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Benzothiazole bipyridine complexes of ruthenium(II) with cytotoxic activity.
Rachel Fletcher
Geometer’s Angle
113 Division St.
Great Barrington, MA
01230 USA
rfletch@bcn.net
Dynamic Root Rectangles Part Two:
The Root-Two Rectangle and Design
Applications
Keywords: descriptive
geometry, diagonal,
dynamic symmetry,
incommensurate values,
root rectangles
Abstract. “Dynamic symmetry” is the name given by Jay Hambidge
for the proportioning principle that appears in “root rectangles”
where a single incommensurable ratio persists through endless spatial
divisions. In Part One of a continuing series [Fletcher 2007], we
explored the relative characteristics of root-two, -three, -four, and five systems of proportion and became familiar with diagonals,
reciprocals, complementary areas, and other components. In Part
Two we consider the “application of areas” to root-two rectangles
and other techniques for composing dynamic space plans.
Introduction
“Dynamic symmetry” is the term given by the early-twentieth-century artist and scholar
Jay Hambidge for the system of incommensurable ratios that appear in root rectangles and
that replicate through endless spatial divisions, while expressing the relationship between
one level of scale and the next.1 In contrast to passive or static symmetry, which Hambidge
associates with the radial subdivisions that characterize regular geometric figures or crystals,
dynamic symmetry governs the spiral-like growth exhibited in plants and shells. Hambidge
observes dynamic symmetry in ancient Egyptian bas reliefs and Classical Greek pottery and
temple plans.2
The root-two rectangle can be formed from the side and hypotenuse of a 45o-45o-90o
triangle. In this article we explore the dynamic properties of this elementary quadrilateral
shape and consider possibilities for design applications.
Review: Root-Rectangles in Series
When a square divides into diminishing root rectangles or when root rectangles expand
from a square, elements of dynamic symmetry become apparent. Fig. 1 illustrates how
expanding root rectangles develop from a square. The diagonal of the preceding square or
rectangle equals the long side of the succeeding four-sided figure.3 The short side of each
root rectangle is 1. The long sides progress in the series ¥2, ¥3, ¥4, ¥5, ¥6… .
Nexus Network Journal 10 (2008) 149-178
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 149
1590-5896/08/010149-30 DOI 10.1007/ S00004-007-0060-Z
© 2008 Kim Williams Books, Turin
D
C
F
H
J
L
N
B
A
E
G
I
K
M
2
3
4
5
6
DB : BA :: 1 : 1
DB : BE :: 1 : ¥2
DB : BG :: 1 : ¥3
DB : BI :: 1 : ¥4
DB : BK :: 1 : ¥5
DB : BM :: 1 : ¥6
Fig. 1
Another method for obtaining root rectangles is from the radii of arcs that increase in
whole number increments.4
x
x
x
x
x
x
x
Draw a horizontal baseline AB equal in length to one unit.
From point A, draw an indefinite line perpendicular to line AB that is slightly
longer in length.
Place the compass point at A. Draw a quarter-arc of radius AB that intersects
line AB at point B and the indefinite vertical line at point C.
Place the compass point at B. Draw a quarter-arc (or one slightly longer) of
the same radius, as shown.
Place the compass point at C. Draw a quarter-arc (or one slightly longer) of
the same radius, as shown.
Locate point D, where the two quarter-arcs (taken from points B and C)
intersect.
Place the compass point at D. Draw a quarter-arc of the same radius that
intersects the indefinite vertical line at point C and line AB at point B (fig. 2).
***
150 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
x
Connect points D, B, A, and C.
The result is a square (DBAC) of side 1 (fig. 3).
D
C
D
C
B
A
B
A
Fig. 2
x
x
x
x
Fig. 3
Add a length of ½ unit to line DB, so that the line DM equals 1.5.
Place the compass point at M. Draw an arc of radius MD that intersects the
extension of line BA at point E.
From point E, draw a line perpendicular to line EB that intersects the
extension of line DC at point F.
Connect points D, B, E, and F.
The result is a root-two rectangle (DBEF) with short and long sides of 1 and ¥2.
x
x
x
x
Add a length of ½ unit to line DM, so that the line DN equals 2.0.
