MODULE 6: Shapes Flashcards
Etymology of Geometry (Greek).
Geo (earth) and Metron (measurement) –> study of Earth’s measurement
What does Geometry study?
sizes, shapes, angles, positions, and dimensions of things
Three Mathematics before Euclidean Mathematics
Babylonian
Egyptian
Greek
Babylonian Mathematics were written in ______ on _______.
Cuneiform
Clay Tablets
The Mathematics that gave us formulas on the areas and volumes of circles and cylinders.
Babylonian
The circumference of a circle according to Babylonian Mathematics.
three times the diameter (3D)
The approximation of pi in Babylonian Mathematics.
3
The area of a circle according to Babylonian Mathematics.
1/12 of C^2 (square of the circumference)
First to use the Pythagorean Theorem/Triples.
Babylonians
Gave us formulas in finding areas and volumes that they used in constructing pyramids and determining food supply.
Egyptian
The Mathematics that discovered irrational numbers.
Greek
First mathematician to calculate the circumference of the Earth.
Eratosthenes (40,000 km)
Discovered the Pythagorean Theorem.
Pythagoras
Contributed to finding the volumes of irregular shapes.
Archimedes of Syracuse
Accurately approximated the value of pi using the method of exhaustion developed by Eudoxus of Cnidus.
Archimedes of Syracuse
Golden Rectangles
rectangle with the most pleasing proportions
responsible for Euclidean Geometry, the mathematical system used globally
Euclid of Alexandria
Euclid’s Textbook
The Elements
Undefined terms according to Euclid
points
lines
planes
Axioms are
logical mathematical statements that do not need to be proven
Euclid’s Five Axioms
- Things that are equal to the same things are equal
- If equals are added to equals, then the wholes are equal
- If equals are subtracted from equal, then the remainders are equal
- Things that coincide with one another are equal to another
- The whole is greater than the part
Postulates are
Mathematical statements are considered true as long as it is not disproven.
Euclidean Postulates
- A straight line can be drawn from any point to any point
- A finite straight line can be produced continuously in a straight line
- A circle may be drawn with any point as the center and any distance as a radius
- All right angles are equal to one another
- Parallel Postulate
Playfair’s Axiom states
Only one line that passes through point P can be parallel to line I
What are Euclidean Triangles?
Triangles whose interior angles’ sum is always 180 degrees
Congruent triangles are
Triangles that have the same size and shape
Similar triangles are
Triangles that have the same shapes and angles but different lengths
Three Congruence Criteria for Euclidean Triangles
SSS
SAS
ASA
A Non-Euclidean Geometry that negated Euclid’s 5th postulate and allowed curved lines, and assumed that there can be at least two lines through point P parallel to line I.
Hyperbolic Geometry
First to publish Hyperbolic Geometry
Nikolai Ivanovich Lobachevsky
Diameters are
line segments passing through the center of the disk
Arcs
intersect yhe disk at rigt angles
Arcs
intersect the disk at right angles
Model used in Hyperbolic Geometry
Poincare’s Disk Model
Hyperbolic Triangles
Triangles whose interior angles are less than 180 degrees
A Non-Euclidean Geometry that negated Euclid’s 5th postulate by proving that there are no parallel lines to I that pass through point P.
Elliptic Geometry
The model used in Elliptic Geometry
Spheres with great circles
Studied Elliptic Geometry
George Friedrich Bernhard Riemann
Elliptic Triangle
Triangles whose interior angles are more than 180 degrees
Topology is
The study of space where there are abstract relations of points and geometry is seen as the theory of space of points
Responsible for Topology
Jules-Henri Poincare
Topological Transformation includes
stretching
shrinking
twisting
bending
Topological Transformation does not include
cutting
tearing
puncturing
merging
Topologically equivalent
same number of holes
Homeomorphism
equivalence
Homeomorphic
when shapes are topologically equivalent to each other
