Maths in Art and Architecture Why Maths? Comenius project

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Maths in Art and Architecture
Maths in Art and
Architecture

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THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWHHYY MMAATTHHSS??
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This project has been funded with support from the European Commission.
This publication reflects the views only of the author, and the
Commission cannot be held responsible for any use which may be made of the
information contained therein.

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Mathematics and art have a long historical relationship. We know that line, shape, form,
pattern, symmetry, scale, and proportion are the building blocks of both art and Maths.
Geometry offers the most obvious connection between the two disciplines. Both art and
Mathematics involve drawing and the use of shapes and forms, as well as an understanding
of spatial concepts, two and three dimensions, measurement, estimation, and pattern.
The parallels between geometry and art can be seen in many works of art.
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1. The Fibonacci sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946,
17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, ….
This famous sequence of numbers is present in a variety of fields: in art, in nature, botany,
zoology, but especially in relation to the golden ratio phi and the golden spiral. Made its
appearance in the "Liber Abaci", but centuries earlier had already been considered by the
Indian mathematician Virahanka and described in 1133 by the scholar Gopal, as a solution
to a problem of metrics related to poetry.
Fibonacci developed his sequence to solve the following problem concerning the breeding
of rabbits:
"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many
pairs of rabbits can be produced from the initial torque in a year assuming that in a month
each pair produces a new pair can reproduce itself in the second month? "
To solve this question Fibonacci assumed that each pair of rabbits:
a) starts to generate after the first month of age
b) generates a new pair every month
c) never dies.
He proceeded by considering a single pair that
after the first month becomes mature and
generates another couple. After the second
month in a mature couple produces another
young couple while the former becomes mature
young couple (couples are then three). After the
third month in each of the two mature couples
generates a new request while the young couple
becomes mature, so couples are five. After the
fourth month the three mature couples each
generate a new pair and the two young couples
become mature. At this point, it is now clear
how one can calculate the total number of pairs

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in each month but also the number of couples young and adult ones. In turn, the number
of young couples to mature couples generate a Fibonacci sequence.
At this point analyzing the diagram below we can see how the numbers of pairs in each
month go to form the Fibonacci sequence:
Pattern inherent to the problem of rabbits in orange are represented mature couples,
young ones in blue.
2. The definition of Fibonacci sequence
Taking the cue from the previous issue of the rabbits, and extending, the Fibonacci
sequence can be defined as follows:
the first two elements are 1, 1;
every other element is the sum of the two preceding it.
Calling F (n) the Fibonacci sequence, we have the following mathematical definition:
F(1) = 1
F(2) = 1
F(n) = F(n-2)+F(n-1) per n = 3, 4, 5, ...
According to this definition it is assumed conventionally F(0) = 0.
So the sequence of Fibonacci:
0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Note that the function F (n) is recursive, that is defined in terms of the function itself.

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3. The particularities of the sequence
The sequence is characterized by numerous and curious feature:
3.1 The square of a Fibonacci number less than the square of the second number is always
a previous number of the sequence
3.2 The greatest common divisor of two Fibonacci numbers is still a Fibonacci number
3.4 Adding an odd number of products of successive numbers in the sequence, the three
products as 1x1, 1x2, 2x3, you get the last square Fibonacci number present in the products
in question. Indeed (1x1) + (1x2) + (2x3) = 2 + 1 + 6 = 9, is the square of the last number
that appears in the previous product (in this case 3). Similarly, we can analyze the series of
seven products: (1x1) + (1x2) + (2x3) + (3x5) + (5x8) + (8x13) + (13x21) = 1 + 2 + 6 + 15 +
40 + 273 +104 = 441 which is just the square of the last number that appears in the
product. This property can be represented geometrically as shown by the figure:
An odd number of rectangles with sides equal to a number of terms of the Fibonacci
sequence are exactly placed in a square the side of which coincides with a side of the larger
rectangle.
3.5 The sequence is also connected with the triangle Tartaglia which is a geometric
arrangement in the shape of a triangle of binomial coefficients, is the coefficients of the
expansion of the binomial (a + b) raised to any power n.

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From this triangle can be drawn Fibonacci numbers,
adding the numbers of the diagonals as shown in the
figure: so we get from the first line 1, from the second
still 1, then 2, 3, 5, 8, 13, ...,
The sequence has many other features and even today
many mathematicians try to find new properties
connected to it.
4. The Fibonacci sequence and the golden section
With the golden section indicates, usually, in art and mathematics the relationship
between two unequal magnitudes of which the largest is the mean proportional between
the child and their sum:
((a + b): a = a: b).
This ratio is approximately 1.618. Apparently an irrational number like everyone else, but
its mathematics and geometry and the abundant presence in various natural settings have
made a canon of harmony and beauty that has always attracted artists and intellectuals of
all time.
It is thought that the first to run into this relationship (1.618), also referred to by the Greek
letter φ (phi), was Hippasus from Metaponto, one of the members of the Pythagorean
school, that around the fifth century BC discovered the existence of this number that
belonged neither the integers nor to those that can be expressed as a ratio of integers
(fractions, rational numbers). This news was a real shock to the followers of Pythagoras, so
that the discovery that there are numbers that, as the golden ratio, extending indefinitely
without any repetition or pattern caused a real philosophical crisis. He welcomed this
discovery with great anguish, so much to consider, probably, as an imperfection cosmic
secret to keep as much as possible.
a b

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The golden section is closely related to the Fibonacci sequence in fact, the relationship
between a term and its previous closer and closer to 1.618.
AB:AP=AP:BP=1.618
PB:AP=AP:AB=0.618
5. Figures with the Golden Section
There are various figures that can be built with the golden section (rectangles, triangles,
pentagons ...); among these the most important is surely one of the golden rectangle,
a rectangle constructed with the particular parameters of the Fibonacci sequence.
5.1 The golden rectangle
With the use of the golden section it is possible to build a very special type of rectangle of
enjoying unusual geometric properties. This rectangle is called the golden rectangle and
has a side that is the golden section of the other. Aureus that is the only rectangle that
allows, by removing a square from his area, to obtain a rectangle similar to the first; a
procedure which can be repeated many times until converging at a point which is exactly
the intersection between the first and the second golden rectangle. this point has been
called "the eye of God," alluding to the divine properties attributed to f.

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DEMONSTRATION: If ABCD is a golden rectangle, then by definition we have:
AD: DC = DC (AD-DC)
If we divide the rectangle in order to obtain a square then you have: ED = DC from which
we get:
AD: ED = ED: AE
Applying the property of decomposing is obtained:
(AD-ED):ED=(ED-AE):AE
Knowing that ED = EF we can write the following proportion:
AE: EF = (EF-AE): AE
And finally from the property of the inverting you get:
EF: AE = AE: (EF-AE)
where AE is the golden section EF AEFB then the new rectangle is a golden rectangle.

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5.2 Fibonacci spiral and golden spiral
Since the golden rectangle is constituted
by the infinite square exists the
possibility to create inside an endless
succession of square and then a spiral,
said spiral Fibonacci, able to
approximate the golden.
Often, inaccuracy , we tend to share
that with authentic spiral golden spiral,
but it is a mistake : the Fibonacci spiral,
in fact , is given by the union of an
infinite number of quarters in
circumference, the true mind golden
spiral is a special type of a logarithmic
spiral , which overlaps only partially to that of the Fibonacci sequence. The degree of
approximation, however , is so good that it hardly be noticed by eye the difference between
the two.
What , however , have in common is the fact both spirals of screwing asymptotically
towards the intersection between the diagonals that can be obtained within the golden
rectangles ; a meeting that was called by Clifford Pickover the eye of God, just for the fact
that everything seems to focus around this point , from the spirals to the diagonals and the
sequence of squares. Interestingly , then, as not only the diagonals real intertwine in this
particular point of the golden rectangle , but also other more straight line connecting major
points of this swirling centralization.

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6. The Fibonacci sequence and the Golden Section around us
Both the geometric and mathematical properties of this relationship, the frequent
repetition of the proportion in various natural settings, seemingly unrelated to each other,
have impressed the centuries the mind of man, who arrived in time to overtake an ideal of
beauty and harmony , going to look for it and, in some cases, to re-create the environment
as a canon of beauty; testimony is perhaps the story of the name in more recent times has
assumed the titles of "gold" (the golden section) or "divine" (divine proportion), just to
demonstrate the fascination.
In architecture and paintings
Famous is the representation of the
“Uomo Vitruviano” by Leonardo in which
a person is inscribed in a square and a
circle. In the square, the height of man
(AB) is equal to the distance (BC) between
the ends of the hands with arms
outstretched. The straight line passing
through the xy navel divides the sides AB
and CD in exactly the golden ratio to one
another. The navel is the center of the
circle that inscribes the human person
with arms and legs outstretched. The
position corresponding to the navel is in
fact considered to be the center of gravity
of the human body.
The Egyptian pyramid of Cheops has a base of 230
meters and a height of 145: the ratio height / width
corresponds to 1.58 very close to 1.6.
In the megaliths of Stonehenge, the theoretical
surfaces of the two circles of blue and Sarsen stones, are
to one another in the ratio of 1.6.
A famous representation of the human figure in the golden ratio is
also the "Venere” by Botticelli in which you can find several report
aureus (1:1.618). In addition to the height from the ground and the
total height of the navel, is aureus also the relationship between
the distance of the neck of the femur at the knee and the length of
the entire leg or the ratio between the elbow and the tip of the
middle finger and the length of the entire arm.