Place the compass point at N. Draw an arc of radius ND that intersects the
extension of line BE at point G.
From point G, draw a line perpendicular to line GB that intersects the
extension of line DF at point H.
Connect points D, B, G, and H.
The result is a root-three rectangle (DBGH) with short and long sides of 1 and ¥3.
x
x
x
x
Add a length of ½ unit to line DN, so that the line DO equals 2.5.
Place the compass point at O. Draw an arc of radius OD that intersects the
extension of line BG at point I.
From point I, draw a line perpendicular to line IB that intersects the extension
of line DH at point J.
Connect points D, B, I, and J.
The result is a root-four rectangle (DBIJ) with short and long sides of 1 and ¥4.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 151
5
4
3
2
D
C
F
H
J
L
A
E
G
I
K
B
Rad
.5
ius 1
M
di
Ra
us
2.0
s
diu
Ra
N
2.5
u
di
Ra
s3
.0
O
P
Fig. 4
x
x
x
x
Add a length of ½ unit to line DO, so that the line DP equals 3.0.
Place the compass point at P. Draw an arc of radius PD that intersects the
extension of line BI at point K.
From point K, draw a line perpendicular to line KB that intersects the
extension of line DJ at point L.
Connect points D, B, K, and L.
The result is a root-five rectangle (DBKL) with short and long sides of 1 and ¥5. The
process can continue infinitely (fig. 4).
152 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
Fig. 5 illustrates how diminishing root rectangles develop from a quarter-arc that is
drawn within a square.5 The long side of each root rectangle is 1. The short sides progress
in the series 1/¥2, 1/¥3, 1/¥4, 1/¥5… .
D
C
G
F
J
M
P
I
L
O
1/ 2
1/ 3
1/ 4
1/ 5
A
B
DB : BA :: 1 : 1
GB : BA :: 1 : ¥2
JB : BA :: 1 : ¥3
MB : BA :: 1 : ¥4
PB : BA :: 1 : ¥5
Fig. 5
Review: Diagonal, Reciprocal, and Complementary Patterns
At the heart of Hambidge’s system is the fact that root rectangles produce reciprocals of
the same proportion when the diagonal of the major rectangle and the diagonal of its
reciprocal intersect at right angles.6 The diagonals divide into vectors of equiangular
distance apart, progressing in spiral-like fashion, while locating endless spatial divisions in
continued proportion. The middle of any three adjacent vectors is the mean proportional
or geometric mean of the other two.
A root-two rectangle divides into two reciprocals in the ratio 1 : ¥2. The area of each
reciprocal is one-half the area of the whole (fig. 6a).
A root-three rectangle divides into three reciprocals in the ratio 1 : ¥3. The area of each
reciprocal is one-third the area of the whole (fig. 6b).
A root-four rectangle divides into four reciprocals in the ratio 1 : ¥4. The area of each
reciprocal is one-fourth the area of the whole (fig. 6c).
A root-five rectangle divides into five reciprocals in the ratio 1 : ¥5. The area of each
reciprocal is one-fifth the area of the whole (fig. 6d).7
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 153
Fig. 6a
1
2
Fig. 6b
1
3
Fig. 6c
1
4
Fig. 6d
1
5
154 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
How to divide a root rectangle into reciprocals
For this demonstration we use the root-two rectangle, but the principle applies to all
rectangles of dynamic proportions.
x
Draw a square (DBAC) of side 1 (repeat figures 2 and 3).
D
C
D
C
B
A
B
A
Fig. 2
x
Fig. 3
Draw the diagonal BC through the square (DBAC).
The side (DB) and the diagonal (BC) are in the ratio l : ¥2.
x
Place the compass point at B. Draw an arc of radius BC that intersects the
extension of line BA at point E (fig. 7).
D
C
B
A
E
Fig. 7
x
x
From point E, draw a line perpendicular to line EB that intersects the
extension of line DC at point F.
Connect points D, B, E, and F.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 155
The result is a root-two rectangle (DBEF) with short and long sides of 1 and ¥2 (1 and
1.4142…) (fig. 8.)
