1 Solving the Vitruvian Man Patrick M. Dey & Damian ‘Pi’ Lanningham The Problem and the Geometry Solving the Vitruvian Man 1 Problem seems to have eluded mathematicians and geometricians for almost 2000 years. Some probably never knew there was problem. It uses that mysterious, and yet very familiar number phi (ϕ), a number that seems to lurk in the shadows of all mathematics. We will show how the human in the square and circle is created, and how the same geometries can account for all anthropometrics, whether the human is in motion or not. Furthermore, we will illuminate how the same geometric process is related to other geometries, such as the pentagram, and how it is a natural self‐ replicating geometry. Vitruvius describes the geometrical proportions of the human body as: “… In the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of the circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of planes and surfaces which are perfectly square.” De Architectura, Book III, 1: 3 Although Vitruvius does not present the question at hand, the question still exists: how does the human body fit into both a circle and a square? This problem has been searched and worked out by many geometricians, such as Agrippa, Cesariano, Dürer, and Da Vinci. The problem is somewhat of a latent problem in that we can map the proportions and geometries of the human body and discover numerous relationships and proportions. But we must ask then: how is this geometry created in the first place? In looking at Leonard Da Vinci’s Vitruvian Man, the most notable interpretation of Vitruvius’s Canon of Proportions, we can begin to understand some general problems: 1 Throughout history the Vitruvian Man has been a study of the ideal male body. This makes gender neutrality difficult to work into this paper. We would like to apologize for the use of “his” rather than “his/her”. In that respect, the problem of idealization and the feminine body is addressed in this paper on pages 12 ‐ 13.
Newly revised and edited version of the Solving the Vitruvian Man paper uploaded in the Spring of 2011. This explores the the various Vitruvian Man geometries proposed throughout history, and corrects Le Corbusier's flawed geometries that create the Modulor System.
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Solving the Vitruvian Man Patrick M. Dey & Damian ‘Pi’ Lanningham
The Problem and the Geometry
Solving the Vitruvian Man1 Problem seems to have eluded mathematicians and geometricians for almost
2000 years. Some probably never knew there was problem. It uses that mysterious, and yet very familiar
number phi (ϕ), a number that seems to lurk in the shadows of all mathematics. We will show how the
human in the square and circle is created, and how the same geometries can account for all
anthropometrics, whether the human is in motion or not. Furthermore, we will illuminate how the same
geometric process is related to other geometries, such as the pentagram, and how it is a natural self‐
replicating geometry. Vitruvius describes the geometrical proportions of the human body as:
“… In the human body the central point is naturally the navel. For if a man be placed flat on his
back, with his hands and feet extended, and a pair of compasses centered at his navel, the
fingers and toes of his two hands and feet will touch the circumference of the circle described
therefrom. And just as the human body yields a circular outline, so too a square figure may be
found from it. For if we measure the distance from the soles of the feet to the top of the head,
and then apply that measure to the outstretched arms, the breadth will be found to be the same
as the height, as in the case of planes and surfaces which are perfectly square.”
De Architectura, Book III, 1: 3
Although Vitruvius does not present the question at hand, the question still exists: how does the human
body fit into both a circle and a square? This problem has been searched and worked out by many
geometricians, such as Agrippa, Cesariano, Dürer, and Da Vinci. The problem is somewhat of a latent
problem in that we can map the proportions and geometries of the human body and discover numerous
relationships and proportions. But we must ask then: how is this geometry created in the first place?
In looking at Leonard Da Vinci’s Vitruvian Man, the most notable interpretation of Vitruvius’s Canon of
Proportions, we can begin to understand some general problems:
1 Throughout history the Vitruvian Man has been a study of the ideal male body. This makes gender neutrality difficult to work into this paper. We would like to apologize for the use of “his” rather than “his/her”. In that respect, the problem of idealization and the feminine body is addressed in this paper on pages 12 ‐ 13.
2
The first indication that there is a geometrical problem is the fact that the upper vertices of the square
do not meet the boundary of the circle, but rather overlap it. This is no mild error, as the overlap is in
error by roughly 1.5% of the circle’s radius. Considering this error, one of two things must be accepted :
either the human body is not perfect and nature is not a perfect geometrician, or there is a natural and
geometrically perfect means of constructing the Vitruvian Canon. Since the former is antithetical to
humanist ideals of the body being perfect, as well as the belief that nature is the most perfect (and
beautiful) geometrician, we have to argue for the latter. But once again: attempted solutions need
examination.
In order to tackle this problem, a bit of history and other attempts to solve it should be looked at. This
question was considered and attempted by the architect Charles ‘Le Corbusier’ Jeanneret, one of the
fathers of Modernist architecture. From 1943 to 1945 Le Corbusier worked with a young collaborator,
who is only known as Hanning, as well as Mlle Elisa Maillard. Le Corbusier, working with AFNOR on
standardizing products, goods, and buildings, presented a proportioning problem to Hanning:
“Take a man‐with‐arm‐upraised, 2 ∙ 20 m. in height; put him inside two squares 1 ∙ 10 by 1 ∙ 10
meters each, superimposed on each other; put a third square astride these first two squares. This
third square should give you a solution. The place of the right angle should help you decide
where to put this third square.”2
Logically, this process should incorporate and be developed from the golden mean, since this property is
inherent in human proportions. The first proposal given by Hanning (Figure A) was to begin with a
square and then produce golden rectangle a. Next, from the initial square draw root‐two rectangle b on
the opposing side of the square from the golden rectangle. The resulting rectangle is almost two
adjacent squares (Figure B), therefore the angle (ϴ) of the “regulating” lines3 of the overall rectangle is
greater than 90° (Figure C) :
Figure A Figure B Figure C
The product of the root‐two solution rectangle is in error by 1.6%, and, therefore, not a viable solution.
About two years later, meeting with Mlle Elisa Maillard, another solution was proposed. A golden
rectangle is constructed from a square. Next a line is constructed from point a of the golden rectangle to
the opposing mid‐point of the initial square, point b. A 90° is drawn from line ab to intersect with the
base of the geometry at point c (Figure D). The resulting rectangle is closer to two adjacent squares
(Figure E), but still in error by 0.63% and produces “regulating” lines that have an angle (ϴ) greater than
90° (Figure F):
2 Le Corbusier. The Modulor. Trans. Peter de Francia and Anna Bostock. Basel, Switzerland: Birkhäuser Publishers. 2004. P. 37. 3 Regulating lines is a term used by Le Corbusier to denote lines that intersect at 90° angles.
