CHARACTERISTICS AND
INTERRELATIONS OF FORMS
ARCH 115
Instructor: F. Oya KUYUMCU
Classification of
Geometrical Shapes for
Architectural Design
• In Geometry, we can make classification of geometrical shapes
according to their dimensions for architectural design.
In this, there are six main groups;
-
O Dimensional Forms with no surface (Point)
1 Dimensional Forms with 1Dimensional Surface (Line, Angle)
2 Dimensional Forms with 1Dimensional Surface (Circle, Ellipse,
Parabola, Hyperbola, Polygons)
3 Dimensional Forms with 1 Dimensional Surface (Helix,
Hemihelix)
3 Dimensional Forms with 2 Dimensional Surfaces (Platonic
Solids, Quadrics)
4 Dimensional Forms with 3 Dimensional Surfaces (Polychrons)
2 Dimensional Forms with
1Dimensional Surface
(Conic Sections -Circle, Ellipse, Parabola, Hyperbolaand Polygons)
POLYGONS
Polygons are two-dimensional shapes with straight
lines.
There are two kinds of Polygons:
. Irregular Polygons have at least one angle or side
different than the other like rectangle, parellelogram,
trapezoid and kite etc.
. Regular Polygons have equal angles and sides.
We can subdivide these into two another group;
. A Convex Polygon has no angles pointing inwards.
(No internal angle more than 180º.)
. A Concave Polygon has any internal angle more
than 180º.
We can subdivide all of these into two another group;
. A Simple Polygon has only one boundary. It
doesn’t cross over itself.
. A Complex Polygon intersect itself.
According to this classification, we can give more
examples;
Irregular Hexagon
Concave Octagon
Complex Polygon
TYPES AND NAMES OF REGULAR POLYGONS
In this course, when we say Polygons, we generally talk about the
regular polygons.
Regular polygons are the basis of prisms, antipirisms, pyramids, and
the classical solids as well as numerous plane figures , patterns and
designs important in architecture.
The characteristics of regular polygons are well known.
. Symmetrical
. Equilateral
. Equiangular
. can be inscribed in a circle,
. can any number of sides from two to infinity.
When the number of sides becomes very great, a regular polygons
becomes a circle, the shape that encloses a given area with the
least perimeter.
We will give the list below, for showing all polygons in a one table.
On the other side, we will learn the way of giving names, to the
polygons. If we try to memorize all of this without using logic, we
can’t.
After the giving properties of triangle is the base of the other
polygons, we will construct all of these polygons from the triangle
(trigon) to the tetrakaidecagon (12-gon).
Hendecagon
Didecagon
Tridecagon
Making Polygon Names
• You can make names using this method:
⟹
Example: A 76 sided polygon is a Heptacontahegzagon
However, for polygons with more than 13 sides,
we can write “13-gon” or “14-gon”…. “100-gon”.
TRUNCATION
In geometry, the term means the changing of one
shape into another by altering the corners.
Truncation is a process by which a smooth
transition with any number of intermediate steps can
be made between one shape and another.
. The equilateral triangle and hegzagon (3/6 series)
. The square and the hegzagon (4/8 series)
. Pentagon and decagon (5/10 series)
Are primary and secondary polygons linked by the
process of truncation.
TRIANGLE
There are three special names given to
triangles that tell how many sides are equal.
. Equilateral Triangle; three equal sides
three equal angles
. Isosceles Triangle; two equal sides
two equal angles.
. Scalene Triangle; no equal sides
no equal angles.
Triangles can also have names that tell you
what types of angle is inside.
. Acute Angles; all angles are less than 90º
. Right Angles; has a right angle 90º
. Obtuse Angles; has an angle more than 90º
Sometimes, a triangle will have two names;
For example; right isosceles triangle has a right
angles and two equal angles.
FORMULAS FOR TRIANGLE
The perimeter of triangle is; just add up three
sides.
The area of a triangle is ; The area is half of the
base times height.
area= ½ X b x h
The Triangle Inequality Theorem;
Any side of a triangle must be shorter than the
other two sides added together.