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The Parthenon contains many golden
rectangles. The result is a harmonious
aspect, which inspires a deep sense of
balance. The projection shows how it has
been built on
a golden rectangle, so that the width and
height are in the ratio: F: 1 (the F is such
in honor of Phidias, architect of the
Parthenon).
The plan of the Parthenon in Athens is
a rectangle with sides of size such that the length is equal to the root of 5 times the
width, while the architrave in front the golden rectangle is repeated several times.
His plan shows that the Parthenon was built on a rectangle 'square
root of 5', is that the length of the root
is 5 times the width.
Golden Rectangles in The Mona Lisa
• the length and the width of the painting itself
• the rectangle around Mona's face (from the top of the
forehead to the base of the chin, and from left cheek to
right cheek). Subdivide this rectangle using the line
formed by using her eyes as
a horizontal divider to divide the Golden Rectangle.
• the three main areas of the Mona Lisa, the neck to
just above the hands, and the neckline on the dress to
just below the hands.

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Link to this interactive poster: LINK prepared by Polish students.

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IIII.. GGEEOOMMEETTRRYY AANNDD GGOOTTHHIICC ((WWIINNDDOOWWSS))
Mathematics, in particular geometry, always played a major
role in architecture. In early civilizations the tombs of leaders
had shapes derived from a prism with a square base or
halfsphere.
A real sophistication of geometric forms in architecture can
be found in ancient Greek, Indian or Chinese architecture.
The windows in gothic churches are normally divided in two
sections: one rectangular area which is covered by a second
area formed by two crossed arches.
The basic pattern in Gothic Architecture is the pointed
arch.
Its geometric construction is based on the intersection of two
circles. The circles are tangent continuous to the sides of an arch or a window, given
as two vertical line segments.
Gothic arch with varying excess parameters
a) Four-centered (0.75) b) Pointed arch (1.25)
c) Equilateral (1.0)
1. Construct the baseline AB, and
extend your compass out to the exact
same length.
2. With your compass needle at point
B, construct arc AC.
3. With your compass needle at point
A, construct arc BC.

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The recognizable feature of Gothic is the stonework tracery that decorates vaulting,
rose windows, arcaded cloisters, to simple windows and doorways. Many of the
shapes grow from an interesting variety of other shapes - including triangles,
pentagons, hexagons, circles, or circles within circles.
Window tracery is the very particular type of window decoration found in any
building of Gothic style. Gothic architecture, and especially window tracery, exhibits
quite complex geometric shape configurations. But this complexity is achieved by
combining only a few basic geometric patterns, namely circles and straight lines,
using a limited set of operations, such as intersection, offsetting, and
extrusions.
In the presentation and in the film you can see how these
objects can be created using pure Euclidean geometric
constructions with a straightedge and compass.
Proposition 11.
If two circles touch one another internally, and their
centers are taken, then the straight line joining their
centers, being produced, falls on the point of contact of the
circles.

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Proposition 12.
If two circles touch one another externally, then the
straight line joining their centers passes through the
point of contact.
1. Set out an equilateral triangle. Measure half the length between A-C to find
point D. Now measure half the length of line B-C to find point F. Draw a line
from points B-D and A-F, to find center, O.
2. From center O, extend your compass to point A. Swing around and return to
point A to complete the outer circle. Extend lines B-D and A-F.
3. To construct a horizontal center line, divide A-B to find point E. At point C,
extend the center line down through O-E-N.
4. Now use centers A, B and C to form the three arcs. Extend your compass from
O-S to complete the outer circle.

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The outer, triangular 'piercing' that surrounds the arcs is accomplished by using
center O and one center of each of the three 'eyes'; for example: A, C and F as shown
in the left piercing, above.

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Trefoil - a three-lobed circle or arch formed by cusping. It was used in windows and
arches.
A trefoil combined with an equilateral triangle was a moderately common symbol
of the Christian Trinity during the late Middle Ages in some parts of Europe.
A stylized shamrock, symbol of perpetuity, with the three leaves representing the
past, present and future. It is also sometimes a symbol of fertility and abundance.