3
Figure D Figure E Figure F
The difference in the errors of the above solutions can be seen here :
Although this system is in error, Le Corbusier used this geometric structure to devise his infamous
Modulor System, which is a system of measurements built on the principle of anthropometrics that
proportionately grow (or diminish) from the initial geometric construct. This was his means to humanize
architecture. We must admit that the geometries are approximated, and at certain times completely
false and probably forced (for instance a tangential line produced from the regulating lines, which is not
tangential to its circle4). But given that these geometries were constructed with T‐squares, triangles,
compasses, and pencils, it seems logical that a geometrical error would be considered an error on the
draftsman’s behalf. More surprisingly, though, is the fact that Le Corbusier did know that this geometry
was flawed! In 1948, a Monsieur R. Taton informs Le Corbusier: “Your two initial squares are not
squares; one of their sides is larger by six thousandths of the other [gross miscalculation].” Le Corbusier
being economical considers: “In everyday practice, six thousandths of a value are what is called a
negligible quantity… it is not seen with the eye.” But Le Corbusier also being a mystic further adds: “… I
suspect that these six thousandths of a value have an infinitely precious importance: the thing is not
open and shut, it is not sealed; there is a chink to let in the air; life is there, awakened by the recurrence
of a fateful equality which is not exactly, not strictly equal… … And that is what creates movement
(sic).”5
We, on the other hand, argue that there is no error in the initial problem and that Le Corbusier just
happens to be dead wrong (which is certainly not the first time he was dead wrong, but 20th Century
paradigms on urbanism do not need to be accounted here). As mathematicians and geometricians we
claim that if there is an error, no matter how trivial and small, it is an error, and therefore incorrect. 4 Ibid., p. 64, Fig. 21. 5 Ibid., p. 234.
4
Furthermore, we argue that the solution has to be elegant. Another look at Le Corbusier’s geometry will
show that it is not elegant. For his geometry to be in accordance with the human body, the portion of
the rectangle created by the regulating lines (established at point c) shown in Figure D has to be
removed from its initial side of the square to be placed next to the golden rectangle (Figure G). This
creates the distance between the top of the head to the tip of the fingers when the arm is raised above
the head (Figure H). The final problem with Le Corbusier’s solution is that it only accounts for the
proportions of vertical human measurements, that is, it only attempts to solve the height. The Modulor
never takes into account horizontal measurements, such the proportions of the arm span to the whole.
This we will account for shortly.
Figure G Figure H (Drawing by Le Corbusier,
from The Modulor)
Looking at Le Corbusier’s Modulor we find that it is not only incorrect, but it is also not elegant. Nature’s
geometries do not disassemble and then reassemble a construction. Humans are the ones, with their
“reason” and “logic”, construct geometries and reassemble them arbitrarily in whatever manner suits
their liking. On the other hand, the geometries of nature are built up from an initial geometry to
construct the whole. Man disassembles and reassembles harmony to create “beauty,” while nature
simply constructs harmony that is beautiful, nature just exists. Now we return to the initial question, but
with more constraints:
1) Construct a geometry that continuously builds upon itself
2) The geometry constructs the human inside both a circle and square in accordance with the
Vitruvian Canon of Proportions
3) The geometry accounts not only for the vertical anthropometrics, but also for the horizontal
measurements and proportions
4) It is devised from the golden mean
5) And, finally, the construction of the geometry has an elegant solution.
In considering this problem, we find that it is justifiable that Le Corbusier would assume that starting
with a square and constructing a golden rectangle that could inevitably produce two perfect squares
through some other form of geometrizing. Also, we assume he must also be right in assuming that the
critical dividing lines created by an overlapping third square must be found in accordance with the
human body, such as the height of the head.
5
When we began approaching this problem, we segmented a square abcd with a line op so that the
segments of line ab are a golden mean (Figure 1). We did this assuming that the golden mean had to be
present with a square, but not necessarily forming a golden rectangle. If the length of segment op that
divides the square abcd into a golden mean is squared6, this naturally creates a golden rectangle bcef
for the whole geometry thus far (Figure 2). Therefore, Figure 1 = Figure 2, and the squares abcd and
opfe are equal. Finally, if the initial square is squared, this naturally creates two perfect and adjacent
squares, as well as giving the position of the third square overlapping the two adjacent squares (Figure
3).
Figure 1 Figure 2 Figure 3
Although this does solve the problem of the two squares created from a golden mean and is rather
elegant, it simply is not elegant enough. It is not elegant enough simply because sacred geometries, and,
therefore, natural geometries, do not typically segment a square with a golden mean without being
created initially from another golden geometry. Additionally, it does not account for horizontal
anthropometrics. So the construction of this geometry must be reconsidered.
The easiest remedy to justifying the golden mean segmentation of the square is to form a square from
two golden rectangles on perpendicular axes. Essentially, starting with square abcd whose edges all
equal to (φ ‐ ϕ) create a golden rectangle cdfe so that line xd equal to line xe, in that point x is the
midpoint of line bc. Then construct another golden rectangle adgh on the perpendicular axis so that line
yd is equal to line yh, in which point y is the midpoint of line ab. We now have a square that is
segmented into two lines, instead of just one, so that the segments are golden means. Therefore create
the square beih. The new square’s area is φ2.
Square Golden Rectangle Two Golden Rectangles Square
Area = φ2 ‐ 2φϕ + ϕ2 Area = φ2 ‐ φϕ Line be = bh = φ Area = φ2
6 In this paper we will use terms in drafting with T‐squares, compasses, and triangles. The term “squared” or “squaring” refers to making a square from a given line segment, and not to be confused with x2, unless stated so.
