Lecture Note 6 Flashcards
he was known to be the first mathematician to calculate the circumference of the earth
erathosthenes
he gave us the most popular version of the Euclid’s 5th postulate
playfair
The point of intersection is called the?
center of perspectivity
Two triangles are ___________ if they have the same shape but can be different sizes
Similar
Gerard considered the concept of parallel line intersecting in a plane and called this the?
ideal point
The model of this geometry conceived by the great universal mathematician Henri Poincare is now called the?
Poincare Disk of Hyperbolic Geometry
other term for golden ratio
divine proportion
published his work on this geometry first in an article entitled “Geometrische Untersuchungen zur Theorie der Parellenien
(Geometrical Researches on the Theory of Parallels)” in 1829
Lobachevsky
elliptic geometry was studied deeply by the German mathematician _____________________________________
George Friedrich Bernhard Riemann
he was another Greek mathematician who contributed a handful geometric concepts like the volumes of irregular shapes
archimedes
includes the mathematical study of vanishing
points and perspective
projective geometry
it is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing
topology
mathematicians who restated Euclid’s 5th postulate
posidonius, proclus, legendre
A model for the hyperbolic geometry can be seen by considering not the usual plane,
but a?
circular disk
In _________________, we consider geometry as the theory of space of points
Topology
this is synonymous with the term axiom
postulate
This geometry existed for quite some time before Euclid formalized it in his book “The Elements”
Euclidean geometry
they were the main contributors
of Hyperbolic geometry
- Nikolai Ivanovich Lobachevsky (Russian mathematician and
geometer) - Janos Bolyai (Hungarian mathematician)
- Johann Carl Friedrich Gauss (German mathematician)
it is all about discovering and analyzing the essential similarities and differences between sets and figures
topology
Two Euclidean triangles are ______________ if and only if they have the same size and shape.
Congruent
This shape, caused by the flattening at the poles, is called an?
oblate spheroid
The sum of the angles of a (hyperbolic) triangle are?
less than 180
method of exhaustion developed by?
Eudoxus of Cnidus
known as the father of projective geometry
Gerard Desargues
gave us the concept that given a right triangle with lengths of sides a and b; and, length of side opposite the right angle is c, then a2 + b2 = c2
Pythagoras
Alexander Henry Rhind purchased the papyrus in?
1885 in Luzor, Egypt
method of finding area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the
containing shape
method of exhaustion
He stated that any two lines on a plane intersect. This gave rise to the creation of?
projective geometry
he published a book entitle “Vorstudien zur Topologie (Introductory Studies to Topology)” having this notion of space and geometry in a symmetric development
Jules-Henri Poincare
it deals with points and set of points
geometry
________________ of Alexandria was one of the most famous Greek mathematician
Euclid
One main idea is ____________________________ that include stretching, shrinking, twisting or bending the figure in any way that allows the points to remain distinct
Topological Transformations
Other consequences of Euclid’s Axiomatic System on triangles are on?
congruency and similarity
Through a point P not on a line l , there exists exactly one line passing through a point P parallel to l .
playfair’s axiom
The sum of the angles of a (elliptic) triangles are?
more than 180
One of the most significant contributions of the Greek was the discoveries of?
irrational numbers and golden ratio
They used this geometry to create works of art having three-dimensional perspective
projective geometry
Two triangles ABC and abc are perspective from a line l if corresponding sides of the triangles meet at points on l , called the?
axis of perspectivity
textbook of Euclid
The Elements
the most popular version of the Euclid’s 5th postulate is now called?
playfair’s axiom
concept of Pythagoras
Pythagorean theorem
There are concepts in geometry which are called?
undefined terms
These formulas are recorded in the famous
Ahmes Papyrus (also known as the Rhind Mathematical Papyrus)
Two figures are called __________________________ if one figure can be stretched, shrunk, twisted or bent into the same shape as the other
Topologically Equivalent
By assuming the negation of Euclid’s 5th Postulate, we get a new geometry called
hyperbolic geometry
a French mathematician and engineer, revisited Euclid’s second postulate: A finite straight line can be produced continuously in a straight line.
Gerard Desargues
it is the oldest geometry
Euclidean Geometry
Rhind Papyrus was named after
Alexander Henry Rhind
a Scottish antiquaria, who purchased the papyrus in 1858 in Luzor, Egypt
Alexander Henry Rhind
The 5th Postulate is also called the?
parallel postulate
He was also known for deriving an accurate approximation of the value of π using the method of exhaustion developed by Eudoxus of Cnidus.
archimedes
