
circle. But so
pentagram se
the hand raise
of the great
er that the sh
gible error. W
geometry of
r did and crea
f the flexibili
ossibly in a fu
this is possib
m is nonethel
the circle insc
r of the body
(point B). Of
error in the s
he human hei
se the human
ometries the
m and see a h
ed a humani
m have any
m is a corolla
same propor
rough of the
omething app
eems to align
ed to the heig
circle’s radiu
houlder is a fl
We can there
the pentagra
ate excuses f
ty of the sh
uture paper.
ble, Agrippa’s
less wrong fo
cribing the pe
y in this stanc
f course we
tudy or lab o
ght lower tha
n would not t
The H
e pentagram
head, arms, le
st symbol, a
further impli
ry to the crea
rtioning syste
circle that ins
pears to work
n at the inters
ght of the he
us. Although
lexible mome
efore establis
am. Although
for this mild
oulder blade
s and Fludd’s
or one reason
entagram exc
ce is actually a
can imagine
of a 16th Cent
an the groin i
ake on the fo
Human Geom
is the only o
egs, arm pits,
an esoteric g
cations towa
ation of the h
m as the asym
scribes the p
k in this insta
section of the
ad. Though t
there is a v
ent in the bo
h that it is p
h this is a neg
error. There
e with human
depictions o
: the groin is
eeds the bou
about 1/64 o
that these g
tury alchemis
s easily corre
orm of a pent
metries and th
one that is a
, a groin, and
geometry th
ards the hum
human geome
mptotic‐anth
pentagram lie
ance, as the le
e great squar
his may seem
ery small err
dy, so the ar
possible for th
gligible error
could be som
n geometries
of the Vitruvia
s not the cent
undaries of th
f the body’s h
geometries w
st. Of course
ected by not e
agram.
he Pentagram
anthropomor
a torso. Mys
hat represent
man geometry
etric construc
ropometric g
es outside the
eft and right
re and circle (
m trivial, it is a
ror in this g
rms can easily
he human to
we do not w
mething mor
s, and we w
an Man in ac
ter of the bod
he great squa
height lower
would appear
e, the center
extending the
m
rphic. We ca
stically the pe
ts the huma
y? Indeed it
ct described a
grid.
e circumferen
vertices (i.e.
(point z), the
actually in err
eometry we
y adjust to co
o be in accord
wish to do wh
re profound t
will investigate
ccordance wit
dy (point A). T
re. The posit
than the cen
to be perfec
of the body
e legs outward
n easily look
entagram has
an. But could
does. In fac
above, as it fo
nce of
point
point
ror by
must
orrect
dance
hat Le
to the
e this
th the
This is
ion of
nter of
ct and
being
d. But
k at a
s been
d the
t, the
ollows
It is alread
A and C a
therefore
that line A
In looking
means in
golden m
on two ax
It appears
two adjac
where are
to be a b
possibility
but it mus
If we take
line br is
Then line
aefd from
of the pen
dy well know
are segmente
line AC = ϕ.
AC = ϕ –φ and
g at this geo
the same m
ean ACD, so t
xes into golde
s that the me
cent squares
e the other v
bit of a riddle
y of the two
st be noted th
e a pentagram
drawn so tha
ab is square
m square abcd
ntagram, so t
wn that the pe
ed into a gold
. Likewise lin
d line CD = φ,
metry some
manner descri
that squares A
en means. Sou
eans to create
from a golde
ertices? Whe
e, but the ot
adjacent squ
hat there are
m with a point
at br is perpe
ed in order to
d. In the cons
hat points e,
entagram has
den mean by
e AD is segm
, and therefo
things can r
ibed above.
