13.4: Math Models and Geometry - Mathematics

We are surrounded by all sorts of geometry. Architects use geometry to design buildings. Artists create vivid images out of colorful geometric shapes. Street signs, automobiles, and product packaging all take advantage of geometric properties. In this chapter, we will begin by considering a formal approach to solving problems and use it to solve a variety of common problems, including making decisions about money. Then we will explore geometry and relate it to everyday situations, using the problem-solving strategy we develop.

  • 13.4.1: Solve Money Applications
    Solving coin word problems is much like solving any other word problem. However, what makes them unique is that you have to find the total value of the coins instead of just the total number of coins. For coins of the same type, the total value can be found by multiplying the number of coins by the value of an individual coin. You may find it helpful to put all the numbers into a table to make sure they check.
  • 13.4.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 1)
    An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. If the sum of the measures of two angles is 180°, then they are supplementary angles. But if their sum is 90°, then they are complementary angles. We will adapt our Problem Solving Strategy for Geometry Applications. Since these applications will involve geometric shapes, it will help to draw a figure and label it with the information from the problem.
  • 13.4.3: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 2)
    Triangles are named by their vertices. For any triangle, the sum of the measures of the angles is 180°. Some triangles have special names such as the right triangle which has one 90° angle. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. To solve problems that use the Pythagorean Theorem, we will need to find square roots.
  • 13.4.4: Use Properties of Rectangles, Triangles, and Trapezoids (Part 1)
    Many geometry applications will involve finding the perimeter or the area of a figure. The perimeter is a measure of the distance around a figure. The area is a measure of the surface covered by a figure. The volume is a measure of the amount of space occupied by a figure. A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and the adjacent side as the width, W.
  • 13.4.5: Use Properties of Rectangles, Triangles, and Trapezoids (Part 2)
    Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of a triangle is one-half the base times the height. An isosceles triangle is a triangle with two sides of equal length is while a triangle that has three sides of equal length is an equilateral triangle. A trapezoid is four-sided figure with two sides that are parallel, the bases, and two sides that are not. The area of a trapezoid is one-half the height times the sum of the bases.

Figure 9.1 - Note the many individual shapes in this building. (credit: Bert Kaufmann, Flickr)

Intersection of Algebraic Topology/Geometry and Model Theory/Set Theory

Is there any intersection between the ideas of Algebraic Topology/Geometry (I know that there is most certainly a non-trivial intersection between Algebraic Geometry, Algebraic Topology, Arithmetic Geometry, etc.) and the ideas of Model Theory, Set Theory and more fundamental topics? I have heard that there are some powerful tools from commutative geometry being applied to topology (correct me if I'm wrong of course) like Andre-Quillen Cohomology, and I remember once seeing a talk about applying some ideas from model theory to commutative algebra (something like applying model theory to put upper bounds on Betti numbers of Cohen-Macaulay rings?). So I'm just wondering if any of these foundational ideas are ever relevant (in the non-obvious ways, i.e. we can't do math without foundations) to these other subjects.

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Represent transformations in the plane using, e.g., transparencies and geometry software describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Models of Geometric Surfaces

The Mathematical Institute has a large collection of historical mathematical models, designed and built over a hundred years ago. While the models retain their aesthetic appeal despite showing the scars of more than a century of use, their purpose can now appear obscure. Oxford undergraduate Adam Barker undertook a summer project (supervised by Sam Howison) to catalogue the models and put together these webpages describing the models with their history and underlying mathematics. The project was made possible with financial support from Christ Church, Oxford, which is gratefully acknowledged. We hope this web resource will make the material accessible to a wide audience.

Much of the content should be accessible to somebody about to start a mathematics degree, occasionally touching on more stretching topics (the harder material can be ignored). The aesthetic beauty of the models should be enjoyable for anyone with an interest in mathematics, art or history, regardless of your level of mathematical training.

Some theory relating to the models (e.g. projective geometry) is fairly standard undergraduate material, but some models exhibit curious properties which are not now commonly studied (e.g. Dupin Cyclides). The models give a snapshot of geometric concepts that were considered important over a hundred years ago. Looking back at them, we see how the study of the subject has since evolved. At the start of the 19th century, geometry was mostly still a visually understandable topic. In line with common intuition about geometry, familiar shapes, curves, surfaces, and other constructions were studied at the forefront of research. However, mathematicians such as Riemann developed more general and abstract structures which extended geometry beyond familiar 3-dimensional Euclidean space. This gave rise to modern differential geometry. Other mathematicians such as Hilbert and Noether developed the algebraic aspects of the subject, giving rise to what is now known as algebraic geometry. Both strands are central to modern mathematics and a host of applications in physics and elsewhere.