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Quatrefoil - a four-lobed circle or arch formed by cusping.
Quatrefoils are not the same as shamrocks, though they do have four leaves; the
leaves of a quatrefoil are more circular and they appear without the stem of a trefoil,
except for very rarely.

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A circle is drawn within the square, the square is divided crossover in four sections.
Half the radius of the inner circle is used as measure for each of the smaller four
circles with overlapping areas.

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Cinquefoil - a five-lobed circle or arch formed by
cusping.

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Sixfoil - a six-lobed circle or arch formed by cusping .
Fish bladder (fischblase)
An ornamental motif of the late Gothic tracery,
reminiscent in form of the air-bladder of a fish.
Despite its organic appearance it results from geometrical
construction by circle. Its simplest shape is two fish-
bladders within one circle that can be constructed
quartering the diameter of the surrounding circle.
Though their construction is easy the effect is amazing.

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The Reuleaux triangle is a constant width curve based on an equilateral triangle.
All points on a side are equidistant from the opposite vertex.
A curve of constant width constructed by drawing arcs from
each polygon vertex of an equilateral triangle between the other
two vertices. The Reuleaux triangle has the smallest area for a
given width of any curve of constant width.
To construct a Reuleaux triangle begin with an equilateral
triangle of side a, and then replace each side by a circular arc
with the other two original sides as radii.

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Tracery is usually the stonework elements that support the glass in a Gothic window
but it may also appear simply as a design element on other surfaces, in which case it
is called blind.
We can find a lot of tracery painted decorations in many buidings in Toruń – the
capital of our province.

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Here you can find the film with the constructions of the elements of gothic windows
in GeoGebra: LINK

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IIIIII.. TTIILLIINNGG AANNDD TTEESSSSEELLLLAATTIIOONNSS
Over 2,200 years ago, ancient Greeks were decorating their homes with tessellations,
making elaborate mosaics from tiny, square tiles. Early Persian and Islamic artists
also created spectacular tessellating designs. More recently, the Dutch artist M. C.
Escher used tessellation to create enchanting patterns of interlocking creatures, such
as birds and fish.
A tessellation is a tiled pattern created by repeating a shape over and over again, with
no overlaps or gaps.
 A classic example of a tessellation is a tile floor in which the floor is covered in
square tiles.
Tessellations appear in numerous works of art in addition to architecture, and
they are also of mathematical interest.
These patterns can be found in a variety of settings, and once we start looking
for tessellations, we start seeing them everywhere, including in nature.

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When tiling it is important that the shape of the tile when repeated should cover the
whole surface or plane without any gaps or overlaps. A repeating pattern is then
formed and in mathematics we call a tiling like this a tessellation.
Let’s first consider a regular tessellation.
Only three regular polygons tessellate:
Equilateral triangles
Squares
Hexagons
Here is a table with the internal angles for regular polygons starting with an
equilateral triangle.
Regular polygon Internal angle
equilateral triangle 60°
square 90°
pentagon 108°
hexagon 120°
heptagon 102.6°
octagon 135°
more than eight sides more than 135°
For shapes to fill the plane without gaps or overlaps, their angles, when arranged
around a point, must have measures that add up to exactly 360°. If the sum is less
than 360°, there will be a gap. If the sum is greater, the shapes will overlap.

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What about regular pentagons? Each angle in a regular pentagon measures 108°, and
360° is not divisible by 108°.
A tessellation of equilateral triangles.
The interior angle of each equilateral
triangle is 60°
60° + 60° + 60° + 60° + 60° + 60° = 360°
Six 60° angles from six equilateral triangles
add up to 360°
A tessellation of squares.
What happens at each vertex?
90° + 90° + 90° + 90° = 360°
Four 90° angles from four squares add up to
360°.
The interior angle of a pentagon is 108°
108° + 108° + 108° = 324°
Not tessellated at all.
So regular pentagons cannot be
arranged around a point without
overlapping or leaving a gap.
A tessellation of regular hexagons.
What happens at each vertex?
120° + 120° + 120° = 360°
Three 120° angles from three regular
hexagons add up to 360°.