When one side is longer, the other two sides
don't meet!
When one side is equal to the other two sides it
is not a triangle.
THE CENTERS OF A TRIANGLE
• The centroid is the point of intersection of the
medians ( is a line from a vertex to the mid point
of opposite side).
• The orthocenter is the point of intersection of
the altitutes.
• The incenter is the point of intersection of the
angle bisectors.
• The circumcenter is the point of intersection of
the perpendicular bisectors of the sides.
GOLDEN TRIANGLE
• Golden Triangle is an isosceles triangle in
which the ratio between legs and base is the
golden ratio (ɸ).
• There are two types of golden triangles;
acute and obtuse depending on whether we
take the ratio of leg to base or the ratio of
base to leg.
• Euclid explain how to construct a golden
triangle. He states that the object is “To
construct an isosceles triangle having each
of the angles at the base double measure
remaining one”.
AO/s = AO/AB = ɸ
• In isosceles triangle AOB, the ratio of side to base is
ɸ. It is called an acute golden triangle.
• Further, triangle OBD is also isosceles. The ratio of
its base OB to a side OD is ɸ. It is called an obtuse
golden triangle.
• Triangle ABD is also an acute golden triangle.
•
There is another way look at this figure. Start with golden
acute triangle ABD and append the obtuse golden triangle
OBD to it. This will create acute golden triangle OAB.
Therefore a obtuse golden triangle can be thought of as a
“gnomon”, a figure added to another to get a larger figure
having the same shape. You could than append obtuse
golden triangles OAE, BEF and so forth to get ever larger
acute golden triangles. Sometimes this figure is called,
wbirling golden triangles.
SUM OF THE ANGLES OF TRIANGLE BY PAPER FOLDING
• ABC, as shown in Figure, perform these steps.
• Fold an altitude BD.
• Fold vertices A, B, and C onto D.
• Does this make clear how A, B, and C, are
related.
• What do you conclude?
• What are the final shape and dimensions of
the figure after folding?
SOME CONCEPTS RELATED WITH TRIANGLE
“CONGRUENT TRIANGLES”
• Two triangles that are identical except for location
and orientation are said to be congruent.
• One of the triangles can be made exactly overlay the
other by sliding, turning or flipping over, or by any
combination of these motions.
turning
• Two triangles are congruent if any one of the
following is true.
flipping
a.
b.
c.
Two angles and side of one are equal to two angles and a side of
the other.
Two sides and the included angle of one are equal, respectively
to two sides and the included angle of the other.
Three sides of one are equal to the three sides of the other.
sliding
SOME CONCEPTS RELATED WITH TRIANGLE
“REFLECTIONAL AND ROTATIONAL SYMMETRY IN TRIANGLES”
•
When half of a figure is the mirror image
of the other half, the figure has “Mirror
Symmetry”
or
“Reflectional
Symmetry”. If we reflect the triangle an
axis of symmetry, this called reflectional
symmetry in a triangle.
Reflectional Symmetry
•
When a figure has rotated about a point
O, if its appearance is unchanged by a
rotation, we can talk about “Rotational
Symmetry” of triangle.
Rotational Symmetry
SOME CONCEPTS RELATED WITH TRIANGLE
“TRANSFORMATION or ISOMETRY in TRIANGLE”
•
Transformation is a motion that carries each triangle from
P point in a plane into, to another P’ point into same
plane.
•
There are two kind of transformations:
a. Congruent Transformations (or Isometries)
b. Similarity Transformations
rotation
ISOMETRIES
Rotation: A turning through an angle
Reflection: A mirror image about some mirror line
Translation: A shift in position of a given distance
Glide Reflection: A combination of translation and reflection
reflection
translation
SOME CONCEPTS RELATED WITH TRIANGLE
“SIMILAR TRIANGLES”
• Two angles are similar if the angles of one
triangle equal the angles of the other.
• Two triangles are said to be similar if they
have the same shape, even if one is larger
than the other or is rotated, translated, or
reflected abou some axis.
