Scientists Create Formula for Perfect Parking (Original paper)
You will need a compass and straightedge for the exam.
The final exam is an opportunity for you to review and deepen your understanding of material that we covered during this semester and to improve your grade from homework and exam items that you may have originally missed.
Questions on the final exam will be drawn from the following sources:
- Exams 1 and 2
- Axioms and theorems for incidence geometry
- Prove independence
- Incidence Theorem 1
- Classification of geometries (p. 30 and §1.3 #6, 11)
- Euclid's,
Hilbert's, Birkhoff's, and the SMSG axiom systems for Euclidean
geometry (See the Chapter 2 Study Guide)
- HW 4 #4, 5
- Theorem 3.2.5 (as Pasch's Axiom in Hilbert's system and as a consequnce of the plane separation postulate in the SMSG system)
- Neutral geometry
- §3.2 #12, 13
and §3.6 #1
- Theorems 3.2.8, 3.3.1 (ASA), 3.3.3 (AAS), 3.3.8 (The Hinge
Theorem), 3.3.9 (SSS)
See the Section 5.3 Study Guide
- Parallel lines in neutral geometry
- Theorem 3.4.1 (AIAT)
- Theorems 3.4.5 (EFP ⇔ PPP), 3.4.6 (PPP ⇔ Converse of AIAT)
- Definitions,
constructions, and results about Saccheri and Lambert quadrilaterals
- Euclidean constructions and their proofs
- Euclid's
Propositions 1, 2, 3, 9, 10, 11, 12, 22, 23, 31, and 46
- Partition a segment (Construction 4.9.6), Construct a product and quotient of two lengths given a unit (pp. 213-214)
- Know which of
these are strictly Euclidean and why
- Transformations
- Definitions of the
Euclidean isometries, reasons they are well-defined, and proofs they
are isometries
See the Chapter 5 Study Guide
- Corollary 5.3.8
- Proof that the
isometries form a group under composition
- Hyperbolic geometry
- Definition 6.2.1
(angle of parallelism) including the definition of d0 on pp. 325-326
- Theorems 6.2.1,
6.2.3, 6.2.6, 6.3.6
See the Sections 6.2-6.3 Study Guide
- Holonomy and Area:
Proof that H(Δ) = K Area(Δ)
See Study Guide 6.4-6.5 to do this in 2 parts: K Area(Δ) = Σαi – π and H(Δ) = Σαi – π
- Be able to generate and
explain the Banach-Tarski deconstruction of the Poincaré disk.
- Elliptic geometry
- The Polar Property Theorem and Theorems 6.8.2, 6.8.3,
6.8.4, 6.8.5, 6.8.6
- Differential geometry
- Know the meaning of the covariant derivative, why it is
an intrinsic derivative, and its relationsship to "straight lines"
(geodesics) and parallel transport.
- Be able to do any portion of
the computations in HW 13.
Tim Duncan Calls Out Geometric Angle Needed To Make Bank Shot
Important: You will need a compass and straightedge for the exam.
In studying for this exam, you should be well-prepared in each of the following areas:
Be prepared to respond to any question from Exam 1.
Euclidean Constructions
- Be able to construct a copy of a given angle on a given ray (4.9.5), partition a segment into a given number of congruent pieces (4.9.6), construct segments equal in length to the sum, difference, product, and quotient of given lengths provided a unit length (pp. 213-214).
- Be able to produce the constructions in Euclid's Propositions 22, 23, 31, 46.
- Be able to prove that the above constructions have the required properties.
- Know the definition of an isometry (Definition 5.3.7).
- Be able to define translations, relfections, rotations, and glide reflections using the SMSG axioms (as in Definitions 5.3.9, 5.3.10, 5.3.11, and 5.3.15).
- Be able to explain why the these transformations are well-defined (i.e., show the existence and uniqueness of the image of an arbitrary point).
- Be able to prove that translations, relfections, rotations, and glide reflections are isometries of the plane (included in the theorem list below).
- Be able to construct the image of a point under each of the isometries of the plane using the constructions in their definition.
- Be able to prove that the set of isometries of the plane form a group.
- Be
able to give a transformational proof that the perpendicular bisectors
of the sides of any triangle are concurrent.
- Be familiar with the Poincaré disk model of hyperbolic geometry.
- Understand the definition of the angle of parallelism, its geometric interpretation, and its relationship to the distinction between the Euclidean and hyperbolic parallel postulates.
- Be able to prove that the hyperbolic parallel postulate implies that the sum of the angles in a triangle is less than 180°.
- Be able to define inversion (either in Euclidean terms as in §5.5 or on the Poincaré disk)
- Be able to prove that all 2/3-ideal triangles are congruent.
- Be able to prove that the area of a 2/3-ideal triangle is a linear function of its exterior angle.
- Be able to explain why the area of all ideal triangles on the Poincaré disk are the same.
- Be able to show that the area of a triangle Δ on the Poincaré disk must be given by A(Δ) = k d(Δ) for some constant k.