D
C
F
B
A
E
DB : BE :: l : ¥2
Fig. 8
x
x
x
Locate the diagonal BF of the rectangle DBEF.
Locate the line EF. Draw a semicircle that intersects the diagonal BF at point
O, as shown.
From point E, draw a line through point O that intersects line FD at point G.
The diagonal (BF) of the major rectangle (DBEF) and the diagonal (EG) intersect at
right angles (fig. 9).
D
G
F
O
B
E
EG : BF :: l : ¥2
Fig. 9
x
x
From point G, draw a line perpendicular to line FD that intersects line BE at
point H.
Connect points H, E, F, and G.
156 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
The result is a smaller root-two rectangle (HEFG) with short and long sides of 1/¥2
and 1 (¥2/2 : 1 or 0.7071… : 1). Rectangle HEFG is the reciprocal of the major rectangle
DBEF.
The major 1 : ¥2 rectangle DBEF divides into two reciprocals (HEFG and BHGD)
that are proportionally smaller in the ratio 1 : ¥2. The area of each reciprocal is one-half the
area of the whole (fig. 10).
D
G
F
O
B
E
H
GF : FE :: FE : EB :: l : ¥2
Fig. 10
The diagonal (BF) of the major rectangle (DBEF) and the diagonal (EG) of the
reciprocal (HEFG) locate endless divisions in continual proportion. A root-two rectangle of
any size divides into two reciprocals in the ratio l : ¥2. If the process continues, the side
lengths of successively larger rectangles form a perfect geometric progression (l ,¥2, 2,
2¥2…). The side lengths of successively smaller rectangles decrease in the ratio of 1 : 1/¥2
toward a fixed point of origin known as the pole or eye (point O). (See figure 11.)8
D
G
F
O
B
H
E
Fig. 11
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 157
x
x
x
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2, and its diagonal BF.
From point F, draw an indefinite line perpendicular to the diagonal FB.
Extend the line BE until it intersects the indefinite line at point I, as shown.
Connect points B, F, and I.
The result is a right triangle (BFI).
x
x
From point I, draw a line perpendicular to line IB that intersects the extension
of line DF at point J.
Connect points E, I, J, and F.
The rectangle EIJF is the reciprocal of the root-two rectangle DBEF.
If the long side (BF) of a right triangle (BFI) equals the diagonal of a major rectangle,
the short side (IF) of the triangle equals the diagonal of the reciprocal (fig. 12).9
D
F
J
B
E
I
EI : IJ :: DB : BE :: l : ¥2
IF : FB :: l : ¥2G
Fig. 12
Application of Areas
Definition:
Application of areas is Jay Hambidge’s term for dividing rectangles into proportional
components, where shapes applied on the short and long sides of a figure are equal in area.
Hambidge’s technique for proportioning areas, which he traces to ancient Egyptian and
Classical Greek design, produces harmonious compositions that are based on the
proportions of the enclosing rectangle. An area of any rectangular shape may be “applied”
on the short and long sides of a rectangle. When a square is applied on the short side, then
“applied” on the long side, the new area is the same as the square, but the shape becomes
rectangular. The reciprocal of the major rectangle is accomplished by locating the point
where the diagonal of the major rectangle and the inside edge of the true square intersect
[Hambidge 1960, 28–29; 1967, 35, 60–72].
158 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
How to apply areas to a root-two rectangle
x
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2, and its diagonal BF.
From point B, draw a quarter-arc of radius BD that intersects line BE at point
K.
x From point K, draw a line perpendicular to line BE that intersects line FD at
point L.
The result is a square (DBKL) “applied” on the short side (DB) of the root-two
rectangle (DBEF).
x
x
x
x
Locate the diagonal BF of the root-two rectangle (DBEF).
Locate point M where the diagonal (BF) intersects line KL.
Draw a line through point M that is perpendicular to line KL and intersects
line DB at point N and line EF at point P.
Locate the rectangle NBKM.
The rectangle (NBKM) is the reciprocal of the root-two rectangle DBEF. Line BM is
the diagonal of the reciprocal (NBKM) (fig. 13.)