• The operations that transform one triangle into
another similar triangle are called “Similarity
Transformations”.
• That means that the angles of one of the
triangles must equal the angles of the other.
SCALE FACTOR FROM THE SIMILAR TRIANGLE
a
• If two triangles are similar, their
corresponding sides are in proportion. The
ratio between corresponding sides is called
the scale factor.
a/b = e/f
Measuring inaccessible distances by
means of similar triangle is not new. Figure
is taken from the Leonardo, shows an
instrument being used to measure facade.
.
Areas of similar triangles are proportional
to the square of the scale factor.
e
f
b
SOME THEOREMS RELATED WITH TRIANGLE
“NAPOLEON’S THEOREM”
• If equilateral triangles are drawn externally or
internally on the sides of any triangle, their
centers form an equilateral triangle.
• The inner Napoleon triangle has same center
as the external triangle.
SOME THEOREMS RELATED WITH TRIANGLE
“MORLEY’S THEOREM”
• The lines of three section of vertices of a
triangle intersect the form an equilateral
triangle.
( Morley theorem was discovered in 1899 by Frank
Morley, whose son Christopher is well known fiction
author.)
• Homework: Try to demonstrate
theorem by constructing methods.
this
SOME THEOREMS RELATED WITH TRIANGLE
“PAPPUS’ THEOREM”
• If point A, C, and E lie on one straight line,
and B, D,and F lie on another line, then
the points of intersection of AB with EF, BC
with DE, and CF with AD lie on a third
straight line.
SOME THEOREMS RELATED WITH TRIANGLE
“DESARGUES’ THEOREM”
•
Desargues’ Theorem concerns two triangles
that are in perspective shown in upper figure.
•
Two triangles are said to be perspective
(Triangles in perspective) from a point if the
three lines joining pairs of corresponding
points meet in a single point.
•
Desargues theorem goes on to say that;
•
If two triangles are perspective from a point,
then the three points of intersection of
corresponding sides extended, lie on a
straight line.
SOME CONSTRUCTIONS RELATED WITH TRIANGLE
“SQUARE-ROOT SPIRAL”
• Construct the square root spiral as shown
in Figure.
• Start with an isosceles right triangle with the
legs equal to 1.
• On its hypothenuse, construct another right
triangle with the other leg equal to 1.
• Continue adding right triangles as desired.
• Shows that the lenghts of the segments
radiating from the center to the spiral are
equal to the square roots of 1, 2, 3, and so
on.
SOME CONSTRUCTIONS RELATED WITH TRIANGLE
“TRIANGLE WITHIN A SQUARE FROM LAVLOR”
• Do the following construction from Lavlor of a
triangle within a square, as shown in Figure.
Bisect two adjacent sides of the square and
connect each to the opposite vertex of the
square.
a. If the sides of the square is one unit, what is
the length of each diagonal?
b. How is the length of each diagonal related to
the golden ratio?
c. Identify the kind of triangle formed.
SOME CONSTRUCTIONS RELATED WITH TRIANGLE
“THE LUTE OF PYTHOGORAS”
• Do the following construction from Lute of
Pythogoras shown in Figure.
• Given Golden Triangle ABC, perform the
following.
a. From B, draw an arc with radius BC to locate
E.
b. Similarly locate D.
c. From D, and E, draw arcs of radius DE to
locate F and G.
d. Repeat the construction upwards a few more
times.
A
B
C
SOME CONSTRUCTIONS RELATED WITH TRIANGLE
“PAPPUS CONSTRUCTION FOR THREE MEAN””
TRIGONOMETRY
Trigonometry; is all about triangle.
From Greek Trigonometry from Trigon (triangle)
+Metron (measure)
The triangle of most interest is the right-angled
triangle.
The right angle is shown by the little box in the
corner.
We usually know another angle θ (Theta).
And we give names to each side:
. Adjacent is adjacent (next to) to the angle θ.
