Homework 1
13 problems
Section 2.1 4, 10, 12, 14
Section 2.2 6, 10, 12,
Enrichment 1 Find two more models for the Sibley Geometry.
Section 2.3 4,
9. Explain why the answer in the back of the book is true.
10.
Enrichment 2 Provide a model and prove the first 2 theorems in
Fano’s Geometry (attached). What is the situation with respect to parallel lines in Fano’s Geometry?
Enrichment 3 Provide two models for Eve’s Geometry (attached). Hint: substitute point and line for dabba and abba.
Fano’s Geometry
Gino Fano (1871 – 1952) began all the work in finite geometries.
He began exploring finite geometries when he was 21 years old.
Undefined Terms: point, line, and on
Axioms
1. There exists at least one line.
2. Every line has exactly 3 points on it.
3. Not all the points are on the same line.
4. There is exactly one line on any two distinct points.
5. There is at least one point on any two distinct lines.
Definitions:
Intersecting lines share points.
Parallel lines share no points.
Concurrent lines are three lines that share a point.
Model: please provide this. This is a categorical geometry.
Theorems:
T1 Two distinct lines intersect in exactly one point.
T2 There are exactly 7 points.
T3 Each point is on exactly 3 lines.
T4 There are exactly 7 lines.
Eve’s Geometry
Undefined terms: abba, dabba
Axioms:
A1 Every abba is a collection of at least two dabbas.
A2 There exist at least two dabbas.
A3 If D and E are two dabbas, then there exists one and only
one abba containing both of them.
A4 If A is an abba, then there exists a dabba, P, not in A.
A5 If A is an abba and D is a dabba not in A, then there exists
one and only one abba containing D and not containing any
dabba that is in A.
Models:
Find a model that has 4 dabbas and 6 abbas.
Find a non-isomorphic model that has 9 dabbas, 12 abbas, and 3 dabbas on each abba.
Theorems:
T1 Every dabba is contained in at least 2 abbas.
T2 There exist at least 4 distinct dabbas.
T3 There exist at least 6 distinct abbas.