The site includes crash-courses in algebraic and differential geometry, with links to sites such as Wolfram MathWorld and Wikipedia. We suggest you start with a look at the Theory section, but it is also possible to jump straight in to look at the models - any unfamiliar terminology is linked back to Theory anyway. To explore the catalogue, just click the blue links on the left of this page. We hope you will enjoy the site!

Adam Barker and Sam Howison

Please contact us for feedback and comments about this page. Last update on 28 September 2015 - 15:00.

Geometric modeling

Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.

The shapes studied in geometric modeling are mostly two- or three-dimensional, although many of its tools and principles can be applied to sets of any finite dimension. Today most geometric modeling is done with computers and for computer-based applications. Two-dimensional models are important in computer typography and technical drawing. Three-dimensional models are central to computer-aided design and manufacturing (CAD/CAM), and widely used in many applied technical fields such as civil and mechanical engineering, architecture, geology and medical image processing. [1]

Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an opaque algorithm that generates its appearance. [ citation needed ] They are also contrasted with digital images and volumetric models which represent the shape as a subset of a fine regular partition of space and with fractal models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred: for instance, a digital image can be interpreted as a collection of colored squares and geometric shapes such as circles are defined by implicit mathematical equations. Also, a fractal model yields a parametric or implicit model when its recursive definition is truncated to a finite depth.

Notable awards of the area are the John A. Gregory Memorial Award [2] and the Bézier award. [3]

  • Jean Gallier (1999). Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann. This book is out of print and freely available from the author.
  • Gerald E. Farin (2002). Curves and Surfaces for CAGD: A Practical Guide (5th ed.). Morgan Kaufmann. ISBN978-1-55860-737-8 .
  • Michael E. Mortenson (2006). Geometric Modeling (3rd ed.). Industrial Press. ISBN978-0-8311-3298-9 .
  • Ronald Goldman (2009). An Integrated Introduction to Computer Graphics and Geometric Modeling (1st ed.). CRC Press. ISBN978-1-4398-0334-9 .
  • Nikolay N. Golovanov (2014). Geometric Modeling: The mathematics of shapes. CreateSpace Independent Publishing Platform. ISBN978-1497473195 .

For multi-resolution (multiple level of detail) geometric modeling :

  • Armin Iske Ewald Quak Michael S. Floater (2002). Tutorials on Multiresolution in Geometric Modelling: Summer School Lecture Notes. Springer Science & Business Media. ISBN978-3-540-43639-3 .
  • Neil Dodgson Michael S. Floater Malcolm Sabin (2006). Advances in Multiresolution for Geometric Modelling. Springer Science & Business Media. ISBN978-3-540-26808-6 .

Subdivision methods (such as subdivision surfaces):

  • Joseph D. Warren Henrik Weimer (2002). Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann. ISBN978-1-55860-446-9 .
  • Jörg Peters Ulrich Reif (2008). Subdivision Surfaces. Springer Science & Business Media. ISBN978-3-540-76405-2 .
  • Lars-Erik Andersson Neil Frederick Stewart (2010). Introduction to the Mathematics of Subdivision Surfaces. SIAM. ISBN978-0-89871-761-7 .

This applied mathematics-related article is a stub. You can help Wikipedia by expanding it.