ACFE and BD
und familiar?
e the human
en mean pro
ere is the oth
ther vertices
ares and the
a few other p
t a and a poin
ndicular to li
o create a sq
struction of g
f, and q all la
s golden mea
y point B, so
mented into a
re line AD = ϕ
eadily be est
In this respe
HG will be eq
geometry (th
oblem) is inhe
er square, so
are on the p
golden mea
possibilities.
nt r as two ne
ne ab, so tha
quare abcd (F
golden rectan
y on the sam
n properties.
that line AB
a golden mea
ϕ.
tablished : li
ct the square
qual. In this in
hat is our pro
erently embe
o that this is t
pentagram. W
n and their r
eighboring ve
at points b, c,
Figure I). Nex
ngle aefd the
me line (Figure
. For instance
B = ϕ – φ and
an by point C
nes ABC and
e of line ABC
nstance squar
oposal to corr
edded into th
two adjacent
We will now
relationships
ertices on the
, and r all lay
xt construct a
line ef is alig
e II).
e, the line seg
d line BC = φ
C, so consequ
d ACD are g
C will produc
re ACFE is bis
rect Le Corbu
he pentagram
squares? It s
demonstrate
to the penta
e pentagram,
y on the same
a golden rect
gned with ver
gment
φ, and
uently
golden
ce the
sected
usier’s
m. But
seems
e one
gram,
and a
e line.
tangle
rtex q
Fi
Following
aegi, whi
lines bh a
rectangle
points j, k
the penta
Fi
Finally the
Therefore
of forming
as was es
with the a
Fi
igure I
g the process
ch is segmen
and df are e
ajki from sq
k, and s all lay
agram (Figure
igure III
e original squ
e squares aeg
g the two squ
tablished abo
anthropomet
igure V
established e
ted into two
equal (Figure
uare aegi, so
y on the sam
e IV). Therefor
uare is double
gi = bjkh = elm
uares from th
ove. This is a
ric‐asymptoti
earlier, squar
golden mean
e III). Still fol
o that line jk
e line. It can
re squares ae
ed, that is sq
mg (Figure V)
he golden me
critical meas
ic grid.
Figur
re the golden
ns on two pe
lowing the e
are aligned w
therefore be
egi and bjkh a
Figur
uare line eg,
). We have ch
an for a reas
urement in o
Figur
re II
n rectangle ae
erpendicular a
exact same p
with the vert
e established
are equal.
re IV
which lies u
hosen to dem
on : if length
order for the
re VI
efd in order t
axes by lines
process, then
tex s on the p
that line ks i
pon vertex l
monstrate thi
al is ϕ ‐ φ, th
pentagram to
to create a s
bh and df, in
n render a g
pentagram, in
s the centerl
of the penta
s particular m
hen length au
o be in accord
quare
n that
golden
n that
ine of
gram.
means
u is ϕ,
dance
In Figure
is ϕ. Ther
moved to
pentagram
Fi
Although
does not
pentagram
pentagram
originally
minimum
that lengt
Therefore
If a third
following
adjacent
equal to ϕ
(Figure XI
and D are
to pentag
divided by
VII if the leng
efore pentag
the other sid
ms Y and Z ar
igure VII
this is techn
do, this is a
m was create
m that either
pentagram
number of p
th al is equal
e it can be est
Figure VIII
pentagram
illustrates th
squares geom
ϕ – φ, and p
), then the in
e proportiona
grams B, C, a
y ϕ). One can
gth of one line
ram X is crea
de of pentagr
re equal.
nically disasse
actually a fo
ed and then
r is enlarged
are used to
points necess
to length rt,
tablished that
is added tha
hat fractalizin
metry (Figure
pentagram C e
itial two‐squa
ately similar t
and D. This n
n easily see tw
e on pentagra
ted from pen
am X as pent
embling and
rm of fracta
moved, both
by the golden
construct t
sary for proof
and that the
t the followin
at is the gol
ng pentagram
e X). Therefor
equals φ. If t
ares‐by‐golde
to pentagram
new two‐squa
wo‐squares BC
am Y is ϕ – φ,
ntagram Y via
tagram Z is in
rearranging t
lizing the pe
are actually
n mean or di
he new pen
f. In the case
two line seg
ng Figures VIII
Figure IX
den mean o
ms by the gol
re in Figure X
this process c
en‐mean no lo
ms A, B, and C
ares BCD is d
CD’s relations
, then the len
the golden m
relation to p
the geometr
entagram. Alb
y logical and
iminished. In
ntagram, and
e of pentagra
ments are pa
I and IX of pe
of one of the
lden mean a
X if pentagra
continues and
onger apply.
