MAT 360

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MAT 360 . Lecture 10 Hyperbolic Geometry. What is the negation of Hilbert’s Axiom?. There exists a line l and a point P not on l such that there are at least two parallels to l through p. Hyperbolic axiom.

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Presentation Transcript
MAT 360
• Lecture 10 Hyperbolic Geometry
What is the negation of Hilbert’s Axiom?
• There exists a line l and a point P not on l such that there are at least two parallels to l through p.
Hyperbolic axiom
• There exist a line l and a point P not in l such that at least two parallels to l pass through P.
Lemma:
• If the hyperbolic axiom holds (and all the axioms of neutral geometry hold too) then rectangles do not exist.
Proof of lemma
• If hyp axiom holds then Hilbert’s parallel postulate does not hold because it is the negation of hyp. Axiom.
• Existence of rectangles implies Hilbert’s parallel postulate
• Therefore, rectangles do not exist.
Recall:
• In neutral geometry, if two distinct lines l and m are perpendicular to a third line, then l and m are parallel. (Consequence of Alternate Interior Angles Theorem)
Universal Hyperbolic Theorem
• In hyperbolic geometry, for every line l and every point P not in l there are at least two distinct parallels to l passing through p.
• Corollary: In hyperbolic geometry, for every line l and every point P not in l there are infinitely many parallels to l passing through p.
Theorem
• If hyperbolic axiom holds then all triangles have angle sum strictly smaller than 180.
• Can you prove this theorem?
Recall
• Definition: Two triangles are similar if their vertices can be put in one-to-one correspondence so that the corresponding angles are congruent.
Similar triangles
• Recall Wallis attempt to “fix” the “problem” of Euclid’s V:
• Add postulate: “Given any triangle ΔABC, and a segment DE there exists a triangle ΔDEF similar to ΔABC”
• Why the words fix and problem are surrounded by quotes?
Theorem
• In hyperbolic geometry, if two triangles are similar then they are congruent.
• In other words, AAA is a valid criterion for congruence of triangles.

Therefore,

all triangles are congruent

The distance

we “see” in this picture is not the one given by Theorem 4.3

Corollary
• In hyperbolic geometry, there exists an absolute unit of length.
Definition
• Angles <A and <B are right angles
• Sides DA and BC are congruent
• The side CD is called the summit.
Lemma
• In a Saccheri quadrilateral □ABCD, angles <C and <D are congruent

18

Definition
• Let l’ be a line.
• Let A and B be points not in l’
• Let A’ and B’ be points on l’ such that the lines AA’ and BB’ are perpendicular to l’
• We say that A and B are equidistant from l’ if the segments AA’ and BB’ are congruent.
Question
• Question: If l and m are parallel lines, and A and B are points in l, are A and B equidistant from m?
Theorem
• In hyperbolic geometry if l and l’ are distinct parallel lines, then any set of points equidistant from l has at most two points on it.
Lemma
• The segment joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both the base and the summit and this segment is shorter than the sides
Theorem
• In hyperbolic geometry, if l and l’ are parallel lines for which there exists a pair of points A and B on l equidistant from l’ then l and l’ have a common perpendicular that is also the shortest segment between l and l’.
Theorem
• In hyperbolic geometry if lines l and l’ have a common perpendicular segment MM’ then they are parallel and MM’ is unique. Moreover, if A and B are points on l such that M is the midpoint of AB then A and B are equidistant from l’.
Hyperbolic Geometry Exercises
• Show that for each line l there exist a line l’ as in the hypothesis of the previous theorem. Is it the only one?
• Let m and l be two lines. Can they have two distinct common perpendicular lines?
• Let m and n be parallel lines. What can we say about them?
Theorem

Where in the proof are we using the hyperbolic axiom?

• Let l be a line and let P be a point not on l. Let Q be the foot of the perpendicular from P to l.
• Then there are two unique rays PX and PX’ on opposite sides of PQ that do not meet l and such that a ray emanating from P intersects l if and only if it is between PX and PX’.
• Moreover, <XPQ is congruent to <X’PQ.
Crossbar theorem (Recall)
• If the ray AD is between rays AC and AB then AD intersects segment BC
Dedekind’s Axiom
• Suppose that the set of all points on a line is the disjoint union of S and T,
• S U T
• where S and T are of two non-empty subsets of l such that no point of either subsets is between two points of the other. Then there exists a unique point O on l such that one of the subsets is equal to a ray of l with vertex O and the other subset is equal to the complement.
Definition
• Let l be a line and let P be a point not in l.
• The rays PX and PX’ as in the statement of the previous theorem are called limiting parallel rays.
• The angles <XPQ and X’PQ are called angles of parallelism.
Question
• Given a line l, are the “angles of parallelism” associated to this line, congruent to each other?
Theorem
• Given lines l and m parallel, if m does not contain a limiting parallel ray to l then there exist a common perpendicular to l and m.
Definition
• Let l and m be parallel lines.
• If m contains a limiting parallel ray (to l) then we say that l and m are asymptotic parallel.
• Otherwise we say that l and m are divergently parallel.
Janos Bolyai
• I can’t say nothing except this: that out of nothing I have created a strange new universe.