6
Thus, the same process of creating Figures 1 – 3 can be reiterated as follows. With square beih construct
a golden rectangle ejki by squaring line segment cg, that is, the golden mean of square beih, so that line
cg is equal to line cj. This gives the same result as if the midpoint of line be was rotated, so that line zi is
equal to line zj. As a result squares beih and cjkg are equal. Subsequently, take line ei and square it so
that lines ei and el are equal. The result is two perfect and adjacent squares, in that squares beih, cjki,
and elmi are all equal, this locates the placement of the third square. The newly created square is
equally segmented into a golden mean as the initial square. The overall geometry thus far has an area of
2φ2.
Golden Rectangle Proof of Third Square Two Squares
Area = 2φ2 ‐ φϕ Squares beih = cjki = φ2 Area = 2φ2
Perimeter = 6φ ‐ 2ϕ Perimeter = 6φ
To reiterate the steps and the mathematics:
1) The initial square with an area of φ2 ‐ 2φφ ‐ ϕ2 creates a golden mean.
2) The area of the golden rectangle equals φ2 – φϕ.
3) Squaring off the golden rectangle creates a new square with an area of φ2.
4) From this new square create another golden rectangle with an area of 2φ2 – φϕ.
5) From the square created in step 2 square one of its lengths to produce two equal squares each
with an area of φ2, so that the overall geometry has an area of 2φ2.
6) The rhythmic pattern of the perimeter of this geometry is : φ ‐ ϕ, ϕ, φ ‐ ϕ, ϕ, φ ‐ ϕ, ϕ, φ ‐ ϕ, ϕ… …
7) Thus the two adjacent squares are equally segmented into golden means.
7
To emphasize that this is correct, the following drawings will illustrate the squares and their placements:
Placement of Third Square Creation of Second Square Two Perfect Adjacent Squares
Here it may be argued that we merely established our own geometry, but did not solve Le Corbusier’s
problem as he posed it. If we were to solve the problem as Le Corbusier posed, we would have to
account for the regulating lines and the 90° angle. As he said : “The place of the right angle should help
you decide where to put this third square.” At this point, the placement of the right angle is rather
trivial. If we were to correct Le Corbusier’s problem as he posed it, the results will still be the same, and
follows as such :
First construct golden rectangle aefd. Then square line ef back toward the circle, which segments square
abcd into a golden mean by the line segment xy. But if this construction is supposed to use regulating
lines, then the easiest way to create two squares is to draw two successive 45° angles that are
perpendicular to each other. So if we draw a 45° (ϴ) from point e toward the opposing edge of the
square the line will intersect at point y.
Next draw a line from point y that is perpendicular to line ey so that it creates point g. This creates the
two perfect and adjacent squares, so that squares abcd = xefy = gxyh. This construction not only gives
the vertical anthropometrics, it also can create the horizontal measurements. Lines bc and ey intersect
at point o. If a line pq is drawn to intersect at point o as well, then the golden rectangle befc and the
square abcd are bisected into golden means.
Thus, the above essentially creates a reiteration of the same geometry we have already proposed.
Although this does solve Le Corbusier’s problem as he posed it, it is not necessary to continue using the
regulating lines to express the rest of the creation of the Vitruvian Man.
8
At this point, the very structure of the our proposed geometric construct incorporates the golden mean,
thus not only accurately giving the human height in the square and circle, but also horizontal
measurements that are the result of squaring the golden mean. It seems rather logical at this point to
assume that line jk is the top of the human head, so we can complete the rest of the geometry by
squaring line bj, based on Vitruvius’s description. Assumption is rather inappropriate here. It is better to
understand the logical construction of the human geometry. It can be seen that this process starts with a
square that grows into another square by a golden mean, and this square grows into a golden mean.
Thus it can be concluded that the next step is to repeat and square off the largest golden mean thus far.
This is not a justification but a completely rational next step, since the first constraint established above
states that the geometries must build upon themselves, much like the way in which a nautilus shell
continuously builds upon itself with the same proportional geometry, i.e. a golden mean.
So, if the process continues as before, the largest golden rectangle bjkh is squared establishing point o.
This is done so that lines bj and bo are equal, thus creating the square bjno, being the square within
which the Vitruvian Man rests. Then the whole array of construction lines and geometries are
orthogonally completed to create blpo, a grid of squares and golden rectangles. This is the primary grid
of the human proportions and measurements, and henceforth will be referred to as the “geometric
construct.”
Squaring the Height Completing the Grid
Line bj = bo
From this geometry we find several aspects that are related to Vitruvius’s description of the human
geometries. For instance lengths jn and jb are equal, and therefore a square. Lengths eb, ei, and el are
equal as well. Given these equalities it can therefore be established that line eq segments the construct
equally vertically, and line vw segments the construct equally horizontally. Therefore, point u is the
midpoint and point of intersection of lines eq and vw. Likewise, if a diagonal line bisecting the construct
were drawn, for instance a line bp or lo, then their midpoints and point of intersection would still be
point u. Thus it can be established that point u is the center of the construct, or, in a sense is the center
9
of inertia. According to Vitruvius, it can logically be concluded that this is the location of navel, as well as
the center of the circle that encompasses the human figure.
Vitruvian Canon of Proportions Vitruvian Man
Area = 4φ2 ‐ 2φϕ
Perimeter = 4φ ‐ ϕ
Not only is this solution elegant and builds upon its own geometries, it also accounts for horizontal
anthropometrics. The initial square, golden mean, and the reflected square establish the breadth of the
shoulders. As opposed to Le Corbusier’s geometry, whose defining geometry of the Canon of
Proportions was slightly irregular, here the solution provides a regular grid that can be repeated, i.e.
construct another square from the initial (or previous) golden mean, then square the initial (or previous)
square, then square the previous golden mean, et cetera.
Creating all Anthropometrics
Logically from this basic geometrical construct the rest of the measurements of the human body should
be able to be constructed from the overall geometry. In essence, we have to account for creating, or,
another way of saying it, correcting Le Corbusier’s Modulor System and achieve all the anthropometrics
of the human body. Furthermore, we should be able to account for all body parts (i.e. the hands and
digits) and bodily locations (i.e. location of the elbow) by building off of the same process established
above. Since these proportions can be built outward, i.e. grow upon themselves, likewise they should be
able to be constructed inward, or fractalized7.