C then a smal
diminished fr
ship to two‐s
ngth of one lin
mean. So if pe
pentagram X,
ry, something
beit it seems
legitimate m
both cases t
d three point
m Z it can be
arallel in the p
ntagrams pai
e other pent
re in accorda
m A is ϕ, the
d a fourth pe
But, since the
ler two‐squa
rom two‐squa
quares ABC a
ne of pentagr
entagram Y is
so that
g we argue n
s that the sm
eans of creat
three points o
ts is typicall
e seen in Figu
parallelogram
irs are true :
tagrams, the
ance with the
en pentagram
entagram is a
e pentagrams
res can be ap
ares ABC by
as a fractal.
ram X
s
nature
maller
ting a
of the
y the
ure VI
m altr.
n the
e two
m B is
added
s B, C,
pplied
φ (or
Fi
If a fifth p
asymptot
Fi
Since it c
proportio
pentagram
(Figure X
numerous
For instan
square th
width cre
lays upon
pentagram
length of
finger tip
vertical ce
igure X
pentagram E
ic grid create
igure XII
can now be
ns construed
ms are relate
XIV). In lookin
s anthropome
nce, if two ne
hat created th
ated by the g
the vertical c
m A, then the
a line segme
s (line ab). P
enterline of t
is added (Fig
d earlier (Fig
established t
d in the hum
ed to the hu
ng at how t
etrics can be
eighboring ve
his geometry
great circle! (
centerline of
e centerline o
ent from pent
Pentagram C
the whole ge
ure XII) then
ure XIII).
that the pro
man body, at
uman geome
the pentagra
corroborated
ertices a and
), then vertex
(Finally!) Like
the whole ge
of pentagram
tagram B is e
is the golde
eometry, and
Figur
a pattern em
Figur
portions of a
this point it
etry, particul
ms relate to
d, and some n
d b of pentag
x c of the pe
ewise, the cor
eometric cons
B is aligned
equal to the l
en mean of p
d therefore th
re XI
merges, name
re XIII
a pentagram
t will be inte
arly on the
o the geome
new ones can
gram A lay on
entagram is v
rresponding v
struct. If pent
with the sho
length of one
pentagram B,
he vertical li
ely that this p
m are a direc
eresting to se
anthropo‐ge
etric construc
n be accounte
n the initial s
vertically align
vertex of c is
tagram B is th
ulder line. Fu
e arm from th
, and its cen
ne of symme
process echoe
ct corollary t
ee how thes
eometric con
ct in this m
ed for.
square (that
ned with the
s d, and verte
he golden me
urthermore th
he shoulder t
nterline is als
etry for the w
es the
o the
e five
struct
anner
is the
extra
ex d is
ean of
he full
to the
so the
whole
body. The
shoulders
also aligne
Thus far
meaningle
somethin
then the a
Fi
Thus in co
squares g
square ge
approach
the fracta
other wor
e full length
s. And so cen
ed with the s
is seems tha
ess and just
g on the bod
angle of axc i
igure XV
omparing the
eometry in Fi
eometries are
ing point x is
alizing pentag
rds they both
of one of t
terline of pe
houlder.
Figure XI
t the point i
sits somewhe
dy? Indeed it
s 36°, which i
angle of the
igure VIII, it c
e approaching
54° (half the
rams (36°) is
h approach po
the line segm
ntagram E (t
IV
n which thes
ere randomly
does. If the f
is also the int
approaching
an be establi
g is also point
interior angle
complement
oint x at angle
ments of pen
he golden me
se diminishin
y on the con
fractlizing pen
erior angle of
Figur
pentagrams
shed that the
x. The angle
e of a pentag
tary to the an
es that add up
ntagram C is
ean of the go
ng pentagram
struct. Or do
ntagrams con
f any pinnacle
re XVI
towards poin
e point towar
in which the
gon, or half of
gle of the fra
p to 90° (Figu
s equal to th
olden mean o
ms approach
oes length ax
ntinue ad infi
e on a pentag
nt x to their re
rds which the
fractalizing t
f 108°). There
actalizing two
ure XVII).