- Be able to show that the area of a triangle on a sphere of
radius ρ is given by A(Δ) = –ρ2d(Δ).
- Know what parallel transport along a line means.
- Be able to find and illustrate the holonomy around a spherical or hyperbolic triangle.
- Be
able to show that for a geometric space M with curvature K, we have K ·
area(M)
= 2π ·
χ(M).
- Corollary 5.3.8.
- Theorem 5.3.13.
- Theorem 5.3.15.
- Theorem 5.3.16.
- Theorem 6.2.1.
- Theorem 6.2.2.
- Theorem 6.2.3.
- Corollary 6.2.4.
- Theorem 6.2.5.
- Theorem 6.2.6.
- Theorem 6.3.3.
- Theorem 6.3.6.
- Theorem 6.4.1.
- Euclid's Propositions 9, 10, 11, 12, 22, 23, 31, 46
Axiomatic systems.
- Understand the meaning and purpose of the basic elements of an axiomatic system: undefined terms, axioms/postulates, definitions, propositions/theorems, and proofs.
- Be able to prove theorems or provide counterexamples using simple axiomatic systems such as the Fe-Fo system, four-point geometry, Fano's geometry, Young's geometry, and incidence geometry.
- Understand the role of models in an axiomatic system and
the meaning of an isomorphism betweem models. Be able to check whether
a 1-1 mapping between two models is an isomorphism and be able to
create an isomorphism between simple isomorphic models.
- Understand the meaning of independence of axioms in an axiomatic systems and be able to demonstrate independence by providing and explaining appropriate models.
- Be able to explain which of the following geometries are examples of others: incidence, finite, four-point, Fano's, Young's, neutral, Euclidean, hyperbolic, spherical, and elliptic.
- Understand the historical development and relationships among the 4 systems.
- Know what the undefined terms are for each system (or treated as undefined for Euclid).
- Know which axiom(s) are used to establish the first congruences between different segments and angles in each system.
- Know how Euclid began to establish additional congruences
through his first 12 propositions.
- Be able to paraphrase each of Euclid's 5 postulates and know the basic ideas captured by his common notions.
- Know the order of and be able to paraphrase each of Euclid's first six propositions.
- Be able to reproduce Euclid's constructions in Propositions
1-3 and Propositions 9-12 and prove that the constructions produce
objects with the desired properties. Be able to explain which previous
propostitions each is used in each of these proofs, and thus why the
propositions are presented in their given order. Be able to prove the
corresponding theorems in neutral geometry using the SMSG axioms and
describe what is different about the proofs in the two systems.
- Be able to reproduce Euclid's proofs of the Isosceles Triangle Theorem and it's converse (Propositions 5 and 6).
- Be able to describe the key features of each of the four
axiomatic systems, the strengths and weaknesses of each, the goals each
one was trying to meet, and examples of how each system met their goals
(or at least came close to meeting).
- Know what Pasch's Axiom is and be able to describe what important role it plays in Hilbert's axiomatic system.
- Know what the Plane Separation Postulate is and be able to describe what important role it plays in the SMSG axiomatic system.
- Understand why the SMSG axiomatic system is not independent and how it borrows key aspects from each of Hilbert's and Birkhoff's axiomatic systems.
- Be able to explain why Birkhoff's geometry is Euclidean
despite not having an explicit parallel postulate (see pp. 63-64).
- Be able to describe general and
specific flaws in Euclid's Elements and how Hilbert's, Birkhoff's, and
the SMSG systems addressed these issues in more rigorous ways.
- Know the 3 possible options for adding a parallel axiom to incidence geometry (leading to Euclidean, projective, or hyperbolic geometries).
- Know why the possibility of an elliptic geometry is excluded by the axioms of neutral geometry.
- Know what statements are equivalent to Euclid's fifth (parallel) postulate.
- Be able to describe what axioms are needed to prove the Alternate Interior Angle Theorem and the implications for which axiomatic systems it is a theorem.
- Know what a Saccheri quadrilateral is, why Saccheri was
trying to prove that any Saccheri quadrilateral is a rectangle (defined
as a quadrilateral with four right angles), and why he was unable to do
so.
- Be able to identify the errors in the "proof" of the
parallel postulate given by Proclus described on p. 126.
- Incidence Theorems 1-3 in §1.4.
- Theorem 3.2.2.
- Theorem 3.2.5 (Pasch's Axiom, which is really a theorem in the SMSG system).
- Theorem 3.2.8.
- Theorem 3.3.1 (ASA).
- Theorem 3.3.3 (AAS).
- Theorem 3.3.8 (The Hinge Theorem which directly implies SSS).
- Theorem 3.4.1.
- Corollary 3.4.2.
- Corollary 3.4.3.
- Corollary 3.4.4.
- Theorem 3.4.5.
- Theorem 3.4.6.
- Theorem 3.4.7.
- Theorem 3.4.8.
- Theorem 4.2.2.
- Corollary 4.2.3.
- Theorem 4.2.4.
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