Le Corbusier’s Modulor is a proportionately growing (or diminishing) system of measurements that are
derived from anthropometrics that grow or diminish by either squaring (or doubling) the measurement
or produced from the golden mean. Since the solution proposed involves the same process of either
squaring or producing a golden mean following this exact same pattern, our solution can provide the
most accurate Modulor System.
7 Here and throughout the paper the term “fractal” will be used to denote shapes and measurements that proportionally augment or diminish in size, but remain proportional to their similar counterpart in relation.
10
One minor flaw on Le Corbusier’s part is that his geometric construct only accounts for a few primary
positions of body parts. For instance, the position of the third square in the Modulor provides the crown
of the head, and the “common” edge of the two squares provides the position of the belly button. Of
course, this only accounts for their vertical positions and not their horizontal positions. All other
positions on the body are accounted for secondarily by either squaring or producing the golden mean of
various numbers, and therefore are rather arbitrary means of accounting for these positions. (See Figure
H above).
Thus, our solution’s “Modulor system,” so to say, has to be able to provide both horizontal and vertical
positions of body parts. The solution to defining the lesser anthropometrics are just as elegant as
constructing the overall geometric construct, as it follows an identical means of construction. The
solution is as follows:
Since rest of the bodily positions and measurements must be defined by a golden mean, the solution
must start with a square. The square surrounding the groin (henceforth will be referred to as the groin
square) is to be chosen, since it is the smallest square that lies within the human body. If one of the
larger squares, say square abcd is chosen, then the construction process will only repeat the overall
geometric construct, and therefore produce no new results. (Since square abcd creates the groin square
through a golden mean). If the square above the head is chosen (in the region created by the arm raised
above the head) to create golden means, then the geometries will lie outside the static human body,
and therefore is not a prime square to choose. The groin happens to be the center of the body when the
arms are held perpendicular to the body or lower. If the arms are higher than the shoulder line then
they fall into the great circle and the navel becomes the center, thus producing geometries outside the
body. So the groin square appears to be the most logical choice.
Construct golden rectangles on all four sides of the groin square. Then take the edges produced by each
golden rectangle and continue them to the edges of the great square, therefore creating a grid. The
measurements produced account for the vertical height of the knees and the center of the breasts (i.e.
the nipples in males, but center of the breasts in adult females), as well as the horizontal position of the
elbow.
11
The next smallest square – that is, one of the smaller squares produced by squaring the golden means of
the groin square – is then used to produce four golden means on each side of the square. This is done in
just the same manner as in the previous step illustrated above. The measurements produced here
account for the vertical height of the groin (i.e. the height of the pubic triangle and the fulcrum point of
the femur) and the position of the collar bone, as well as the horizontal position in which the eyes are
offset from the nose (i.e. the outer edge of the eye).
Again, the next smallest square then produces four golden means. The measurements created accounts
for the vertical position of the mouth and the trough of the breastplate’s center (i.e. bottom of the
sternum), and the horizontal position of the wrist.
The same construction repeated for the next smallest square accounts for the vertical height of the
centerline of the arms and the height of the eyes, as well as the horizontal position of the center of the
palm. Doing this once more will account for the horizontal location of the moment where the fingers
meet the palm.
12
This is, of course, an iterative construction with no end. Logically, because of the continuous squaring of
the golden means the asymptote approaches the opposing vertex of the square. And, since it is an
iterative process, any square on any side of any square can be chosen to create another bodily
measurement or position. Speculatively, though most probably, this asymptotic grid of anthropometrics
accounts for every human measurement from the length of the small intestines to the thickness of the
earlobe. The asymptotic grid formed here echoes a similar grid Le Corbusier constructed to devise both
vertical and horizontal Modulor measurements of the human. 8 Although, his is ad hoc and is trying to
force the Modular to account for horizontal bodily measurements, and it is not constructed from the
overall construct.
Now, Le Corbusier with his Modulor System, and even Albrecht Dürer before him, devised numerical
measurements for the human body for application in the visual arts, furniture design, architecture,
product design, and anything with ergonomic applications. So as a final note to Le Corbusier’s Modulor,
8 Ibid., p. 85 ‐ 87, Figures 33 – 35.
13
we do not wish to correct his Modulor dimensions. 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 used the six foot tall man based on
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.9 In the U.S. the average Caucasian man is 5 foot 9 ½ inches tall.10 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” opened when 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
provide a “most accurate” Modulor. Really, we are just providing the most accurate means for creating
the Modulor. There really is nothing wrong with the values of the 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 or produced a golden mean
from them. In short, his original 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 circle11. 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
9 National Center for Social Research. Health Survey for England 2008. United Kingdom: National Center for Social Research. 2009. 10 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. 11 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, relatively speaking, their specific relations to the human body.
14
will be the radius). But where does it come from? How is it created from the rest of the human
geometry? Where does the extra width on either flank of the square from the circle originate? How
much is that extra length? These may seem like trivial questions. It may be argued that the great circle
has already been accounted for. We established that the navel is the center of the geometric construct
(that is, so to say, the “center of inertia”), and therefore we can draw a circle whose circumference is
tangential to the bottom of the feet and the tips of the fingers when the arms are upraised. In this
sense, the circle has been accounted for vertically and not horizontally. In answering these questions,
we will demonstrate that the geometries of nature are not designed in statics, but that nature designs in
dynamics. The fact that nature designs in dynamics is critical to the animal kingdom.
The term “static human” would be assumed to mean a person standing with their arms at their side,
which will satisfy vertical anthropometrics, but not horizontal. Although this is completely legitimate, we
will refer to the static human as a person with his arms held outright and his feet are flat on the ground
in order to create the square. Therefore, any instance in which the arms are raised or lowered from the
static position, or the legs are swung outward, new geometries are created that are in accordance with
the great circle. This may already be obvious with Da Vinci’s sketch.
In order to further address this problem, it is important to consider what other intellectuals besides
Vitruvius, Da Vinci, and Le Corbusier had to say on the matter. Of particular importance are Cornelius
Agrippa, Cesare Cesariano, and Albrecht Dürer. In examining the Vitruvian Canon of Proportions from
these individuals we will find that there is more information embedded in Di Vinci’s sketch than
originally assumed (we will reveal Da Vinci’s secrets as we progress).
