Number 1 on the picture **Homework Set 7: Incidence Geometries, Part 2**1. **Consider the following models of finite geometries.** You may assume that each dark circle is a point and each line that looks like a line is a line in the model. Determine which of the following models are Incidence Geometries, Projective Planes, and/or Affine Planes. Be sure to clearly explain your answer. **Diagrams:** - **Diagram 1:** Three points connected in a triangular shape with one additional point inside, connected to the vertices. - **Diagram 2:** Six points forming a rectangle with an additional two points inside, aligned centrally and connected to the opposite points of the rectangle. - **Diagram 3:** Six points in a triangular formation, with each point connected to every other point. - **Diagram 4:** Seven points forming a hexagon with intersections, creating an inner star and connections among all points.2. **Consider the infinite geometry R², where points are (x, y) ∈ R and lines are ax + by + c = 0 with a, b, c ∈ R but not both a and b zero at the same time.** Show that R², as defined above, is an Affine plane. Note, this means that you need to verify each of the 4 axioms: (I1), (I2), (I3), and (P’). (This is similar to exercises 6.2 & 6.5c in your textbook.)3. **(Exercise 6.3c)** Show that the 4 axioms that define a Projective Plane are independent (see page 71). Note, I am not expecting a formal proof. A small diagram and description / reason is sufficient.4. **Consider the following interpretation of a geometry.** Begin with a punctured sphere in Euclidean 3-space. This is a sphere with one point, P, removed, where everything else about the sphere “looks normal”. Let “points” be points in the normal sense on the surface of the punctured sphere. Let “straight lines” be defined as circles on the surface of the sphere that pass through the point P (note: these are the only “infinite straight lines” for this infinite geometric model).

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my.charlotte.edu FH X Homework Set 7 ← → Cuncc.instructure.com/courses/175603/assignments/1660252 Type here to search X M Inbox (1,497) - zbessant@uncc.ex X hw-7-questions.pdf MATH-3181-001-10824 Assignments > Homework b Home | bartleby Homework Set 7 incidence geometries, part 2 x + 1. Consider the following models of finite geometrics. You may assume that each dark circle is a point and cach line that looks like a line is a line in the model. Determine which of the following models are Incidence Geometrics, Projective Planes, and/or Affine Planes. Be sure to clearly explain your answer. 2. Consider the infinite geometry R2, where points are (x, y) ER and lines are ar+by+ c = 0 with a, b, c ER but not both a and b zero at the same time. Show that R², as defined above, is an Affine plane. Note, this means that you need to verify each of the 4 axioms: (11), (12), (13), and (P¹). (This is similar to exercises 6.2 & 6.5c in your textbook.) 3. (exercise 6.3c) Show that the 4 axioms that define a Projective Plane are independent (see page 71). Note, I am not expecting a formal proof. A small diagram and description / reason is sufficient. 4. Consider the following interpretation of a geometry. Begin with a punctured sphere in Euclidean 3-space. This is a sphere with one point, P, removed, where everything else about the sphere "looks normal". Let "points" be points in the normal sense on the surface of the punctured sphere. Let "straight lines" be defined as circles on the surface of the sphere that pass through the point P (note: these are the only "infinite straight lines" for this infinite geometric model). 9 Download O 71°F Sunny ℗ Info ZOOM + 0 0 X Close 11:58 AM 10/16/2022 x I B

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT