Section 6 - Projective Planes Flashcards
state the projective pappus thm
six points, lying alternately on two straight lines, form a hexagon whose three pairs of opposite sides meet on a line
state the projective desargues thm
if two triangles are in perspective from a point, then their pairs of corresponding sides meet on a line
state the little desargues thm
if two triangles are in perspective from a point P, and if two pairs of corresponding sides meet on a line L through P, then the third pair of corresponding sides also meets on L
state the converse projective desargues thm
If corresponding sides of two triangles meet on a line, then the two triangles are in perspective from a point.
state the scissors thm
if ABCD & A’B’C’D’ are quadrilaterals with vertices alternating on two lines, and if AB//A’B’, BC//B’C’, AD//A’D’, then CD//C’D’
what is the set up for projective arithmetic
- assume desargues & pappus hold
- chose any 2 different lines to be the x and y axes, their pt on intersection is the origin
- chose a line (not on the axes) and call it the line at ∞
- declare l₁//l₂ iff their intersection is on the line at ∞
how do we do addition on the x axis?
- take a,b on x axis
- draw L // to x axis
- construct line from a to where L meets the y axis
- construct line from b up to L // to y axis
- construct line from where the previous line meets L to the x axis // to the 1st constructed line
- a + b is where it meets the x axis
how do we do multiplication on the x axis?
- choose a pt on each axis and call it 1, take a,b on x axis
- draw 1x1y
- draw 1ya
- draw line from b // to 1x1y
- draw line from where the above line meets y // to 1ya
- ab is where it meets the x axis
note: by scissors thm the position of ab depends on the position of 1x but not 1y
show that sums on the x axis correspond to sums on the y axis
(1) compute a+b on x axis using:
- line from ax to ay
- line L // to x axis at ay
- line from bx to by // to axay
- line m from bx // to y axis
- line from intersection of m and L down to x axis // to axay & bxby
(2) compute a+b on y axis using:
- line from ay to bx
- line m // to y axis at bx
- line from by // to x axis
- line from intersection of m and above line to y axis // to aybx
(3) there is a pappus config. between m and y axis
- so a+b on y // to bxby
- the line from intersection of m & L to a+b on x is // to bxby
- so line from a + b on x to a + b on y is a single straight line so the sums correspond
how can we show that products correspond?
using the scissors thm
state the commutative laws
a + b = b + a
ab = ba
we can use pappus to show these
state the associative laws
a + (b + c) = (a + b) + c
a(bc) = (ab)c
state the identity laws
a + 0 = a
a(1) = a
state the inverse laws
a + (-a) = 0
a(a⁻¹) = 1
state the distributive laws
a(b + c) = ab + ac