Introduction xix
Jean-Michel TANGUY

Chapter 1. Reminders on the Mechanical Properties of Fluids 1
Jacques GEORGE

1.1. Laws of conservation, principles and general theorems 1

1.2. Enthalpy, rotation, mixing, saturation 13

1.3. Thermodynamic relations, relations of state and laws of behavior 20

1.5. Dynamics of geophysical fluids 30

Chapter 2. 3D Navier-Stokes Equations 35
Véronique DUCROCQ

2.1. The continuity hypothesis 35

2.2. Lagrangian description/Eulerian description 36

2.3. The continuity equation 37

2.4. The movement quantity assessment equation 38

2.5. The energy balance equation 41

2.6. The equation of state 41

2.7. Navier-Stokes equations for a fluid in rotation 41

Chapter 3. Models of the Atmosphere 43

3.2. The various simplifications and corresponding models 44

3.3. The equations with various systems of coordinates 56

3.4. Some typical conformal projections 61

3.5. The operational models 67

Chapter 4. Hydrogeologic Models 71
Dominique THIÉRY

4.1. Equation of fluid mechanics 71

4.2. Continuity equation in porous media 72

4.3. Navier-Stokes’ equations 74

4.5. Calculating mass storage from the equations of state 80

4.6. General equation of hydrodynamics in porous media 82

4.7. Flows in unsaturated media 84

Chapter 5. Fluvial and Maritime Currentology Models 93
Jean-Michel TANGUY

5.1. 3D hydrostatic model 99

5.2. 2D horizontal model for shallow water 107

5.3. 1D models of fluvial flows 119

5.4. Putting 1D models into real time 131

Chapter 6. Urban Hydrology Models 155
Bernard CHOCAT

6.1. Global models and detailed models used in surface flows 156

6.2. Rainfall representation and rainfall-flow transformation 161

6.3. Modeling of the losses into the ground 164

6.5. Modeling of the hydraulic operating conditions of the networks 177

6.6. Production and transport of polluting agents 189

Chapter 7. Tidal Model and Tide Streams 213
Bernard SIMON

7.2. Non-harmonic methods 215

Chapter 8. Wave Generation and Coastal Current Models 235
Jean-Michel TANGUY, Jean-Michel LEFÈVRE and Philippe SERGENT

8.1. Types of swell models 235

8.2. Spectral approach in high waters 242

8.3. Wave generation models 246

8.4. Wave propagation models 260

8.5. Agitating models within the harbors 266

8.6. Non-linear wave model: Boussinesq model 298

8.7. Coastal current models influenced or created by the swell 320

Chapter 9. Solid Transport Models and Evolution of the Seabed 335
Benoît LE GUENNEC and Jean-Michel TANGUY

9.1. Transport due to the overthrust effect 338

9.3. Bed forms and roughness 344

9.4. Suspension transport 346

9.5. Evolution model of movable beds 357

Chapter 10. Oil Spill Models 371

10.1. Behavior of hydrocarbons in marine environment 371

10.2. Oil spill drift models 372

10.3. Example: the MOTHY model 375

10.4. Calculation algorithm of the path of polluting particles 378

10.5. Example of a drift prediction map 379

Chapter 11. Conceptual, Empirical and Other Models 381
Christelle ALOT and Florence HABETS

11.1. Evapotranspiration 382

Chapter 12. Reservoir Models in Hydrology 397
Patrick FOURMIGUÉ and Patrick ARNAUD

12.3. Mathematical tools 401

12.5. Integration of the spatial information 405

Chapter 13. Reservoir Models in Hydrogeology 409
Dominique THIÉRY

13.1. Principles and objectives 409

13.3. Setting the model up 411

13.4. Data and parameters 412

13.5. Application domains 412

Chapter 14. Artificial Neural Network Models 419

14.1. Neural networks: a rapidly changing domain 420

14.2. Neuron and architecture models 422

14.3. How to take into account the non-linearity 429

14.4. Case study: identification of the rainfall-runoff relation of a karst 434

Chapter 15. Model Coupling 445
Rachid ABABOU, Denis DARTUS and Jean-Michel TANGUY

Chapter 16. A Set of Hydrological Models 493
Charles PERRIN, Claude MICHEL and Vasken ANDRÉASSIAN

What is mathematical modeling?

While there is no consensus yet as to a precise definition of this term, mathematical modeling is generally understood as the process of applying mathematics to a real world problem with a view of understanding the latter. One can argue that mathematical modeling is the same as applying mathematics where we also start with a real world problem, we apply the necessary mathematics, but after having found the solution we no longer think about the initial problem except perhaps to check if our answer makes sense. This is not the case with mathematical modeling where the use of mathematics is more for understanding the real world problem. The modeling process may or may not result to solving the problem entirely but it will shed light to the situation under investigation. The figure below shows key steps in modeling process.

Mathematical modeling approaches can be categorized into four broad approaches: Empirical models, simulation models, deterministic models, and stochastic models. The first three models can very much be integrated in teaching high school mathematics. The last will need a little stretching.

Empirical modeling involves examining data related to the problem with a view of formulating or constructing a mathematical relationship between the variables in the problem using the available data.