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What about regular heptagons?
In any regular polygon with more than six sides,
each angle has a measure greater than 120°, so
no more than two angles can fit about a point
without overlapping.
So the only regular polygons that create monohedral tessellations are equilateral
triangles, squares, and regular hexagons. A monohedral tessellation of congruent
regular polygons is called a regular tessellation.
Only three regular polygons tessellate:
Equilateral triangles
Squares
Hexagons
Tessellations of squares, triangles and hexagons
are the simplest and are frequently seen in
everyday life, for example in chessboards and
beehives.
Tessellations can have more than one type of shape.
You may have seen the octagon-square combination. In this tessellation, two regular
octagons and a square meet at each vertex.
Notice that you can indicate any
vertex and that the point is
surrounded by one square and two
octagons. So you can call this a 4.8.8
or a 4.82 tiling. The sequence of
numbers gives the vertex
arrangement, or numerical
name for the tiling.

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An Archimedean tessellation (also known as a semi-regular tessellation) is a
tessellation made from more that one type of regular polygon so that the same
polygons surround each vertex. There are only 8 semi-regular tessellations.
3.3.3.3.6
3.3.3.4.4
3.3.4.3.4
3.4.6.4
3.6.3.6
3.12.12
4.6.12
4.8.8
To name a tessellation, go around a vertex and
write down how many sides each polygon has, in
order ... like "3.12.12".
And always start at the polygon with the least
number of sides, so "3.12.12", not "12.3.12
The cofiguration at vertex 1 is 3.6.3.6 and the
cofiguration at vertex 2 is 3.6.3.6. This proves
that it is a semi-regular tesselation.
Tiling 3.3.3.3.6 Tiling 3.3.3.4.4

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Often, different vertices in a tiling
do not have the same vertex
arrangement. If there are two
different types of vertices, the tiling
is called 2-uniform. If there are
three different types of vertices, the
tiling is called 3-uniform.
A 2-uniform tessellation:
3.4.3.12 / 3.12.12
Tiling 3.3.4.3.4 Tiling 3.4.6.4
Tiling 3.6.3.6
Tiling 4.8.8Tiling 4.6.12
Tiling 3.12.12

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All triangles tessellate.
All interior angles of all triangles, whether equilateral, isosceles or scalene, will add
up to 180°. Therefore we can fill the space around a vertex, if we use two of each of
the angles of the triangle.
All quadrilaterals tessellate.
Let’s begin with an arbitrary quadrilateral ABCD. Rotate by 180° about the midpoint
of one of its sides, and then repeat using the midpoints of other sides to build up a
tessellation.
The angles around each vertex are exactly the four angles of the original
quadrilateral. Since the angle sum of the quadrilateral is 360°, the angles close up,
the pattern has no gaps or overlaps, and the quadrilateral tessellates.
Irregular Tessellations
Irregular tessellations encompass all
other tessellations, including the tiling in
the main image. Many other shapes,
including ones made up of complex curves
can tessellate. The image below was
prepared using Geogebra is an example of
an irregular tessellation.
The techniques of forming symmetry are called transformations. These include:
translations, rotations, reflections and glide reflections.
Symmetry: exact correspondence of form and constituent configuration on opposite
sides of a dividing line or plane or about a centre or an axis;

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Rotation: a circular movement about a centre of rotation;
Translation: a function that moves an object in a given direction for a particular
distance;
Reflection: a transformation in which the direction of one axis is reversed;
Glide-reflection: a reflection over a line followed by a translation in the same
direction as the line;
One of the simplest types of symmetry is translational symmetry. A translation is
simply a vertical, horizontal or diagonal slide.
Another type of symmetry is rotational symmetry. This is where a shape is moved a
certain number of degrees around a central point, called the centre of rotation. The
amount that the shape is turned is called the angle of rotation. Rotations are used in
tessellations to make shapes fit together like in the image above.

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The most familiar type of symmetry is reflective
symmetry. Reflections occur across a line called
an axis. The distance of a point from this axis
must be the same in the reflection.
The last type of symmetry is glide reflection. A glide reflection is a reflection and a
translation combined together. It does not matter which of the transformations
happens first.
The shape that emerges as a result of a reflection and
translation is simply called the glide reflection of the
original Figure.
In order for a glide reflection to take place an axis is
needed to perform the reflection, and magnitude and
direction are needed to perform the translation.
Penrose tiling is a particular aperiodic tiling.
Roger Penrose in the 1970s discovered
particular aperiodic tilings: he defined two
couples of figures - derived from a pentagon -
which must be set flanking identical sides in
the same direction:
 kite and dart, whose angles are
multiple of 36°;
 rhombus, whose angles are multiple of
36° too.

