MAT 360

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MAT 360

• Lecture 10 Hyperbolic Geometry

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Homework next week (Last hw!)

• Chapter 4 Problem 10.
• Chapter 5 Problem 8
• Chapter 6  Problems 2, 3, 5, 14
• Extra credit (deadline at the end of the course).
• Chapter 4 Problems 8 and 9
• Chapter 5 Problem 1
• Chapter 6 Problem 15.

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• To son Janos
• For Gods sake, please give it work on
  hyperbolic geometry up. Fear it no less than
  sensual passion, because it, too, may take up all
  your time and deprive you of your health, peace
  of mind and happiness in life.
• Wolfgang Bolyai

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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.

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Lemma

• If the hyperbolic axiom holds (and all the axioms
  of neutral geometry hold too) then rectangles do
  not exist.
• Exercise
• Find another formulation of this lemma.
• Prove it

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Proof of lemma

• If hyp axiom holds then Hilberts parallel
  postulate fails because it is the negation of
  hyp. Axiom.
• Existence of rectangles implies Hilberts
  parallel postulate
• Therefore, rectangles do not exist.

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Formulation Assume neutral geometry.

• If rectangles exist then hyperbolic axiom does
  not hold.
• If Euclid v does not hold then rect. do not
  exist.
• (any statement equivalent to Euclic V holds )
  then rect. do not exist.
• If hyp axiom holds then if three angles of a
  quadrilateral are right then the fourth angle is
  not right

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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)

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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.

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Theorem

• If hyperbolic axiom holds then all triangles have
  angle sum strictly smaller than 180.
• Can you prove this theorem?

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Recall

• Definition Two triangles are similar if their
  vertices can be put in one-to-one correspondence
  so that the corresponding angles are congruent.

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Similar triangles

• Recall Wallis attempt to fix the problem of
  Euclids V
• Add postulate Given any triangle ?ABC, and a
  segment DE there exists a triangle ?DEF similar
  to ?ABC
• Why the words fix and problem are surrounded by
  quotes?

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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.

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Corollary

• In hyperbolic geometry, there exists an absolute
  unit of length.

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Recall

• Quadrilateral ?ABCD is a Saccheri quadrilateral
  if
• Angles ltA and ltB are right angles
• Sides DA and BC are congruent
• The side CD is called the summit.
• Lemma In a Saccheri quadrilateral ?ABCD, angles
  ltC and ltD are congruent

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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.

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Question

• Question If l and m are parallel lines, and A
  and B are points in l, are A and B equidistant
  from m?

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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.

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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

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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.

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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.

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Hyperbolic Geometry Exercises

• Show that for each line l there exist a line l
  as in the hypothesis of the previous theorem. Is
  it there 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?

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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, ltXPQ is congruent to ltXPQ.

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Crossbar theorem

• If the ray AD is between rays AC and AB then AD
  intersects segment BC

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Dedekinds 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.

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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 ltXPQ and XPQ are called angles of
  parallelism.

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Question

• Given a line l, are the angles of parallelism
  associated to this line, congruent to each other?

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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.

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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.

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Janos Bolyai

• I cant say nothing except this that out of
  nothing I have created a strange new universe.