First let us look at the former, the alchemist and philosopher Cornelius Agrippa. In his Three Books of
Occult Philosophy Agrippa produced six plates that diagrammatically describe the overall human
geometries.12 It seems easy for us to dismiss his drawings of the human geometries because his
Medieval drawing style does not look proportionate or as well drawn at Di Vinci’s sketch (and it must be
stressed that Di Vinci’s Vitruvian Man is only a sketch). Or we might want to dismiss these for the fact
that there are alchemal symbols inscribed about the human figure, and we know alchemy to be a lot of
mysticism with some science behind it. But as we will prove Agrippa’s diagrams are in accordance with
human geometries. Here are the six plates he produced :
Plate 1 Plate 2 Plate 3
12 Agrippa, Heinrich Cornelius. De Occulta Philosophia Libri Tres. Book II, Chapter XXVII. 1533.
15
Plate 4 Plate 5 Plate 6
Plates 1 ‐3 illustrate three aspects of the human geometry we already understand. Plate 1 illustrates
that the navel is the center of the body when the arms are raised above the shoulder line, and in this
stance the body is inscribed in a circle (the great circle to be exact). The circle above the head in Plate 1
represents the full height of the arms raised above the head. Plate 2 illustrates that the human body is
inscribed in a square (the great square) when the arms are held perpendicular to the body, so that the
arm span is equal to the human height. In this stance, the groin is the center of the body. Plate 3
illustrates that when the arms are raised to their full height above the head the body fits into the great
circle, which, in this illustration, the circle can be inscribed inside of a square; in this stance the navel is
the center of the body. Plate 4 demonstrates the same geometric consequences as Plate 1, but with a
new feature: when the arms are raised above the shoulders and the feet are swung outward away from
the vertical centerline of the body, the human is still inscribed in the same great circle and the navel is
still the center of the body. Therefore, Plate 4 illustrates that when the legs are swung outward the
navel lies on the centerline of the legs, i.e. the legs appear to hinge upon the navel. Di Vinci illustrates
the same geometry in his sketch of the Vitruvian Man. Therefore all these plates logically fit the
geometric construct of the human body we have proposed above.
Plates 5 and 6 illustrate some things that seem rather forced, as if the human body could not actually be
in accordance with another square formed by the arms and legs as diagonals, which means this square is
smaller than the great square; as well as the pentagram. The latter may seem more mystically contrived
rather than an actual human geometry.
Let us see how Plate 5 works first. Agrippa is not the only one to propose this geometry, as the same
geometry is proposed by Cesare Cesariano in his Italian translation of Vitruvius’s De Architectura in
1521, twelve years before Agrippa’s De Occulta. Of the two woodcuts Cesariano illustrates depicting the
Vitruvian Canon of Proportions the following appears to have some geometric logic within it, primarily
that at some point a human can raise the arms and legs to some specific and respective height so that a
square can be placed within the great circle, and the square’s corners lie on the circumference of the
circle.
16
Ceasiano’s woodcut establishes the navel as the center, therefore the circle present here is the same
great circle concerning us. When the legs swing outward they always lay within the great circle, and,
therefore, the distance from the navel to the tips of the fingers will always be equal to the distance from
the navel to the center of the feet. So, even though the navel will not be the center of the square, it will
still be the center of the body due to the limbs’ relationship to the circle (i.e. their equal length / radius).
Logically, if the legs are swung outward within the great circle, then the arms can be raised to form a
square in accordance with the legs. This can be demonstrated through a series of iterative diagrams that
illustrate how the great square changes position and size when the arms and legs are raised or lowered
in relation to the great circle, and vice versa.
If the process starts with a static human, i.e. a human with its arms raised perpendicular to the body,
only the great square is generated. But if only the arms are raised to the height of the head, then the
great circle is generated and the square no longer applies.
Actually, at any moment in which the arms are raised more than exactly perpendicular to the body, the
great square no long applies. It is when the arms are at head‐height or higher that the great circle is
generated. But certainly there comes a point when the arms are raised so high that the circle no longer
applies. This occurs roughly when the arms are raised almost straight up. Once the arms are raised
completely upright then this is the height of the crest of the great circle. Here the finger tips cannot be
on the great circle’s circumference because the crest of a circle cannot lie in two different points that
are more than the shoulder’s breadth apart. In other words, the tangential crest of the circle (assuming
some arbitrary orientation) cannot be determined by points R and L, which could be denoted by the
right and left arms respectively, if they do not share the same position on the circumference of the
circle.
To demonstrate, when the arms are perpendicular to the body the great square is generated, and when
they are raised to the height of the head the square no applies and the great circle is generated.
17
If the arms remain at head‐height and the legs are swung outward to a certain point then the square is
regenerated. This is precisely what Da Vinci is depicting in his sketch: the static human superimposed on
the human with the arms raised to head‐height and the legs swung out. It is clear in this stance that the
new square is smaller than the great square of the static human.
If the arms and legs are raised a little bit more, the square will become even smaller. There then comes
the point in which all for corners of the square will lay on the circumference of the great circle, so that
the diagonals of the square are equal to the diameter of the circle. It can therefore be established here,
with absolute certainty, that the perceived error between the great square and great circle in Da Vinci’s
sketch is only the result of dynamic human geometries. If this iterative process of raising the arms and
legs is viewed in reverse, then the great square naturally and logically falls into place. It is quite clear
here to realize that nature does design in dynamics, and that nature would account for its geometries in
motion as well at a static stance.
18
It is also clear that Agrippa’s human geometry shown in Plate 5 and Cesariano’s woodcut are correct.
Additionally, this is corroborated by Albrecht Dürer’s interpretation of the Vitruvian Man.13 Similar to
Dürer’s illustration is William Blake’s watercolor Glad Day (1794).
Albrecht Dürer’s “Proportion of Man” William Blake’s “Glad Day”
Thus the square is at its maximum size when the human assumes the static stance, when the arms are
held perpendicular to the body and the legs are not swung outward; as well as when the groin is the
center of the body. Likewise, the square is at its minimum size when the arms are legs are raised so that
the center of the square shares the same position as the center of the circle (the navel). It seems that as
if the great circle and square did not overlap in the first place, this problem wouldn’t exist at all. But it
can now be understood the reason for this overlap, as nature designs in motion (and that motion does
not come from a 0.6% error by Le Corbusier).