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PRACTICAL USES
Tessellations are often used by architects to make pavements, floors or wall
coverings: in this case the tiles are made by concrete materials, such as cemented
ceramic squares or hexagons.
These tiles may be decorative patterns or may have a structural function within a
building such as providing durable and water-resistant coverings.
Tessellation and Art
Historically, tessellation was used in Ancient Rome and in Islamic art: the decorative
tiling of the Alhambra palace (Granada) are beautiful examples of this.
Tessellation in Roman buildings floors.

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Escher
M.C. Escher popularized the use of
mathematical tessellations in art with his
fantastical repeating designs and optical
illusions. Escher was born in 1898 in the
Netherlands, and trained as a Dutch graphic
artist, who was obsessed with “filling the
plane”.
His interest began in 1936,
when he traveled to Spain
and saw the tile patterns
used in the Alhambra. The walls, ceilings and floors of this 13th
century fortress built by Islamic moors are covered in tessellating
mosaics. Escher spent days copying the designs in his sketch book
and remarked “...it is a pity that the religion of the Moors forbade
them to make graven images.”

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In Escher's works, we can often find the parent polygon, which has been altered, and
a piece or two of the original polygon flipped, glided or rotated to produce an
irregular tessellation.
He was fascinated by the rich possibilities latent in the rhythmic division of a plane
surface found in Moorish tessellations. He and his wife studied these artworks deeply
and Escher finally came up with a complete practical system that he applied in his
later artworks of metamorphosis and cycle prints.
Impossible constructions
"Relativity" is one of his most famous
lithographies: each part of the image
seems to be logical but the whole it is
impossible.

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“Drawing Hands” is very famous too: he
uses contrast and shading to create the
illusion of texture and dimension in a two-
dimensional work.
Hyperbolic Geometry
Escher created a few designs that could be interpreted as
patterns in hyperbolic geometry. Here he uses Poincaré
model of hyperbolic geometry: the hyperbolic points are
represented by Euclidean points within a bounding circle.
Here you have a nice video describing how you can build an Escher's Pegasus:
https://www.youtube.com/watch?v=NYGIhZ_HWfg
Here you can find an interactive poster about Escher prepared by Polish students:
LINK

43
During the Art lesson Polish students have created the Escher-style tessellation using
an equilateral triangle with rotations and squares/quadrilaterals with translations.
They have created our own tessellation by first making a shape tracer that can be
repeated over and over and over again. Here we can see some examples.

44
The Italians students have made some tilings during our Maths lessons, applying
symmetry, translations and rotations.
We used some tutorials to make tessellations with Geogebra. You can find them here:
https://www.youtube.com/watch?v=Eb36i-FU3NM
https://www.youtube.com/watch?v=NAKzOwQIIfk
Tilings and the art of the Alahambra
Escher was greatly inspired and tried to emulate
a rhythmic theme on a plane surface himself.
However, he was frustrated by his attempts to do
so, as he could only produce some ugly, rigid four
legged beasts which walked upside down on his
drawing paper. It was only during the second visit
in 1937 that he began a more serious study into the art of creating tessellations.

45
Spanish students prepared the presentation about the mosaics in Alhambra.- here is
the LINK.
Polish students prepared the interactive poster about Alhambra - here is the LINK.
ISLAMIC ART AND TILING
In the Islamic world, geometric shapes are symbols for the infinite and God (Allah):
this takes to a form spirituality without using the figurative iconography that other
religions often use: to Muslims, this infinite pattern of forms, taken together, extends
beyond the visible material world and takes to the infinite.
The individual has a direct line to God and the worship of idolatrous images is
therefore both delusive and useless: so representations that do not seek to create an
illusion of reality, are acceptable if kept away from any place of prayer.

46
OORRIIGGAAMMII TTEESSSSEELLLLAATTIIOONNSS
Origami tessellations are geometric designs folded from a single sheet of paper,
creating a repeating pattern of shapes from folded pleats and twists.
Three very basic tessellation patterns, called "regular tessellations„ are used heavily
in origami tessellation designs.
The three tiling patterns are formed with single, repeating shapes: equilateral
triangles, squares, and hexagons.
Often these patterns are referred to as
the 3.3.3.3.3.3, 4.4.4.4, and 6.6.6
tessellations, respectively. Three very
common examples used in origami
tessellations are the 3.6.3.6, 3.4.6.4,
and 8.8.4 tessellations.
Origami tessellations often follow one
of these six tessellation geometries by
employing a sheet of paper precreased
with a geometric grid. Origami
tessellations require very little in the
way of materials or tools-only a sheet
of paper and your hands are needed.
Here you can see some examples of the origami tessellations prepared by Polish
students.