L
D
N
M
B
K
F
D
P
N
E
B
L
M
K
F
P
E
NB : BK :: DB : BE :: l : ¥2
BM : BF :: l : ¥2
Fig. 13
x
Fig. 14
Locate the square (DBKL) and the rectangle BEPN.
The area of the rectangle (BEPN) and the area of the square (DBKL) are equal. In
Hambidge’s system, the “square” is applied on the short and long sides of the root-two
rectangle (fig. 14).10
x
Locate the reciprocal NBKM of the major root-two rectangle DBEF.
x
Locate the rectangles DNML and KEPM.
Rectangle DNML is the complement of the reciprocal NBKM.11
The areas of rectangles DNML and KEPM are equal.
x
Locate rectangles NBKM and LMPF.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 159
The rectangles NBKM and LMPF share the same diagonal and are similar (fig. 15).12
Any quadrilateral figure can be applied to the long and short sides of a root rectangle in
this fashion. In figure 16 the areas of rectangles DBQR and BEUT are equal and the areas
of rectangles DTSR and QEUS are equal.
L
D
N
M
F
D
K
F
P
T
B
R
E
Fig. 15
B
U
S
E
Q
Fig. 16
Definitions:
Hambidge identifies three ways to apply one area to another. The applied area can be
less than, equal to, or in excess of the other. If the applied area is less, it is elliptic; if equal,
it is parabolic; and if in excess, it is hyperbolic.
If a square is applied to the short side of a rectangle, it “falls short” and is elliptic. If a
square is applied to the short side of a root-two rectangle, the excess area contains a square
and a root-two rectangle (fig. 17a). If squares are applied to both short sides of a root-two
rectangle, they overlap and the total rectangle divides into three squares and three root-two
rectangles (fig. 17b). If a square is applied to the long side of a rectangle, it “exceeds” the
rectangle and is hyperbolic. If a square is applied to the long side of a root-two rectangle,
the excess area contains two squares and a root-two rectangle (fig. 17c).13
1
2
Fig. 17a
160 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
1
2
Fig. 17b
2
Fig. 17c
1
The Root-Two Rectangle and Variations on Area Themes
The application of areas permits the division of root rectangles into harmonious
compositions that are based on the ratio of the overall figure. Let us consider the root-two
rectangle and related figures.14
How to divide a square into root-two proportional areas
x Draw a square (DBAC) of side 1. (Repeat figures 2 and 3.)
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 161
D
C
D
C
B
A
B
A
Fig. 2
x
x
x
x
x
Fig. 3
Draw the diagonals AD and CB through the square (DBAC).
Locate point O, where the two diagonals intersect.
Place the compass point at A. Draw an arc of radius AO that intersects line
AC at point E.
Place the compass point at B. Draw an arc of radius BO that intersects line
BD at point F.
Connect points E and F.
The result is a root-two rectangle (AEFB) of sides 1/¥2 and 1 (0.7071… and 1) (fig.
18.)
D
C
F
E
O
A
B
Fig. 18
x
Place the compass point at A. Draw a quarter-arc of radius AO that intersects
line AC at point E and line AB at point G.
162 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
x
Place the compass point at B. Draw a quarter-arc of radius BO that intersects
line BD at point F and line BA at point H.
Place the compass point at D. Draw a quarter-arc of radius DO that intersects
line DC at point I and line DB at point J.
Place the compass point at C. Draw a quarter-arc of radius CO that intersects
line CD at point K and line CA at point L.
Connect points J and L. Connect points I and H. Connect points G and K.
x
x
x
The results are three root-two rectangles (CKGA, DJLC, and BHID). When root-two
rectangles are drawn on all four sides of a square, the result is a composition that contains
one center square, four smaller corner squares, and four root-two rectangles (fig. 19).15
D
K
I
C
F
E
O
J
B
L
G
A
H
Fig. 19
How to divide a root-two rectangle into segments that progress in the ratio 1 : ¥2
x Draw a root-two rectangle (DBEF) of sides 1 and ¥2. (Repeat figures 2, 3, 7,
and 8.)