In order to avoid confusion, it must be noted here that these geometries are possible, but whether or
not if they are natural or comfortable to perform is another matter. If we are trying to map the
geometries of the human body, one thing that may be noticed is that the distance from the humeral
head to the great square is not equal to the distance from the humeral head to the crest of the great
circle. So how can this geometry be possible? Would not the arm arc outside of the great circle at some
13 Dürer, Albrect. Vier Bücher von Menschlicher Proportion. Nüremberg. 1528.
19
point? Although this is true, we must stress that nature designs in motion and not statics. The head of
the shoulder is not a fixed point that the arm swings around about. Because of the shoulder blade the
head of the humerus has flexibility in its position. So when the arm is upraised the head of the shoulder
is higher, and when the arm is down the head of the humerus is lower. If the shoulder was a fixed hinge,
then a human (and all primates) would not be able to lift the arms above the head (at least not easily).
Primates would certainly have had a hard time climbing around in the trees.
This image of the arm upraised above the shoulder line is really nothing new. Aside from literal
depictions of the Vitruvian Man, the same stance is in every crucifixion. The most typical representation
during the Middle Ages and the Proto‐Renaissance depicts Christ with the arms raised to the height of
the head. The Talisman of Orpheus from the 3rd Century BCE depicts Orpheus crucified with arms at
head‐height. Saint Andrew was crucified on crux decussata, or X‐shaped cross, but commonly referred
to as a “saltire”. Depictions of his crucifixions show either the arms raised to head‐height, with the legs
parted in accordance with the angle of the cross, which would not intersect orthogonally; or the beams
of the cross do intersect orthogonally (at a 90° angle) and he assumes the position in which the square is
fully inscribed within the great circle. But whether the crucifixion is of Saint Peter, Jesus, Saint Andrew,
Orpheus / Bacchus, Mithra, Krishna, et cetera, they all depict, in some fashion, the Vitruvian Canon.
Images and statues of Hindu deities with multiple arms and legs depict much the same canon, albeit not
all these cultures were familiar with Vitruvius.
Crucifixion Talisman of Orpheus Crucifixion of Saint Andrew
Raffaello Sanzio (1503 CE) 3rd Century CE 14th Century CE
Before moving forwards and analyzing Agrippa’s human and the pentacle we need to fully address the
nature of the extra lengths flanking outside of the body created by the circle. It has now been
established that the circle is the result of dynamic anthropometrics. But the extra length must somehow
fit into the human geometry and be accounted for in some manner by our proportioning system (i.e.
from the asymptotic grid, or a variation from it). If these dimensions of the circle could be accounted for
with what we have been working with so far, there would be no problem. Unfortunately our proposed
system needs further development in order to do so. Here we will look at another little key to the
human geometry that Da Vinci gives us in his sketch: notice that Da Vinci places lines at several key
20
bodily moments, and notice the distances between them. Although these lines do not always align with
the anthropometric‐asymptotic grid, what Da Vinci is illustrating is the proportions of squares that
proportionally enlarge from the head in order to divide the human in the square into fourths and
eighths.
Da Vinci illustrates that this extra width created by great circle is one eighth of the arms span (or body
height), and therefore its dimension is half the breadth of the shoulder. Although in his sketch this
seems correct, it is actually in error by 5.6% of being one eighth of the arm span. It may have been
noticed earlier in the anthropometric grid that some of the lines do not align with Da Vinci’s lines, but
then again his drawing is only a sketch and is meant to explore an idea, not to finalize it. But if the our
proposed geometric construct is revisited, we discover that Da Vinci is somewhat correct : the extra
breadth of the great circle is half the breadth of the shoulders in the geometric construct proposed
earlier. Here it can be illustrated that length x is half the breadth of the shoulders, as well as the extra
length produced by the great circle and that the two are equal. As a consequence of this geometry, the
final area of this geometry (that is of the square inscribing the great circle) is 4ϕ2.
Breadth of Shoulders = Extra Length of Circle Length AB = x
Is this necessarily so? Indeed it is. If two similar 45 ‐ 45 right triangles are constructed so that each of
their hypotenuses horizontally divide the great square, and the square inscribing the great circle, equally
and respectively, then the vertical distance between the two hypotenuses will be equal to the horizontal
21
distance between the two the corresponding 45° vertices. This is true for all similar squares that share a
common edge, and upon that common edge a common midpoint. But in the human geometry this case
and proof is rather special.
It can then be established here the relationship between the two bodily centers, i.e. the navel and the
groin, and the consequences of the great square in relation to the great circle when the body is in
motion. As the arms rise above the shoulder line and the legs are swung outward, the square diminishes
in size. Consequently as this happens the center of the square approaches the center of circle, until
finally the two centers share the same position. We will account more fully on how the anthropometric‐
asymptotic grid can account for this extra length created by the great circle shortly.
So as the arms rise higher and the legs swing more outward, the square diminishes in size until the two
centers unite at the navel. Can the same be true for the circle? Can the circle diminish in size to the
point that the circle’s center lays upon the center of the groin? This is what is depicted in Plate 6 by
Agrippa : that if the arms are lowered to a certain degree and the legs swung outward to a certain
degree the human will rest upon a pentagram, with the groin as the center. A similar depiction is
featured in Robert Fludd’s alchemal and occult philosophical treatise The Metaphysical, Physical, and
Technical History of the Two Worlds, published in 1617, depicting man as a microcosm of the universe,
and the universe as a macrocosm of man.
Destiny by Robert Fludd (1617)
The question is: does this really work? As the legs swing outward they always fall on the great circle, a
circle whose center is at the navel. How, then, can the legs swing outward and lie within a smaller circle
with the groin as the center?