48
IIVV.. GGAAUUDDII’’SS AARRCCHHIITTEECCTTUURREE AANNDD MMAATTHHSS
Antoni Gaudi (1852- 1926)
Antoni Gaudi was and early 20th century Spanish
architect. He was born in Reus in 1852 and received his
Architectural degree in 1878. From the very beginning his
designs were different from those of his contemporaries.
Gaudí's work was greatly influenced by forms of nature
and this is reflected by the use of curved construction
stones, twisted iron sculptures, and organic-like forms
which are traits of Gaudí's Barcelona architecture. Having
studied geometry he noticed the relationship between
nature and Maths.
Casa Vicens
Casa Batllo
From the outside the façade of Casa Batlló looks like it has been made from skulls and
bones. The "Skulls" are in fact balconies and the "bones" are supporting pillars.
Casa Vicens is a family residence
in Barcelona and built for
industrialist Manuel Vicens. It was
Gaudí's first important work.

49
Casa Milà
Parabolic arches inside
Casa Milà

50
Park Güell
Park Güell is a garden complex
with architectural elements in
Barcelona.
It was designed by Gaudí and
built in the years 1900 to 1914.
Mosaic seating area adorned with multi-coloured tiles
Large organic looking columns made from stone

51
Crypt of the Colonia Guell church
Casa Batllo
Fractals, cones, and hyperbolic paraboloid are all
examples. Gaudi often admired tree trunks and
skeletons being both functional and eye pleasing. No
matter what the intended purpose of the building, it
was still designed with heavy religious tones.
As a child Antoni Gaudi lived close to
nature. He paid attention to organic and
naturalistic geometry, and made it
blended to his distinctive art and
architecture style. His last work was his
magnum opus: Sagrada Familia, which
until now had never been finished.

52
The outside looks skeletal and the inside was designed to look like a forest. Pillars are
meant to be tree trunks and the ceiling like leaves that allows light to shine through
from the stained glass windows.
The nave in the Sagrada Familia with a hyperboloid vault. Inspiration from nature is
taken from a tree, as the pillar and branches symbolise trees rising up to the roof.
Gaudí was a great
innovator in all senses,
but he was particularly
so with regard to
architectural structures,
which he based on the
geometrical forms of
nature. Observe the
following geometric
forms and relate them to
the elements of Gaudí’s
buildings.
http://metalocus.es

53
HYPERBOLOID
A hyperboloid is a quadric – a type of surface in three dimensions – described by
the equation
Hyperboloid of one sheet Hyperboloid of two sheets
12
2
2
2
2
2

c
z
b
y
a
x
12
2
2
2
2
2

c
z
b
y
a
x

54
Paraboloid
A paraboloid is a quadric surface of special kind. There are two kinds of
paraboloids: elliptic and hyperbolic.
Elliptic paraboloid Hyperbolic paraboloid
2
2
2
2
b
y
a
x
c
z
 2
2
2
2
a
x
b
y
c
z

It is shaped like an oval cup It is shaped like a saddle
Gaudi’s research on hyperbolic-paraboloid and hyperboloid structure
Gaudi said that tree trunks, were the best example of natural structure thus he
explored tree trunks’ properties really much. His interest on natural geometrical form
was original, as he said, because he was a Mediterranean - born who lived close to sun
and nature. He had a clear image of nature’s hidden treasure of geometrical structure
and concrete tectonic skills. Example of this concrete natural image, was in his
masterpiece.
Another element widely used by Gaudí was the catenary curve.
Gaudi's catenary model at Casa Milà

55
VV.. GGEEOODDEESSIICC DDOOMMEE
A geodesic dome is a spherical or partial-spherical shell structure based on a
network of great circles (geodesics) lying on the surface of a sphere.
A triangle is the only polygon that holds its shape with
force acted upon it. The smaller the triangles in the
design, the more complex the network is and the more
the dome resembles the shape of a true sphere.
The geometric shape in which these structures form is
called a icosahedron.
The geodesic dome was invented by
R. Buckminster Fuller also known as
Bucky (1895-1983) in 1954. Fuller
was an inventor, architect, engineer,
designer, geometrician, cartographer
and philosopher. He has been called
“the 20th century Leonardo da Vinci”.
Geodesic structures can now be found everywhere. They are present in the structure
of viruses and the eyeballs of some vertebrae. The soccer ball is the same geodesic
form as the 60-atom carbon molecule C60, named buckminsterfullerene in 1985 by
scientists who had seen Bucky’s 250-foot diameter geodesic dome at the 1967
Montreal Expo. This dome was the largest of its time and still stands today.

