06 Topology

​ hapes (​ Pt. 1) Geometry​ ​was developed as a practical way of
dealing with areas, lengths, and volumes.
- geo -​ earth
- metron -​ measurement
Geometry before Euclid
1. Babylonian civilization
- clay tablets with cuneiform script
3. Eratosthenes
- first mathematician to calculate the
circumference of the earth ​using a tower’s
shadow, the distance of Alexandria and
- calculated value: 250,000 stadia or
around 40,000 km
- actual value: 40,075 km
- VIDEO:​ ​http://bit.do/shadowcircum 4. Pythagoras
- Pythagorean theorem
a2 + b2 = c2
wrote formulas for finding areas and
volumes of shapes like circles and
approximated ​π​
π​ to 3
circumference =
​ 3​
​ ​ • diameter
​ iameter
area = 12 circumference​2
had the concept of Pythagorean
​ ythagorean triples
but no sources indicate they know the
Pythagorean theorem
2. Egyptian civilization
- flourished almost at the same time as the
Babylonian civilization
- formulas in finding areas and volumes
were essential in the construction of
famous ​pyramids​
pyramids​ and the determination
of​ food supply​; recorded on the Ahmes
​ hmes
Papyrus a​ ka Rhind
​ hind Mathematical Papyrus
5. Archimedes of Syracuse
- volume of an​ irregularly shaped object
= volume displaced when submerged in
- derived an accurate approximation of the
value of π
​ using the ​method
method of exhaustion
developed by Eudoxus of Cnidus
6. Greeks
- irrational numbers
e.g. √2 , π
​ , golden ratio ( ϕ )
*refer to previous trans ​(04 Module 5)
for explanation of golden ratio; alam ko
namang naprint/nasave niyo na yun :​ ^)
NOTE: ϕ = 1.618…
ϕ = 0.618...
Euclidean geometry - by mathematician Euclid
​ uclid of Alexandria
- first to show how results stated by
earlier mathematicians fit into a
comprehensive deductive and logical
mathematical system
- The Elements ​- Euclid’s book;
collection of postulates, definitions,
proofs, etc
- oldest geometry conceived; geometry studied
- undefined terms in geometry: p
​ oints, lines,
Euclidean axioms formulated to develop a structured mathematical
system 1. Things that are equal to the same thing are
2. If equals are added to equals, then the whole
are equal.
3. If equals are subtracted from equal, the
remainders are equal.
4. Things that coincide with one another are
equal to one another.
5. The whole is greater than the part.
Euclidean postulates govern Euclid’s axiomatic system 1. A straight line can be drawn from any point to
any point.
- point 1, point 2, connect sila → line
2. A finite straight line can be produced
continuously in a straight line.
- maraming line segment sa loob ng
straight line
3. A circle may be drawn with any point as
center and any distance as radius.
- stationary point 1, point 2 somewhere
then ikutan yung point 1 (equal
distance/radius) → charan circle
4. All right angles are equal to one another.
- kahit anong paikot sa mundo o sayo, 90​°
= 90​°
5. Called ​parallel postulate​. If a transversal falls
on two lines in such a way that the interior
angle on one side of the transversal are less
than two right angles, then the lines will
intersect on that side on which the angles are
less than two right angles.
- if interior angles on one side of the
transversal line are both <90​°,​
°,​ the two
lines will intersect when extended
towards the said side
dito galing ang if the sum of the interior
angles are 180​°​°​, the two lines are parallel
“​parallel postulate​” pero hindi directly
tungkol sa parallelism? enter Playfair’s
Playfair’s axiom ​(based on 5th postulate)
Through a point P not on line l , there
exists exactly one line ( m ) passing
through point P parallel to l . Lines s
and q are not parallel to l .
Euclidean triangles - sum of the measures of interior angles is
always 180​°
Congruence (≡) criteria
1. SSS​ (side - side - side)
3 corresponding sides​ are congruent.