he breadth o
of pentagram
(point x) is r
x actually me
initum (Figure
gram (Figure
espective two
diminishing t
wo‐squares a
efore the angl
‐squares (54°
of the
m C) is
rather
easure
e XV),
XVI).
o‐
two‐
are
le of
°). In
Placing t
anthropo
Figure XIV
human he
of creatin
vertical m
So it seem
a half. It c
new mean
the cours
body and
Figure X
the fractalizi
metrics, but
V is also the
eight (from h
ng the anthro
measurements
ms we have co
can be easily s
ns of admirin
e of this essa
anthropome
XVII
ing pentagra
it does not a
height of th
ead to toes),
opometric‐asy
s. And in Figu
Figure XVIII
ome full‐circle
seen the beau
g the proport
y we have int
trics (both st
ams as the
account for ve
he human (or
, and the frac
ymptotic grid
ure XVIII they
e. Actually we
uty and wond
tions, geomet
tensely been
atic and dyna
y are in F
ertical measu
r the arm sp
ctalizing pent
d, then the p
do so indeed
e have come
der in nature’
tries, and me
studying vari
amic). A final
Figure XIV o
urements. Co
an). Since th
tagrams are i
entagrams sh
.
more than fu
’s designs. It s
easurements
ious geometr
aspect left to
only accoun
onsequently t
he length of a
in accordance
hould also be
ull‐circle; mor
seems as tho
of our own b
ries in relation
o admire is th
ts for horiz
the length of
ax is equal t
e with the pr
e able to est
re like a circle
ugh we have
odies. Throug
n to the huma
e numerics o
zontal
ax in
to the
rocess
ablish
e and
a
ghout
an
of
human pr
two‐squa
Fi
If this patt
H
φ
ϕ
φ
ϕ
ϕ
ϕ
ϕ
2ϕ
2ϕ
3ϕ
3ϕ
5ϕ
5ϕ
8ϕ
8ϕ
13
13
2
2
…
…
roportions in
res‐by‐golden
igure XIX
tern continue
orizontal Gro
φ
ϕ ‐ φ
φ
ϕ ‐ φ
ϕ ‐ φ
ϕ
ϕ
ϕ ‐ φ
ϕ – φ
ϕ ‐ φ
ϕ – φ
ϕ ‐ 2φ
ϕ ‐ 2φ
ϕ ‐ 3φ
ϕ ‐ 3φ
3ϕ ‐ 5φ
3ϕ ‐ 5φ
1ϕ ‐ 8φ
1ϕ ‐ 8φ
…
… et cetera
accordance w
n‐mean geom
es then a rath
owth :
with the fract
metries the ra
her interestin
alizing two‐sq
te at which th
g curiosity oc
Vert
ϕ ‐ φ
ϕ
2ϕ ‐
3ϕ ‐
5ϕ ‐
8ϕ ‐
13ϕ
21ϕ
34ϕ
quare geome
hey increase
ccurs :
ical Growth :
φ
φ
φ
2φ
3φ
‐ 5φ
‐ 8φ
‐ 13φ
etry. Starting w
can be seen i
with any of th
in Figure XIX.
he
Horizontally the measurements are increasing by ϕ + n (where n is the previous value established by
increasing by ϕ). Vertically they simply increase by ϕ, and therefore diminish by φ, or 1/ϕ. It is rather
curious that in this numeric system the numbers multiplied with either ϕ or φ follows the Fibonacci
series, in which when any of the numbers of the series is divided by the previous number then the
values are ϕ, or 1.618033989... But then again, phi has a very peculiar habit of repeating itself.
We have to admit that much of what has been discussed above has been the result of correcting very
trivial errors. But then again, it goes to show that the age old cliché “little things do matter” has some
justifications.
We all usually regard our bodies as a mere vessel that only exists to get our brains from place to place,
whether it is from one meeting to another, or from the office to home, or from class to the dorms. We
hope this paper illuminates some of the beautiful (and even mystical) qualities of these vessels we
inhabit for the duration of our lives, and even its relationship to simple ideas, like a pentagram or a
square.
So what is next? Next we can try the geometries of the horse…
Patrick M. Dey
Damian ‘Pi’ Lanningham
5 January 2011
The Open Problem Society