D
C
D
C
B
A
B
A
Fig. 2
Fig. 3
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 163
D
C
B
A
E
D
C
F
B
A
E
Fig. 7
x
Fig. 8
Draw the diagonal (BF) of the major rectangle (DBEF), the reciprocal HEFG,
and its diagonal (EG). (Repeat figures 9 and 10.)
G
D
F
D
G
O
O
E
B
Fig. 9
F
B
H
E
Fig. 10
The diagonal (BF) of the major rectangle (DBEF) and the long side (GH) of the
reciprocal (HEFG) intersect at point I.
x
x
x
x
Connect points G and I.
From point I, draw a line perpendicular to line IG that intersects the diagonal
(EG) of the reciprocal (GHEF) at point J.
From point J, draw a line perpendicular to line JI that intersects the diagonal
BF at point K.
From point K, draw a line perpendicular to line KJ that intersects the diagonal
EG at point L.
The process can continue infinitely.
The lines LK, KJ, JI, IG, GF, FE, and EB compose a rectilinear spiral that increases in
root-two proportion and decreases in the ratio of 1 : 1/¥2 toward the pole or eye (point O)
(fig. 20).
164 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
G
D
F
K
L
O
I
B
J
H
E
KJ : JI :: JI : IG :: IG : GF :: GF : FE :: l : ¥2G
Fig. 20
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2.
x
Draw the diagonals (BF and DE) of the root-two rectangle DBEF, the
diagonals (EG and HF) of the reciprocal HEFG, and the diagonals (HD and
BG) of the reciprocal BHGD (fig. 21).
***
Use the diagonals to repeat the equiangular spiral three times, as shown (fig. 22).
D
G
F
D
G
F
B
H
E
B
H
E
Fig. 21
Fig. 22
How to divide a root-two rectangle into smaller root-two rectangles of equal area
x
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2.
Locate the diagonals (BF and DE) of the root-two rectangle DBEF.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 165
x
x
x
Locate the diagonals (HD and BG) of the reciprocal root-two rectangle
BHGD and the diagonals (EG and HF) of the reciprocal root-two rectangle
HEFG.
Locate the midpoints (G, H, M, and N) of the root-two rectangle DBEF.
Connect midpoints G and H, then midpoints M and N.
The result is a root-two rectangle divided into four smaller root-two rectangles of equal
area (fig. 23).
G
D
F
M
N
B
E
H
Fig. 23
x Locate points O, P, Q, and R, where the diagonals intersect, as shown.
x Through these points, extend two vertical and two horizontal lines to the sides
of the original rectangle, as shown.
The result is a root-two rectangle divided into nine smaller root-two rectangles of equal
area (fig. 24).
G
D
B
F
P
O
Q
R
H
E
Fig. 24
166 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
x
x
From midpoint M, draw the half diagonals ME and MF through the root-two
rectangle, as shown.
From midpoint N, draw the half diagonals ND and NB through the root-two
rectangle, as shown (fig. 25).
G
D
F
M
N
B
H
E
Fig. 25
x
x
Locate points S, T, U, and V, where the diagonals and half diagonals intersect,
as shown.
Through these points, extend three vertical and three horizontal lines to the
sides of the original rectangle, as shown.
The result is a root-two rectangle divided into sixteen smaller root-two rectangles of
equal area (fig. 26).
G
D
F
V
S
U
T
B
H
E
Fig. 26
x
Extend four vertical and four horizontal lines to the sides of the original
rectangle, through the points of intersection, as shown.
The result is a root-two rectangle divided into twenty-five smaller root-two rectangles of
equal area (fig. 27).16
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 167
D
G
F
B
H
E
Fig. 27
How to divide a root-two rectangle into patterns of proportional areas
x Locate the root-two rectangle (DBEF) of sides 1 and ¥2.
x Apply a square (DBAC) to the left side (DB) of the root-two rectangle, as
shown (fig. 28).
D
C
F
B
A
E
DB : BE :: l : ¥2
Fig. 28
x
x
x
From the midpoint G of line FD, draw the half diagonals GB and GE
through the root-two rectangle, as shown.