Congruent vs similar Euclidean triangles
same shape and
same shape but not
necessarily same size
angles and sides
equal corresponding angles,
proportional corresponding
All congruent triangles are similar.
Not all similar triangles are congruent.
△ABC ≡ △XY Z
2. SAS​ (side - angle - side)
2 corresponding sides​ and the ​angle
angle between
them are congruent.
Consider the 2 congruent triangles below.
✓ same shape
✓ same size
- corresponding sides are of the same lengths
AB = X Y
BC = Y Z
AC = X Z
corresponding angles are equal
m∠A = m∠X
△ABC ≡ △XY Z
m∠B = m∠Y
m∠C = m∠Z
3. ASA​ (angle - side - angle)
2 corresponding angles​ and the ​side
side between
them​ are congruent.
△ABC ≡ △XY Z
4. AAS*​ (angle - angle - side)
2 corresponding angles​ and a ​non-included
side​ are congruent.
Non-Euclidean geometry Hyperbolic geometry Poincaré Disk
△ABC ≡ △XY Z
*hindi kasama sa ppt ni sir pero kasama sa
Similarity (~) criteria
*hindi rin kasama sa discussion/ppt ni sir pero for
reference lang kung gusto icompare sa congruence
Through a point P not on a line l, there exists at least
two lines passing through point P parallel to line l.
- sum of hyperbolic triangle’s interior angles is
less than 180°
- similar hyperbolic triangles are congruent
Elliptic geometry
1. SSS​ (side - side - side)
3 corresponding sides​ are in proportion.
XY = Y Z = AC
2. SAS ​(side - angle - side)
2 corresponding sides are proportional,​ and
the ​angle
angle between​ them are congruent.
XY = Y Z
m∠B = m∠Y
3. AA/AAA​ (​internal screams??​)
2 or 3 corresponding angles​ are congruent.
m∠A = m∠X
m∠B = m∠Y
m∠C = m∠Z
Through a point P not on a line l, there exists no line
passing through point P parallel to line l.
- sum of elliptical triangle’s interior angles is
more than 180°
- similar elliptical triangles are congruent
Application​: in sea and air navigation, spherical
model of elliptic geometry is used in calculating
distances and planning flight paths
Projective geometry
- includes mathematical s​ tudy of vanishing
points and perspective
- three-dimensional (3D) visualization
- artists’ portrayal of humans in
creating works of art (Renaissance
- 3D images in games (modern times)
Gerard Desargues
- father of projective geometry
- stated that any two lines on a plane intersect
giving rise to creation of projective geometry
- revisited ​Euclid’s
Euclid’s second postulate​ ​and
considered the concept of parallel lines
interesting in a plane at an ideal point
Two triangles are​ perspective from a point P ​if sides
Aa, Bb and Cc intersect at point p. The point of
intersection is called the​ center of perspectivity.
Two triangles ABC and abc are ​perspective
perspective from line l
if corresponding sides of the triangles meet at points
on l, called the axis
a​ xis of perspectivity.
Desargues Theorem
If two triangles are perspective from a point, then
they are perspective from a line.
Geometries described previously (Euclidean and
non-Euclidean) talked about geometries with very
​ ery
concrete notions of space.
Space ​was conceptualized as ​a set of points​ t​ ogether
with an ​abstract
abstract set of relations i​ n which these points
are involved.
- an invention of many mathematicians (e.g.
Cantor, Euler, Frechet, Huasdorf, Mobius and
- focuses on discovering and analyzing the
essential similarities and differences between
sets and figures
Topological transformations
- stretching, shrinking, twisting or bending the
figure in any way that allows the points to
remain distinct
Two figures are called topologically
t​ opologically equivalent​ ​if one
figure can be stretched, shrunk, twisted or bent in to
the same shape as the other
A square, a circle, and a triangle are topologically
The coffee and the cup are topologically
equivalent, but a coffee cup and a muffin are not.
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