. Opposite is opposite the angle θ
. Hypotenuse is the longest side θ
PHYTAGORA’S THEOREM
Pythagora’s Theorem;
When the triangle has a right angle 90º,.. and
squares are made on each of three sides,..
Then the biggest square has the exact same
area as the other two squares put together.
According to the Pytagorean Theorem,
In a right angled triangle, the square of the
hypotenuse is equal to the sum of the squares
of the other two sides.
The converse of is also true;
If the square of one side of a triangle equals the
sum of the squares of the other two sides, then
we have a right angle.
a2 + b2 = c2
3-4-5 TRIANGLE (ROPE-KNOTTER’S TRIANGLE)
The rope-stretcher’s triangle is also called the
3-4-5 right triangle, the Rope-Knotter’s
Triangle and the Phytagorean Triangle.
One tool they may have used is a ropeknotted into12 sections stretched out to form a
3-4-5 triangle. Does it produce a right angle?
For the 3-4-5 triangle;
5²=3²+4²
25=9+16
It checks, showing a rope knotted like this will
give a right angle.
PROVING THEOREM WITH FORMS
• Draw a right angled triangle on the paper,
leaving plenty of space.
• Draw a square along the hypothenuse.
• Draw the same sized square on the other
side of the hypothenuse.
• Draw lines as shown on the figure
• Arrange them so that you can prove that the
big square has the same area as the two
squares on the other sides.
ANOTHER PROVING METHOD
• Here is one of the oldest proofs that the square on the long side has
the same area as the other squares.
SETTING OUT RIGHT ANGLES: THE 3-4-5 METHOD
The base line is defined by the poles (A) and (B) and a
right angle has to be set out from peg (C). Peg (C) is on
the base line.
Three persons hold the tape the way it has been
explained above.
- The first person holds the zero mark of the tape
together with the 12 m mark on top of peg (C).
- The second person holds the 3 m mark in line
with pole (A) and peg (C), on the base line.
- The third person holds the 8 m mark and, after
stretching the tape, he places a peg at point (D).
The angle between the line connecting peg (C) and
peg (D) and the base line is a right angle.
Line CD can be extended by sighting ranging
poles.
TRIGONOMETRIC IDENTITIES
• The Trigonometric Identities are equations
that are true for Right Angled Triangles.
We are soon going to be playing with all sorts of
functions and it can get quite complex, but
remember it all comes back to that simple triangle
with:
. Angle “θ”
. Hypotenuse
. Adjacent
. Opposite
SINE, COSINE and TANGENT
• Trigonometry is good at find a missing side or angle in a triangle.
• The special functions Sine, Cosine and Tangent help us!
• They are simply one side of a triangle divided by another.
For any angle “θ” :
Sine Function:
sin (θ) =Opposite / Hypotenuse
Cosine Function:
cos (θ) =Adjacent / Hypotenuse
.
Also, if we divide Sine by Cosine we get:
.
So, we can also say:
tan (θ) = sin (θ) / cos (θ) (First Trigonometric Identity)
Tangent Function:
tan (θ) =Opposite / Adjacent
COSECANT, SECANT and COTENGANT
•
We can also divide “the another way around” .(such as
adjacent/opposite instead of opposite/ adjacent)
Cosecant Function:
Secant Function:
Cotangent Function:
csc(θ) = Hypotenuse / Opposite
sec(θ) = Hypotenuse / Adjacent
cot(θ) = Adjacent / Opposite
Because of all that we can say;
sin(θ) = 1/csc(θ)
cos(θ) = 1/sec(θ)
tan(θ) = 1/cot(θ)
And the another way around;
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)
And also we have;
cot(θ) = cos(θ)/sin(θ)
MAGIC HEXAGON FOR TRIG IDENTITIES
• This hexagon is a special diagram
to help you remember some Trigonometric
Identities.
• To building it;
tan(x) = sin(x) / cos(x)
think "tsc !“
• Then add:
cot (which is cotangent) on the opposite side of the
hexagon to tan
csc (which is cosecant) next, and
sec (which is secant) last
•
OK, we have now built our hexagon, what do we get out of it?