56
Buckminster’s dome designs have been proved to be the strongest structures ever
made. They are the only man made structures that get proportionally lighter and
stronger as its size increases, so basically the bigger they are the stronger.
Geodesic domes are an extremely efficient form of architecture.
They are commonly used to cover weather stations and research locations in areas
where harsh weather conditions exist. They have been proven to withstand hurricane
force winds and pounding snow. They are also used to build sport domes because
they do not need any interior bearing points or walls, and leave a completely open
structure.
The examples of geodetic domes
The Złote Tarasy (Golden Terraces) a commercial, office, and
entertainment complex in the center of Warsaw

57
They can also be seen in amusement parks and playgrounds.

58
VVII.. PPEERRSSPPEECCTTIIVVEE
Perspective is a system for representing three dimensional objects, viewed in spatial
recession, on a two-dimensional surface. The simplest form of perspective drawing is
linear perspective, a system that allows artists to trick the eye into seeing depth on a
flat surface.
The study of the projection of objects in a plane is called projective geometry.
One-point perspective uses lines that lead to a single vanishing point; two-point
perspective uses lines that lead to two different vanishing points.
The principles of perspective drawing were elucidated by the
Florentine architect F. Brunelleschi (1377-1446).
Brunelleschi made at least two paintings in correct
perspective, but is best remembered for designing buildings
and over-seeing the building works.
Here you can find the presentations prepared by Belgian students about perspective:
LINK
www.mathworld.wolfram.com/Perspective.html

59
VVIIII.. GGEEOOMMEETTRRYY AANNDD AARRTT FFRROOMM TTHHEE CCOORRDDOOVVAANN PPRROOPPOORRTTIIOONN
The “cordovan proportion” was discovered by Rafael de la Hoz (he was born on
October 9, 1926, Cordoba, and he died on 13th of June, 2000, Madrid). He studied in
the technical college of architecture in Madrid. He carried through study of the
“cordovan proportion” and he used it in his buildings. This proportion is present in
buildings and monuments of Cordoba putting rectangles in such a way that they look
like they were put randomly. In 1951 the students were asked to draw an ideal
rectangle. They thought the students would draw a golden rectangle and they did the
same test with people living in Cordoba and they had got the same result: the most
drew a rectangle with the “cordovan proportion”.
The Cordovan proportion, is the ratio between the
radius of the regular octagon and its side length. The
irrational value of this ratio is known as the Cordovan
number.
Cordovan rectangle and its construction from a regular octagon.
Cordoban proportion in the regular octagon

60
The mentioned architect Rafael de la Hoz Arderius found this rectangle in the plan
and the elevation of the Mosque of Cordova.
The cordovan proportion is represented by the cordovan the polygon related to his
type of architecture is the octagon. Octagon that the cordovan rectangle comes from
appears in Mezquita in the Mihrab.

61
The cordovan proportion appears in the plan of the n “Mezquita of Cordoba” in the
Alhaken II’s door. It is present in other such as: Convent of the “Capuchinos”. The
Mosque’s architecture, is based on the composition with cordovan rectangles.
Here you can find the LINK to the presentation prepared by Spanish students.

62
VVIIIIII.. MMAATTHHSS IINN NNEEWWGGRRAANNGGEE
Newgrange is the best known Irish passage tomb and dates to c.3, 200BC. The large
mound is approximately 80m in diameter and is surrounded at its base by a kerb of
97 stones. The most impressive of these stones is the highly decorated Entrance
Stone.
There are many different types and
examples of art at Newgrange of
different styles and skill levels.
Some of the art and designs and carved
deeper than others and are very detailed.
There is a 10ft long and 14ft high stone
that stands at the door of Newgrange, it
has been called 'one of the most artistic
stones in the history of the earth'.
Here you can see a prezi showing how the New Grange monument in Co Meath,
Ireland was designed LINK
Celtic artwork has always been famous
for its geometric motifs. Some of these
outstanding works date back to 3000 BC
and can still be found on stone carvings
today.

63
IIXX.. CCEELLTTIICC KKNNOOTTSS AANNDD TTHHEE BBOOOOKK OOFF KKEELLLLSS
The Book of Kells, which was created by Irish monks around
the early 9th century, contains the four gospels written in
Latin. Almost all of the folios of the Book of Kells contain small
illuminations like this decorated initial.
The triquetra is found in
the Book of Kells.
Here you can see the presentation prepared about the Book of Kells by Irish students:
LINK

Maths in Art and Architecture Why Maths? Comenius project

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