To consider if this geometry is even correct, it must first be established which points on the pentagram
are constants. First, the crown of the head is constant, as this is the vertex of the top point of the
pentagram (point o). Second, the legs can be swung outward in order to create two lower vertices of the
pentagram, and therefore a constant. In order for two lower vertices of the pentagram to be in
accordance with the geometries established above, these two vertices (i.e. the placement of each foot)
must lay on the circumference of the great circle (points m and n). Given these three points the size and
placement of the pentagram will appear as such:
22
As can be seen, the trough of the circle that inscribes the pentagram lies outside the circumference of
the great circle. But something appears to work in this instance, as the left and right vertices (i.e. point
x) of the pentagram seems to align at the intersection of the great square and circle (point z), the point
in which the hand raised to the height of the head. Though this may seem trivial, it is actually in error by
0.087% of the great circle’s radius. Although there is a very small error in this geometry, we must
remember that the shoulder is a flexible moment in the body, so the arms can easily adjust to correct
this negligible error. We can therefore establish that it is possible for the human to be in accordance
with the geometry of the pentagram. Although this is a negligible error we do not wish to do what Le
Corbusier did and create excuses for this mild error. There could be something more profound to the
nature of the flexibility of the shoulder blade with human geometries, and this will be the subject of a
future paper.
Although this is a possible explanation, Agrippa’s and Fludd’s depictions of the Vitruvian Man in
accordance with the pentagram is nonetheless wrong for one reason : the groin is not the center of the
body (point A). This is because the circle inscribing the pentagram exceeds the boundaries of the great
square. The position of the center of the body in this stance is actually about 1/64 of the body’s height
lower than the center of the groin (point B). Of course we can imagine that these geometries would
appear to be perfect and without error in the study or lab of a 16th Century alchemist. The center of the
body being 1/64 of the human height lower than the groin is easily corrected by not extending the legs
outward. But in that case the human would not take on the form of a pentagram.
The Human Geometries and the Pentagram
Of all geometries the pentagram is the only one that is anthropomorphic. We can easily look at a
pentagram and see a head, arms, legs, arm pits, a groin, and a torso. Mystically the pentagram has been
considered a humanist symbol, an esoteric geometry that represents the human. But could the
pentagram have any further implications toward the human geometry? Indeed it does. In fact, the
pentagram is a corollary to the creation of the human geometric construct described above, as it follows
the exact same proportioning system as the asymptotic‐anthropometric grid.
23
It is already well known that the pentagram has golden mean properties. For instance, the line segment
A and C are segmented into a golden mean by point B, so that line AB = ϕ – φ and line BC = φ, and
therefore line AC = ϕ. Likewise line AD is segmented into a golden mean by point C, so consequently
that line AC = ϕ –φ and line CD = φ, and therefore line AD = ϕ.
In looking at this geometry some things can readily be established : lines ABC and ACD are golden
means in the same manner described above. In this respect the square of line ABC will produce the
golden mean ACD, so that squares ACGE and BDHF will be equal. In this instance square ACGE is
bisected on two axes into golden means.
It appears that the means to create the human geometry (that is our proposal to correct Le Corbusier’s
two adjacent squares from a golden mean problem) is inherently embedded into the pentagram. But
where are the other vertices? Where is the other square, so that this is two adjacent squares? It seems
to be a bit of a riddle, but the other vertices are on the pentagram. We will now demonstrate one
possibility of the two adjacent squares and the golden mean and their relationships to the pentagram It
must be noted that there are a few other possibilities.
If we take a pentagram with a point a and a point r as two neighboring vertices on the pentagram, and a
line br is drawn so that br is perpendicular to line ab, so that points b, c, and r all lay on the same line:
then line ab is squared in order to create a square abcd (Figure I). Next construct a golden rectangle
aefd from square abcd. In the construction of golden rectangle aefd the line ef is aligned with vertex q
of the pentagram, so that points e, f, and q all lay on the same line (Figure II).
24
Figure I Figure II
Following the process established earlier, square the golden rectangle aefd in order to create a square
aegi, which is segmented into two golden means on two perpendicular axes by lines bh and df, in that
lines bh and df are equal (Figure III). Still following the exact same process, then render a golden
rectangle ajki from square aegi, so that line jk are aligned with the vertex s on the pentagram, in that
points j, k, and s all lay on the same line. It can therefore be established that line ks is the centerline of
the pentagram (Figure IV). Therefore squares aegi and bjkh are equal.
Figure III Figure IV
Finally the original square is doubled, that is square line eg, which lies upon vertex l of the pentagram.
Therefore squares aegi = bjkh = elmg (Figure V). We have chosen to demonstrate this particular means
of forming the two squares from the golden mean for a reason : if length al is ϕ ‐ φ, then length au is ϕ,
as was established above. This is a critical measurement in order for the pentagram to be in accordance
with the anthropometric‐asymptotic grid.
Figure V Figure VI
25
In Figure VII, if the length of one line on pentagram Y is ϕ – φ, then the length of one line of pentagram
X is ϕ. Therefore pentagram X is created from pentagram Y via the golden mean. So if pentagram Y is
moved to the other side of pentagram X as pentagram Z is in relation to pentagram X, so that
pentagrams Y and Z are equal.
Figure VII
Although this is technically disassembling and rearranging the geometry, something we argue nature
does not do, this is actually a form of fractalizing the pentagram. Although it seems that the smaller
pentagram was created and then moved, both are actually logical and legitimate means of creating a
pentagram that either is enlarged by the golden mean or diminished. In both cases three points of the
originally pentagram are used to construct the new pentagram, and three points is typically the
minimum number of points necessary for proof. In the case of pentagram Z it can be seen in Figure VI
that length al is equal to length rt, and that the two line segments are parallel in the parallelogram altr.
Therefore it can be established that the following Figures VIII and IX of pentagrams pairs are true :
Figure VIII Figure IX
If a third pentagram is added that is the golden mean of one of the other pentagrams, then the
following illustrates that fractalizing pentagrams by the golden mean are in accordance with the two
adjacent squares geometry (Figure X). Therefore in Figure X if pentagram A is ϕ, then pentagram B is
equal to ϕ – φ, and pentagram C equals φ. If this process continues and a fourth pentagram is added
(Figure XI), then the initial two‐squares‐by‐golden‐mean no longer apply. But, since the pentagrams B, C,
and D are proportionately similar to pentagrams A, B, and C then a smaller two‐squares can be applied
to pentagrams B, C, and D. This new two‐squares BCD is diminished from two‐squares ABC by φ (or
divided by ϕ). One can easily see two‐squares BCD’s relationship to two‐squares ABC as a fractal.
