Download The first book - Architeaching Geometry

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.
Multilateral Comenius
Project
"Architeaching"
Module 3 –
Sciences and Architecture
Maths
Geometry
Lessons by Constantin Jitariu
Theoretical Highschool
“Mihail Kogãlniceanu”
Maths
Multilateral Comenius Project "Architeaching"
Geometry
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metria
"measurement") is a branch of mathematics concerned with questions of
shape, size, relative position of figures, and the properties of space.
Geometry is one of the oldest mathematical sciences. Initially a body of
practical knowledge concerning lengths, areas, and volumes, in the 3rd
century BC geometry was put into an axiomatic form by Euclid, whose
treatment—Euclidean geometry—set a standard for many centuries to
follow.
Archimedes developed ingenious techniques for calculating areas and
volumes, in many ways anticipating modern integral calculus. The field of
astronomy, especially mapping the positions of the stars and planets on the
celestial sphere and describing the relationship between movements of
celestial bodies, served as an important source of geometric problems during
the next one and a half millennia. A mathematician who works in the field of
geometry is called a geometer.
Ancient scientists paid special attention to constructing geometric
objects that had been described in some other way. Classical instruments
allowed in geometric constructions are those with compass and straightedge.
However, some problems turned out to be difficult or impossible to solve by
these means alone, and ingenious constructions using parabolas and other
curves, as well as mechanical devices, were found.
Points, Lines, and Planes
Point, line, and plane, together with set, are the undefined terms that provide
the starting place for geometry. When we define words, we ordinarily use
simpler words, and these simpler words are in turn defined using yet simpler
words. This process must eventually terminate; at some stage, the definition
must use a word whose meaning is accepted as intuitively clear. Because
that meaning is accepted without definition, we refer to these words as
undefined terms. These terms will be used in defining other terms. Although
these terms are not formally defined, a brief intuitive discussion is needed.
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Geometry
Point
A point is the most fundamental object in geometry. It is represented by a
dot and named by a capital letter. A point represents position only; it has
zero size (that is, zero length, zero width, and zero height).
Figure 1
Three points.
Line
A line(straightline) can be thought of as a connected set of infinitely many
points. It extends infinitely far in two opposite directions. A line has infinite
length, zero width, and zero height. Any two points on the line name it. The
symbol ↔ written on top of two letters is used to denote that line. A line
may also be named by one small letter.
Figure 2
Two lines.
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Geometry
Collinear points
Points that lie on the same line are called collinear points. If there is no line on which
all of the points lie, then they are noncollinear points. In figure points M, A, and
N are collinear, and points T, I, and C are noncollinear.
Figure 3
Three collinear points and three noncollinear points.
Plane
A plane may be considered as an infinite set of points forming a connected
flat surface extending infinitely far in all directions. A plane has infinite
length, infinite width, and zero height (or thickness). It is usually represented
in drawings by a four-sided figure. A single capital letter is used to denote a
plane. The word plane is written with the letter so as not to be confused with
a point
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Geometry
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three
corners or vertices and three sides or edges which are line segments. A
triangle with vertices A, B, and C is denoted ABC. In Euclidean geometry
any three non-collinear points determine a unique triangle and a unique
plane
Types of triangles
By relative lengths of sides
Triangles can be classified according to the relative lengths of their sides:
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In an equilateral triangle all sides have the same length. An
equilateral triangle is also a regular polygon with all angles measuring
60°.
In an isosceles triangle, two sides are equal in length. An isosceles
triangle also has two angles of the same measure; namely, the angles
opposite to the two sides of the same length; this fact is the content of the
Isosceles triangle theorem. Some mathematicians define an isosceles
triangle to have exactly two equal sides, whereas others define an
isosceles triangle as one with at least two equal sides. The latter
definition would make all equilateral triangles isosceles triangles. The
45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is
isosceles.
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
Multilateral Comenius Project "Architeaching"
Geometry
In a scalene triangle, all sides are unequal. The three angles are also
all different in measure. Some (but not all) scalene triangles are also right
triangles.
.
Equilateral
Isosceles
Scalene
By internal angles
Triangles can also be classified according to their internal angles, measured
here in degrees:
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A right triangle (or right-angled triangle, formerly called a rectangled
triangle) has one of its interior angles measuring 90° (a right angle). The
side opposite to the right angle is the hypotenuse; it is the longest side of
the right triangle. The other two sides are called the legs or catheti
(singular: cathetus) of the triangle. Right triangles obey the Pythagorean
theorem: the sum of the squares of the lengths of the two legs is equal to
the square of the length of the hypotenuse: a2 + b2 = c2, where a and b
are the lengths of the legs and c is the length of the hypotenuse. Special
right triangles are right triangles with additional properties that make
calculations involving them easier. One of the two most famous is the 34-5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, and 5 are a
Pythagorean Triple. The other one is an isosceles triangle that has 2
angles that each measure 45 degrees.
Triangles that do not have an angle that measures 90° are called
oblique triangles.

A triangle that has all interior angles measuring less than 90° is an
acute triangle or acute-angled triangle.

A triangle that has one angle that measures more than 90° is an obtuse
triangle or obtuse-angled triangle.
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Geometry
A triangle that has two angles with the same measure also has two sides with
the same length, and therefore it is an isosceles triangle. It follows that in a
triangle where all angles have the same measure, all three sides have the
same length, and such a triangle is therefore equilateral.
Right
Obtuse
Acute
Oblique
Triangles in construction
The Flatiron Building in New York is triangularly shaped.
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Geometry
Rectangles have been the most popular and common geometric form
for buildings since the shape is easy to stack and organize; as a standard, it is
easy to design furniture and fixtures to fit inside rectangularly-shaped
buildings. But triangles, while more difficult to use conceptually, provide a
great deal of strength. As computer technology helps architects design
creative new buildings, triangular shapes are becoming increasingly
prevalent as parts of buildings and as the primary shape for some types of
skyscrapers as well as building materials. In Tokyo in 1989, architects had
wondered whether it was possible to build a 500 story tower to provide
affordable office space for this densely packed city, but with the danger to
buildings from earthquakes, architects considered that a triangular shape
would have been necessary if such a building was ever to have been built (it
hasn't by 2011). In New York City, as Broadway crisscrosses major avenues,
the resulting blocks are cut like triangles, and buildings have been built on
these shapes; one such building is the triangularly-shaped Flatiron Building
which real estate people admit has a "warren of awkward spaces that do not
easily accommodate modern office furniture" but that has not prevented the
structure from becoming a landmark icon. Designers have made houses in
Norway using triangular themes. Triangle shapes have appeared in churches
as well as public buildings including colleges as well as supports for
innovative home designs. Triangles are sturdy; while a rectangle can
collapse into a parallelogram from pressure to one of its points, triangles
have a natural strength which supports structures against lateral pressures. A
triangle will not change shape unless its sides are bent or extended or broken
or if its joints break; in essence, each of the three sides supports the other
two. A rectangle, in contrast, is more dependent on the strength of its joints
in a structural sense. Some innovative designers have proposed making
bricks not out of rectangles, but with triangular shapes which can be
combined in three dimensions. It is likely that triangles will be used
increasingly in new ways as architecture increases in complexity.
Square
In geometry, a square is a regular quadrilateral. This means that it has four
equal sides and four equal angles (90-degree angles, or right angles). A
square with vertices ABCD would be denoted ABCD.
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Geometry
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two
pairs of parallel sides. The opposite or facing sides of a parallelogram are of
equal length and the opposite angles of a parallelogram are of equal
measure. The congruence of opposite sides and opposite angles is a direct
consequence of the Euclidean Parallel Postulate and neither condition can be
proven without appealing to the Euclidean Parallel Postulate or one of its
equivalent formulations. The three-dimensional counterpart of a
parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines")
reflects the definition.
Rhombus
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral
whose four sides all have the same length. The rhombus is often called a
diamond, after the diamonds suit in playing cards, or a lozenge, though the
latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is a parallelogram, and a rhombus with right angles is a
square. (Euclid's original definition and some English dictionaries' definition
of rhombus excludes squares, but modern mathematicians prefer the
inclusive definition.)
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Geometry
The English word “rhombus” derives from the Ancient Greek όμβος
(rhombos), meaning spinning top. The plural of rhombus can be either
rhombi or rhombuses.
Trapezium
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides
is referred to as a trapezoid in American English and as a trapezium in
English outside North America. A trapezoid with vertices ABCD is denoted
ABCD or ABCD. The parallel sides are called the bases of the trapezoid.
This article uses the term trapezoid in the sense that is current in the United
States (and sometimes in some other English-speaking countries)[citation needed].
Readers in the United Kingdom and Australia should read trapezium for
each use of trapezoid in the following paragraphs. In all other languages
using a word derived from the Greek for this figure, the form closest to
trapezium (e.g. French 'trapèze', Italian 'trapezio', German 'Trapez', Russian
'трапеция') is used.
The term trapezium has been in use in English since 1570, from Late Latin
trapezium, from Greek trapezion, literally "a little table", diminutive of
trapeza "table", itself from tra- "four" + peza "foot, edge". The first recorded
use of the Greek word translated trapezoid (τραπεζοειδη, table-like) was by
Marinus Proclus (412 to 485 AD) in his Commentary on the first book of
Euclid’s Elements.
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Geometry
In an isosceles trapezoid, the base angles have the same measure, and the
other pair of opposite sides AD and BC also have the same length.
In a right trapezoid, two adjacent angles are right angles.
In architecture the word is used to refer to symmetrical doors, windows, and
buildings built wider at the base, tapering towards the top, in Egyptian style.
Pentagon
In geometry, a pentagon (from pente, which is Greek for the number 5) is
any five-sided polygon. A pentagon may be simple or self-intersecting. The
sum of the internal angles in a simple pentagon is 540°. A pentagram is an
example of a self-intersecting pentagon
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Geometry
Pentagons in nature
Plants

