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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.

II.. TTHHEE FFIIBBOONNAACCCCII SSEEQQUUEENNCCEE IINN AARRTT AANNDD AARRCCHHIITTEECCTTUURREE

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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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 4.8.8 Tiling 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

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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.

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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.

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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.

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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.

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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.

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Casa Milà

Parabolic arches inside

Casa Milà

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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

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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.

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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

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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

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2

2

2

2

2

c

z

b

y

a

x 1

2

2

2

2

2

2

c

z

b

y

a

x

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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

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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.

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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

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They can also be seen in amusement parks and playgrounds.

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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

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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

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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.

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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.

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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.

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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

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