Locate point W where the half diagonal GE intersects the right side (AC) of
the square (DBAC), as shown.
From point W, draw a line perpendicular to line AC that intersects line EF at
point X.
168 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
When a square (DBAC) is “applied” on the short side of a root-two rectangle (DBEF),
it falls short and is elliptic. The excess area (AEFC) is composed of a square (WXFC) and a
root-two rectangle (AEXW) (fig. 29).
G
D
B
C
F
W
X
A
E
Fig. 29
x
Apply a square (EFYZ) to the right side (EF) of the root-two rectangle, as
shown (fig. 30).
D
B
Y
G
Z
C
F
W
X
A
E
Fig. 30
x
x
x
From point D, draw the diagonal DA of the square DBAC. From point F,
draw the diagonal FZ of the square EFYZ, as shown.
Locate point Aƍ where the diagonal DA and the half diagonal GB intersect, as
shown.
Extend the line XW through point Aƍ, as shown, until it intersects the line DB
at point Bƍ.
When squares (DBAC and EFYZ) are “applied” on both short sides of a root-two
rectangle (DBEF), they overlap. The various diagonals reveal a pattern of three squares and
three root-two rectangles (fig. 31).
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 169
D
Y
A'
B'
B
G
C
W
Z
F
X
A
E
Fig. 31
x
x
x
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2.
Locate the squares DBAC and EFYZ applied to the root-two rectangle
(DBEF), as shown.
From point B, draw the diagonal BC of the square DBAC. From point E,
draw the diagonal EY of the square EFYZ, as shown.
Locate point C´ where the two diagonals intersect (fig. 32).
D
Y
C
F
A
E
C'
B
Z
Fig. 32
x
x
Draw a line through point Cƍ that is perpendicular to and intersects line FD at
point G and line BE at point H.
Draw a line through point Cƍ that is perpendicular to line GH and intersects
line DB at point Dƍ and line EF at point Eƍ (fig. 33).
170 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
D
Y
G
C
D'
F
E'
C'
B
Z
H
A
E
Fig. 33
x
x
x
From point D, draw the diagonal DA of the square DBAC. From point F,
draw the diagonal FZ of the square EFYZ, as shown.
Locate point F´ where the two diagonals intersect.
Draw a line through point F´ that is perpendicular to line GH and intersects
line DB at point G´ and line EF at point H´.
The four diagonals reveal a pattern of six squares and six root-two rectangles (fig. 34).
D
Y
G
C
F
D'
E'
C'
F'
G'
B
H'
Z
H
A
E
Fig. 34
x
x
x
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2.
Locate the square DBAC applied to the root-two rectangle (DBEF), as shown.
From point B, draw the diagonal BF of the root-two rectangle DBEF.
Locate point Iƍ where the diagonal BF intersects the right side (AC) of the
square (DBAC), as shown.
x Draw a line through point Iƍ that is perpendicular to line AC and intersects line
DB at point Dƍ and line EF at point Eƍ.
x Locate the rectangle BEEƍDƍ.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 171
The area of the square DBAC and the area of the rectangle BEEƍDƍ are equal. The
“square” is applied on the short and long sides of the root-two rectangle.
x
Locate the rectangle DƍBAIƍ.
Rectangle DƍBAIƍ is the reciprocal of the major root-two rectangle DBEF.
x
Locate the rectangles DDƍIƍC and AEEƍIƍ.
Rectangle DDƍIƍC is the complement of the reciprocal DƍBAIƍ.
The areas of rectangles DDƍIƍC and AEEƍIƍ are equal.
x
Locate the rectangles DƍBAIƍ and CIƍEƍF.
The rectangles DƍBAIƍ and CIƍEƍF share the same diagonal and are similar (fig. 35).
C
D
D'
F
E'
I'
B
A
E
Fig. 35
x
x
x
x
x
x
Locate the square EFYZ applied to the root-two rectangle (DBEF), as shown.
Locate the diagonals (BF and DE) of the root-two rectangle DBEF.