Pentagonal cross-section of okra.

Morning glories, like many other flowers, have a pentagonal shape.

The gynoecium of an apple contains five carpels, arranged in a fivepointed star
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Geometry

Starfruit is another fruit with fivefold symmetry.
Animals

A sea star. Many echinoderms have fivefold radial symmetry.

An illustration of brittle stars, also echinoderms with a pentagonal shape
Circle
A circle is a simple shape of Euclidean geometry consisting of the set
of points in a plane that are a given distance from a given point, the centre.
The distance between any of the points and the centre is called the radius.
Circles are simple closed curves which divide the plane into two regions: an
interior and an exterior. In everyday use, the term "circle" may be used
interchangeably to refer to either the boundary of the figure, or to the whole
figure including its interior; in strict technical usage, the circle is the former
and the latter is called a disk.A circle is a special ellipse in which the two
foci are coincident and the eccentricity is 0. Circles are conic sections
attained when a right circular cone is intersected by a plane perpendicular to
the axis of the cone.
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Geometry
History
The word "circle" derives from the Greek, kirkos "a circle," from the base
ker- which means to turn or bend. The origins of the words "circus" and
"circuit" are closely related.The circle has been known since before the
beginning of recorded history. Natural circles would have been observed,
such as the Moon, Sun, and a short plant stalk blowing in the wind on sand,
which forms a circle shape in the sand. The circle is the basis for the wheel,
which, with related inventions such as gears, makes much of modern
civilization possible. In mathematics, the study of the circle has helped
inspire the development of geometry, astronomy, and calculus.Early science,
particularly geometry and astrology and astronomy, was connected to the
divine for most medieval scholars, and many believed that there was
something intrinsically "divine" or "perfect" that could be found in circles.
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Geometry
The compass in this 13th century manuscript Circles on an old astronomy
is a symbol of God's act of Creation. Notice drawing, by Ibn al-Shatir
also the circular shape of the halo
Some highlights in the history of the circle are:
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1700 BC – The Rhind papyrus gives a method to find the area of a
circular field. The result corresponds to 256/81 (3.16049...) as an
approximate value of π.
300 BC – Book 3 of Euclid's Elements deals with the properties of
circles.
In Plato's Seventh Letter there is a detailed definition and explanation
of the circle. Plato explains the perfect circle, and how it is different
from any drawing, words, definition or explanation.
1880 – Lindemann proves that π is transcendental, effectively settling
the millennia-old problem of squaring the circle.[
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