26
Figure X Figure XI
If a fifth pentagram E is added (Figure XII) then a pattern emerges, namely that this process echoes the
asymptotic grid created earlier (Figure XIII).
Figure XII Figure XIII
Since it can now be established that the proportions of a pentagram are a direct corollary to the
proportions contained in the human body, at this point it will be interesting to see how these five
pentagrams are related to the human geometry, particularly on the anthro‐geometric construct (Figure
XIV). In looking at how the pentagrams relate to the geometric construct in this manner numerous
anthropometrics can be corroborated, and some new ones can be accounted for.
For instance, if two neighboring vertices a and b of pentagram A lay on the initial square (that is the
square that created this geometry), then vertex c of the pentagram is vertically aligned with the extra
width created by the great circle. Likewise, the corresponding vertex of c is d, and vertex d is lays upon
the vertical centerline of the whole geometric construct. If pentagram B is the golden mean of
pentagram A, then the centerline of pentagram B is aligned with the shoulder line. Furthermore the full
length of a line segment from pentagram B is equal to the length of one arm from the shoulder to the
finger tips (line ab). Pentagram C is the golden mean of pentagram B, and its centerline is also the
vertical centerline of the whole geometry, and therefore the vertical line of symmetry for the whole
27
body. The full length of one of the line segments of pentagram C is equal to the breadth of the
shoulders. And so centerline of pentagram E (the golden mean of the golden mean of pentagram C) is
also aligned with the shoulder.
Figure XIV
Thus far, it seems that the point which these diminishing pentagrams approach (point x) is rather
meaningless and just sits somewhere randomly on the construct. Or does length ax actually measure
something on the body? Indeed it does. If the fractlizing pentagrams continue ad infinitum (Figure XV),
then the angle of axc is 36°, which is also the interior angle of any pinnacle on a pentagram (Figure XVI).
Figure XV Figure XVI
Therefore, in comparing the angle of the approaching pentagrams toward point x to their respective
two‐squares geometry in Figure VIII, it can be established that the point toward which the diminishing
two‐square geometries are approaching is also point x. The angle in which the fractalizing two‐squares
are approaching point x is 54° (half the interior angle of a pentagon, or half of 108°). Therefore the angle
of the fractalizing pentagrams (36°) is complementary to the angle of the fractalizing two‐squares (54°).
In other words, they both approach point x at angles that add up to 90° (Figure XVII).
28
Figure XVII
Placing the fractalizing pentagrams as they are in Figure XIV only accounts for horizontal
anthropometrics, but it does not account for vertical measurements. Consequently the length of ax in
Figure XIV is also the height of the human (or the arm span). Since the length of ax is equal to the
human height (from head to heels), and the fractalizing pentagrams are in accordance with the process
of creating the anthropometric‐asymptotic grid, then the pentagrams should also be able to establish
vertical measurements. And in Figure XVIII they do so indeed.
Figure XVIII
So it seems we have come full‐circle. Actually we have come more than full‐circle; more like a circle and
a half. We can easily see the beauty and wonder in nature’s designs. It seems as if we have a new means
of admiring the proportions, geometries, and measurements of our own bodies. Throughout the course
of this essay, we have intensely studied various geometries in relation to the human body and
anthropometrics (both static and dynamic). A final aspect left to admire is the numerics of human
29
proportions in accordance with the fractalizing two‐square geometry. Starting with any of the two‐
squares‐by‐golden‐mean geometries, the rate at which they increase can be seen in Figure XIX.
Figure XIX
If this pattern continues then a rather interesting curiosity occurs :
Horizontal Growth : Vertical Growth :
φ
ϕ ‐ φ ϕ ‐ φ
φ
ϕ ‐ φ ϕ
ϕ ‐ φ
ϕ 2ϕ ‐ φ
ϕ
2ϕ ‐ φ 3ϕ ‐ φ
2ϕ – φ
3ϕ ‐ φ 5ϕ ‐ 2φ
3ϕ – φ
5ϕ ‐ 2φ 8ϕ ‐ 3φ
5ϕ ‐ 2φ
8ϕ ‐ 3φ 13ϕ ‐ 5φ
8ϕ ‐ 3φ
13ϕ ‐ 5φ 21ϕ ‐ 8φ
13ϕ ‐ 5φ
21ϕ ‐ 8φ 34ϕ ‐ 13φ
21ϕ ‐ 8φ
…
… et cetera
30
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 follow the Fibonacci
series, which demonstrates that when any of the numbers of the series are divided by the previous
number then the values are always 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 in this paper has been the result of correcting
very trivial errors. But then again, it supports that the age old cliché “little things do matter” has some
justifications.
We 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 “point A to point B”. 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 human in profile…
Patrick M. Dey
Damian ‘Pi’ Lanningham
5 January 2011
The Open Problem Society
Edited 5 September 2011
31
Bibliography:
Agrippa, Heinrich Cornelius. Three Books of Occult Philosophy. Trans. James Freake. Woodbury, MN:
Llewelyn Publications. 1992.
Dürer, Albrect. Four Books on Human Proportion. Nüremberg, Germany: Hieronymus Formschneyder.
1528.
Fludd, Robert. The Metaphysical, Physical, and Technical History of the Two Worlds. 1617.
Le Corbusier. The Modulor. Trans. Peter de Francia and Anna Bostock. Basel, Switzerland: Birkhäuser
Publishers. 2004
National Center for Social Research. Health Survey for England 2008. United Kingdom: National Center for Social Research. 2009.
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.
Pollio, Marcus Vitruvius. The Ten Books on Architecture. Trans. Morris Hicky Morgan. Mineola, New York
: Dover Publications, Inc. 1960.
Pollio, Marcus Vitruvius. De Architectura. Trans. Cesare Cesariano. Como, Italy. 1521.
Further Reading:
Elam, Kimberly. Geometry of Design. Princeton, New Jersey: Princeton Architectural Press. 2001.
Livio, Mario. The Golden Ration: The Story of Phi, The World’s Most Astonishing Number. New York, New
York: Broadway Books. 2002.
Padovan, Richard. Proportion: Science, Philosophy, Architecture. New York, New York: Spon Press. 1999.
Skinner, Stephen. Sacred Geometry: Deciphering the Code. New York, New York: Sterling Publishing Co.