Well, we can now follow "around the clock" (either direction) to get all the
"Quotient (bölüm,katsayı) Identities":
•
Clockwise
tan(x) = sin(x) / cos(x)
sin(x) = cos(x) / cot(x)
cos(x) = cot(x) / csc(x)
cot(x) = csc(x) / sec(x)
csc(x) = sec(x) / tan(x)
sec(x) = tan(x) / sin(x)
•
Counterclockwise
cos(x) = sin(x) / tan(x)
sin(x) = tan(x) / sec(x)
tan(x) = sec(x) / csc(x)
sec(x) = csc(x) / cot(x)
csc(x) = cot(x) / cos(x)
cot(x) = cos(x) / sin(x)
• The hexagon also shows that a function between any two functions
is equal to them multiplied together (if they are opposite each other,
then the "1" is between them):
• Example: tan(x)cos(x) = sin(x)
• Example: tan(x)cot(x) = 1
• Some more examples:
sin(x)csc(x) = 1
tan(x)csc(x) = sec(x)
sin(x)sec(x) = tan(x)
•
You can also get the "Reciprocal (karşılık,müteakip) Identities", by going
"through the 1“.
•
Here you can see that
sin(x) = 1 / csc(x)
•
Here is the full set:
sin(x) = 1 / csc(x)
cos(x) = 1 / sec(x)
cot(x) = 1 / tan(x)
csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
tan(x) = 1 / cot(x)
• AND we also get these:
• Examples:
sin(30°) = cos(60°)
tan(80°) = cot(10°)
sec(40°) = csc(50°)
• The Unit Circle shows us that
sin2 x + cos2 x = 1
• The magic hexagon can help us remember
that, too, by going clockwise around any of
these three triangles:
• And we have:
sin2(x) + cos2(x) = 1
1 + cot2(x) = csc2(x)
tan2(x) + 1 = sec2(x)
• You can also travel counterclockwise around a
triangle, for example:
1 - cos2(x) = sin2(x)
TRIANGLE IDENTITIES
•
The triangle identities are equations that are true for all
triangles (they don't need to have a right angle). For the
identities involving right angles triangles see
Trigonometric Identities.
•
The Law of Sines (also known as The Sine Rule) is;
•
The Law of Cosines (also known as The Cosine Rule) is;
•
The Law of Tangent is;
SOLVING TRIANGLES PROBLEMS
"Solving" means finding missing sides and angles.
• What is the sine of 35°?
sin(35°) = Opposite / Hypotenuse = 2,8/4,9 = 0,57...
• What is the missing length here?
Sine is the ratio of Opposite / Hypotenuse
sin(45°) = 0,7071...
Opposite length = 20 × 0,7071... = 14,14
SOLVING SIX DIFFERENT TYPES OF TRIANGLES
PROBLEMS
1. AAA Types:
This means we are given all three angles
of a triangle, but no sides.
A AAA triangle is impossible to solve
further since there are is nothing to show
us size ... we know the shape but not how
big it is.
We need to know at least one side to go
further.
SOLVING SIX DIFFERENT TYPES OF TRIANGLES
PROBLEMS
2.
AAS Types:
This mean we are given two angles of a
triangle and one side, which is not the side
adjacent to the two given angles.
Such a triangle can be solved by using
Angles of a Triangle to find the other angle,
and The Law of Sines to find each of the
other two sides.
SOLVING SIX DIFFERENT TYPES OF TRIANGLES
PROBLEMS
3.
ASA Types:
This means we are given two angles of a
triangle and one side, which is the side
adjacent to the two given angles.
In this case we find the third angle by using
Angles of a Triangle, then use The Law of
Sines to find each of the other two sides.
SOLVING SIX DIFFERENT TYPES OF TRIANGLES
PROBLEMS
4. SAS Types:
This means we are given two sides and the
included angle.
For this type of triangle, we must use The
Law of Cosines first to calculate the third
side of the triangle; then we can use The
Law of Sines to find one of the other two
angles, and finally use Angles of a Triangle
to find the last angle.
