Geometry test 2

constructible numbers

a length that can be geometrically built using only an unmarked straightedge, a compass, and starting unit segment length of 1

the only constructible numbers

only constructible if they can be written with the 4 basic algebraic operations (addition, subtraction, multiplication, division) and square roots

field of constructible numbers

the field is a set of 2 operations addition and multiplication (products and sums) F are in F

constructions with compass and straightedge rule 0

start with 2 any distinct points in the plane

constructions with compass and straightedge rule 1

draw a line through any 2 constructible points with straight edge (axioms 1 and 2)

constructions with compass and straightedge rule 2

draw a circle with center at an already constructed point through an already constructed point

constructions with compass and straightedge rule 3

coonstruct points of intersection

regular polygon

polygon is regular if all its sides and angles are respectively congruent

polyhedron

3D analogue of a polygon

regular polyhedron

if is convex, all of its faces are congruent to regular polygons, and there are same number of faces at each vertex

a platonic solid is also known as

a polyhedron

tetrahedron

has 4 triangular faces

cube

has 6 square faces

octahedron

has 8 triangular faces

dodecahedron

has 12 pentagonal faces

icosahedron

has 20 triangular faces

great circles

circles with center also at O and padius rho, cut the surface in half, lines on Sphere

antipodal points

on sphere are points opposite each other on S, endpoints of diameter S

luna

sections of a sphere between 2 half great circles which makes an angle theta between intersections

assuming 1 is constructible, what other numbers are..

ones that can be obtained from the integers by a finite number of basic operations and square root…field of constructible numbers

5 platonic solids

tetrahedron, dodecahderon, cube, icosahedron, octahedron

Euler characteristic formula

X = F + V - E , = faces + vertices - edges

area of sphere formula

2(theta)(rho²) , with angular measure at theta in terms of polar

area of spherical triangle

(rho²)(A + B + C - pi)

changes in axiom 1 for spherical geometry

can draw a segment between any 2 point

changes in axiom 2 for spherical geometry

can extend segments into great circles, can extend length of segments by E along the path of a great circle

changes in axiom 3 for spherical geometry

circles of possible radii can be drawn, typically allows values in (0, pi(rho))

changes in axiom 4 for spherical geometry

no change, any 2 right angles are congruent

changes in axiom 5 for spherical geometry

FAILS in spherical, all pairs of distinct lines intersect at 2 points,

given any great circle l and any point P not on l, expect 2 intersections between l and great circles through P, distinct lines intersect at 2 points

isometries on sphere

can be viewed as a subject of isometries in R³ that fix- O, forces to send S to S (map)

rotations on sphere

can rotate about origin , rotations about diameter and fixes antipodal points at end of diameter

reflections on sphere

can reflect across planes through O, only fixed points are intersections

translations on sphere

only trivial (the idenitity)

spherical pythag theorem

right angle at C, a,b,c are side lengths opposite angles A,B,C … cosc = cosa cosb

Theorem-

sinA = sina/sinc … cosA = (cosa sinb)/sinc

spherical law of sines

sina/sinA = sinb/sinB = sinc/sinC

spherical law of cosines

cosc = cosa + cosb + sin(a)sin(b)cosC

Basic construction rules to fule

→start by constructing two points (typ a unit apart)

→you may construct a segment or line between any 2 existing points

→you may construct a circle with center at ibe existing point through another existing point

→you may construct points of intersection of existing objects

what is constructible using basic construction rules

geometric object, length (segment with that length), number (corresponding point C), real numbers (positive integers and other numbers obtained by add, sub, mult, div and square root), complex values (combinations of integers and i through add, sub, mult, div, square root)

how to bisect an angle

→draw a circle at point B of radius |AB|

→build intersection of this circle with ray BC called point D

→build circles of radius |AD| at A and D

→build the intersection point of these circles called points E and F

→build line (ray) EF, line bisects angle

is this a constructible segment length, 78

yes, positive integer

is this a constructible segment length, cubed root of 5

No, cubed root

is this a constructible segment length, square root of (1+ square root(2))

yes, constrible, uses square roots

is this a constructible segment length, 3/7

Yes rational number, uses division

is this a constructible segment length, circumference of circle with radius of 2

No, 4pi, cannot construct pi

Euler characteristic of cube

faces-6 , edges- 12 , vertices-8 … X = 6+12-8 = 2

What has Euler characteristics of 2

platonic solids (topologically similar → same Euler characteristic)

What shape must the triangle take if two of the vertices of a spherical triangle on the unit sphere are antipodal points, A and A’

degenerate triangle- which resembled a line segment (if third vertex of triangle was on the edge connecting A and A’) Area = 0 (inside) 4pi(rho²) (outside)

luna- the third vertex of triangle lies off of the segment connecting A and A’ so segments BA and BA’ would both lie on the same great circle, triangle ABA’ resembles Luna, angle A = A’ and B = pi (area 2A or 2A’)

Given two points on the sphere, how many great circles exist through those two points?

depends on points location … antipodal, infinitely many, otherwise there is one great circle through the two points.

How many isometries send A to its antipodal point A’

reflections, rotations, reflection followd by rotation