Locate point Jƍ where the diagonal DE intersects the left side (YZ) of the
square (EFYZ), as shown.
Locate point Kƍ where the diagonal BF intersects the left side (YZ) of the
square (EFYZ), as shown.
Locate point Lƍ where the diagonal DE intersects the right side (AC) of the
square (DBAC), as shown.
Draw a line through points Kƍ and Lƍ that intersects line DB at point Gƍ and
line EF at point Hƍ.
The diagonals BF and DE of the major root-two rectangle (DBEF) reveal a pattern of
six squares and five root-two rectangles (fig. 36).
172 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
D
C'
G'
B
Y
G
C
F
J'
I'
K'
L'
Z
H
A
D'
H'
E
Fig. 36
Design applications
x
x
Locate the root-two rectangle (DBEF) of sides 1 and ¥2.
Construct a diagonal grid composed of
o the diagonals of the major root-two rectangle (DBEF)
o the diagonals of the two reciprocals (BHGD and HEFG)
o the diagonals of the two squares (DBAC and EFYZ). (See figure 37.)
D
Y
G
C
F
B
Z
H
A
E
Fig. 37
Figures 38–42 illustrate patterns of proportional areas based on the ratio 1 : ¥2.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 173
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
174 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
Rectangles based on root-two proportions
Besides the square and the root-two rectangle, the designer may use a variety of
quadrilateral figures when composing root-two patterns of dynamic symmetry.
The rectangle formed by a square plus a root-two rectangle is in the ratio 1:1 + ¥2 or
1:2.4142… (fig. 43).
The rectangle formed by the reciprocal of a root-two rectangle plus a square is in the
ratio 1 : 1 + 1/¥2 or 1: 1.7071… (fig. 44).
The rectangle formed by a square plus the reciprocal of a root-two rectangle on either
side is in the ratio 1 : 1 + 2/¥2 or 1 : 2.4141… (fig. 45).
Fig. 43
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 175
Fig. 44
Fig. 45
176 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications
Conclusion
Ratios inherent in geometric forms offer a rich vocabulary for achieving unity among a
diversity of elements. But such systems cannot guarantee good design any more than rules
of scale ensure good musical composition. They are mere theories that must be judged by
how they serve. No ordering system has merit that does not inform the situation at hand.
The capacity of proportional techniques to support and enhance specific cultural and
functional requirements remains a matter of individual talent.
Notes
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
“Dynamic symmetry” appears in root rectangles based on square root proportions. The edge
lengths of such rectangles are incommensurable and cannot divide into one another. But a
square constructed on the long side of the rectangle can be expressed in whole numbers,
relative to a square constructed on the shorter side [Hambidge 1960, 22–24; 1967, 17–18].
See [Fletcher 2007] for more on dynamic symmetry and its fundamental components.
Hambidge believes the artistic application of dynamic symmetry disappeared after the
Classical Greek period of the sixth and seventh centuries BC, but its theoretical expression
remained in Euclidean geometry. He speculates that the dynamic proportioning of
rectangular forms evolved from the ancient Egyptian practice of “cording the temple” to lay
out building plans, including the cord’s division into twelve (3 + 4 + 5) equal units to form a
right angle [Hambidge 1920, 7–12]. Edward B. Edwards argues that the artistic application
of dynamic symmetry persisted well beyond classical Greece, in numerous styles of ornament
[1967, viii–ix]. See also [Scholfield 1958, 116–119].
See [Fletcher 2007, 329–334] to draw the construction.
The construction is taken from [Edwards 1967, 10–11].
See [Fletcher 2007, 345–346] to draw the construction.
The diagonal is the straight line joining two nonadjacent vertices of a plane figure, or two
vertices of a polyhedron that are not in the same face. The reciprocal of a major rectangle is a
figure similar in shape but smaller in size, such that the short side of the major rectangle
equals the long side of the reciprocal [Hambidge 1967, 30, 131]. See [Fletcher 2007, 336]
and [Hambidge 1967, 33–37] for more on these elements.