SOLVING SIX DIFFERENT TYPES OF TRIANGLES
PROBLEMS
5.
SSA Types:
This means we are given two sides and one
angle that is not the included angle.
In this case, use The Law of Sines first to
find either one of the other two angles, then
use Angles of a Triangle to find the third
angle, then The Law of Sines again to find
the final side.
SOLVING SIX DIFFERENT TYPES OF TRIANGLES
PROBLEMS
6. SSS Types:
This means we are given all three sides of a
triangle, but no angles.
In this case, we have no choice. We must
use The Law of Cosines first to find any one
of the three angles, then we can use The Law
of Sines (or use The Law of Cosines again) to
find a second angle, and finally Angles of a
Triangle to find the third angle.
USING TRIANGLE IN ART
•
Triangles and triangular designs can be found
throughout the history of art and architecture.
•
One common usage of triangles is to represent
the Trinity in Christianity. God the Father, the
Son, and the Holy Sipirit. For exmple, a halo, or
a nimbus, is a zone of light often depicted
behind the head or or body of a sacred figure in
a religious paintings and sculptures. A triangular
halo, such as shown in “Creation of Adam”
Jacoba della Quercia is used only when
representing God the Father.
•
Triangular windows are common in churches,
perhaps also representing the Trinity.
USING EQUILATERAL TRIANGLE IN ART
• The right triangle of equilateral is the 30°60° used for a multitude of patterns, grids
and designs, often with the illusion of depth.
• The grid of equilateral triangle, the 60° grid,
is the geometric basis for many patterns
including the well-known hegzagonal pattern
of tiles.
• For centuries, people have found equilateral
triangle to be spiritually significant and a
powerful symbol. If there is a mystique
associated with the equilateral triangle, it
must also be the mystique of the square root
of three.
• There are seemingly endless variations on
the rectangle of equilateral triangle, the
equilateral triangle itself, and the hexagon.
The common denominator is the square root
of three which, like the key of C minor in a
symphony, can be the principal tonality in
an architectural composition.
1
1
√7
√3
2
ARCHITECTURAL USE OF
TRIANGLE
• Equilateral triangle is the first of geometric
primaries, is an isosceles triangle.
• They are part of regular polygons, the sides
of pyramids and antipirism, and the interior
partitions of the classical solid.
• In architecture, they are seen as roof slopes,
low for rain, steep for snow.
• A frame-dwellings, ancient and modern are
prime examples.
• Of the isosceles triangles, the equilateral
triangle is the most compact, most versatile,
the most frequently used.
• Equilaterals, and
shapes.
all
triangles
are
rigid
• When shapes are cut from cardboard this
aspect is of no significance, but when the
shapes are framed in steel, aluminum, or
wood, rigidity is important. The simplest
connection of the framing members is a pin,
which permits rotation. When pin connections
are used, an equilateral is rigid, but a square,
pentagon or other polygon will collapse under
pressure from the side.
• For this reason, equilateral and other triangles
frequently are seen in bridges and trusses and
as an alternative method of bracing in
building.
Like in Alcoa Office Building in San Fransisco by Skidmore, Owings,
1963.
The Robe-stretchers Triangle and Kepler Triangle in
Egptian Pyramid
•
Let's look to the pyramids. If we take a crosssection through a pyramid we get a triangle. If
the pyramid is the Great Pyramid, we get the socalled Egyptian Triangle. It is also called The
Triangle of Price, and the Kepler Triangle.
•
Kepler Triangle is a right triangle in which the
lenghts of two sides and hypothenues form a
geometrisc progression i.e.
hypotenues / larger side=larger side /
smaller side
height = 146.515 m, and base = 230.363 m
half the base is; 230.363 ÷ 2 = 115.182 m
So ; s 2 = 146.515 + 115.182 2 = 34,733 m2
s = 186.369 m
186.369/146.515 = 146.515 / 115.182
1.27201= 1.27203