A root-two rectangle is in the ratio 1 : 1.4142… . Its reciprocal (¥2/2 : 1) is in the ratio
0.7071… : 1. A root-three rectangle is in the ratio 1 : 1.732… . Its reciprocal (¥3/3 : 1) is in
the ratio 0.5773… : 1. A root-four rectangle is in the ratio 1 : 2.0. Its reciprocal (¥4/4 : 1) is
in the ratio 0.5 : 1. A root-five rectangle is in the ratio 1 : 2.236… . Its reciprocal (¥5/5 : 1) is
in the ratio 0.4472… : 1.
For more on the pole or eye, see [Fletcher 2004, 105–106].
The construction applies Euclid’s method of obtaining the mean proportional of two given
lines [1956, II: 216 (bk. VI, prop. 13)].
For the geometrical proof, see [Hambidge 1967, 46].
In Hambidge’s system, each rectangle has a reciprocal and each rectangle and reciprocal have
complementary areas. The complementary area is the area that remains when a rectangle is
produced within a unit square. If the rectangle exhibits properties of dynamic symmetry, its
complement will also. See [Fletcher 2007, 347–354] and [Hambidge 1967, 128].
Rectangles are similar if their corresponding angles are equal and their corresponding sides are
in proportion. Similar rectangles share common diagonals.
See [Hambidge 1920, 19–20; 1960, 35].
See [Hambidge 1967, 40–47].
See [Fletcher 2005b, 56–62] for similarities to the Sacred Cut.
See [Schneider 2006, 30-32, 34] for similar constructions.
NEXUS NETWORK JOURNAL – VOL. 10, NO. 1, 2008 177
References
EDWARDS, Edward B. 1967. Pattern and Design with Dynamic Symmetry: How to Create Art Deco
Geometrical Designs. 1932. New York: Dover.
EUCLID. 1956. The Thirteen Books of Euclid’s Elements. Thomas L. Heath, ed. and trans. Vols. I–
III. New York: Dover.
FLETCHER, Rachel. 2004. Musings on the Vesica Piscis. Nexus Network Journal 6, 2 (Autumn
2004): 95–110.
———. 2005a. Six + One. Nexus Network Journal 7, 1 (Spring 2005): 141–160.
———. 2005b. The Square. Nexus Network Journal 7, 2 (Autumn 2005): 35–70.
———. 2007. Dynamic Root Rectangles. Part One: The Fundamentals. Nexus Network Journal 9,
2 (Spring 2007): 327–361.
HAMBIDGE, Jay. 1920. Dynamic Symmetry: The Greek Vase. New Haven: Yale University Press.
______. 1919–1920. The Diagonal. 1, 1–12 (November 1919–October 1920). New Haven: Yale
University Press: 1–254.
______. l960. Practical Applications of Dynamic Symmetry. 1932. New York: Devin-Adair.
______. l967. The Elements of Dynamic Symmetry. 1926. New York: Dover.
SCHNEIDER, Michael S. 2006. Dynamic Rectangles: Explore Harmony in Mathematics and Art.
Constructing the Universe Activity Books. Vol. IV. Self-published manuscript.
SCHOLFIELD, P. H. 1958. The Theory of Proportion in Architecture. Cambridge: Cambridge
University Press.
About the Geometer
Rachel Fletcher is a theatre designer and geometer living in Massachusetts, with degrees from Hofstra
University, SUNY Albany and Humboldt State University. She is the creator/curator of two museum
exhibits on geometry, “Infinite Measure” and “Design By Nature”. She is the co-curator of the
exhibit “Harmony by Design: The Golden Mean” and author of its exhibition catalog. In
conjunction with these exhibits, which have traveled to Chicago, Washington, and New York, she
teaches geometry and proportion to design practitioners. She is an adjunct professor at the New York
School of Interior Design. Her essays have appeared in numerous books and journals, including
Design Spirit, Parabola, and The Power of Place. She is the founding director of Housatonic River
Walk in Great Barrington, Massachusetts, and is currently directing the creation of an African
American Heritage Trail in the Upper Housatonic Valley of Connecticut and Massachusetts.
178 RACHEL FLETCHER – Dynamic Root Rectangles Part Two: Root-Two Rectangles and Design Applications