Simulation modeling involve the use of a computer program or some technological tool to generate a scenario based on a set of rules. These rules arise from an interpretation of how a certain process is supposed to evolve or progress.

Deterministic modeling in general involve the use of equation or set of equations to model or predict the outcome of an event or the value of a quantity.

Stochastic modeling takes deterministic modeling one further step. In stochastic models, randomness and probabilities of events happening are taken into account when the equations are formulated. The reason behind this is the fact that events take place with some probability rather than with certainty. This kind of modeling is very popular in business and marketing.
Examples of mathematical modeling can be found in almost every episode of the TV hit drama series The Numbers Behind NUMB3RS: Solving Crime with Mathematics

The series depicts how the confluence of police work and mathematics provides unexpected revelations and answers to perplexing criminal questions. The mathematical models used may be way beyond K-12 syllabus but not the mathematical reasoning and thinking involve. As the introduction in each episode of Numbers says:

We all use math every day
to predict weather, to tell time, to handle money.
Math is more than formulas or equations
it’s logic, it’s rationality,
it’s using your mind to solve the biggest mysteries we know.

The Mathematics Department of Cornell University developed materials on the mathematics behind each of the episodes of the series. You can find the math activities in each episode here.

The challenge in mathematical modelling is “. . . not to produce the most comprehensive descriptive model but to produce the simplest possible model that incorporates the major features of the phenomenon of interest.” -Howard Emmons

100- and 200- Level courses

  • Math 092/094/098 & 102/103/105/108 – offered most every Fall, Spring, Summer, WinteriM
  • Math 116, 117 – every Fall, Spring, Summer, UWinteriM
  • Math 175, 176 – every Fall and Spring
  • Math 205 – every Fall and Spring
  • Math 211 – every Fall, Spring, Summer
  • Math 221/222 sequence – every academic year (fall-spring)
  • Math 231, 232, 233, 234 – every Fall, Spring, Summer
  • Math 275, 276, 277, 278 – each at least once during the academic year, possibly more often

Question: Do I need to take a math class at UWM?

  • All UWM students must satisfy the two-level Quantitative Literacy requirement in order to graduate: Quantitative Literacy Parts A (QLA) and B (QLB). Many departments offer QLB courses (ask your advisor which is right for you), but QLA is only satisfied by a Math curricular area course or equivalent placement exam score.
  • If you earned a Math Placement Test Code of 30 or higher, you satisfied your QLA requirement via this test!
  • Earning a grade of C or higher in Math 102 or 103 (or 105, 108, 175) will also satisfy QLA – and these courses also satisfy the L&S math requirement for the BA degree.

Question: Who can take Math 092/102 or Math 103 to satisfy their degree requirements?

    • Primarily students in the Arts, Humanities, and Social Welfare can take 102 or 103, as they do not need to take any further math or science courses.
    • Most students in the School of Education who do not place directly into Math 175 are advised to take 103 or 092+102 first.
    • Anyone in a STEM, Business, or Health Sci program, or anyone interested in the BS in L&S, will need different math courses for their program, and should consult their advisor to find the right courses (Math 094, 098, 105, 115, 116, 117, 211, 231, etc.).
    • All students should double check their program requirements and/or consult their academic advisor when choosing which math courses to take each semester.
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    Question: Do I qualify for Math 103?

    • The prerequisite of Math 103 is either an ACT-math subscore of 18 or higher, or Math Placement Level 10—attained by earning a level 10 on the Math Placement Test, a grade of C or better in Math 90, or a grade of D or better in Math 94.

    Question: When can I take Math 103?

    • Students are expected to complete QLA within their first 60 credits, so that they can complete their QLB course without delays to their graduation.
    • Starting AY 17-18, Math 103 will primarily shift to Spring semesters.
    • Math 103 will be offered in summer and UWinterm (if sufficient enrollment).
    • There will be a limited number of sections of 103 offered each Fall (either online, in evenings and/or in early mornings), for students who couldn’t complete 103 earlier in their program due to special circumstances

    Question: Should/Could I take Math/Philos 111 to satisfy my QLA?

    • Math or Philos 111 is a good QLA choice for students who intend to take Philos 211 as their QLB course. (Note: Philos 211 is a logic course and is NOT the same as Math 211 Survey of Calculus).
    • Math 111 is “jointly offered” with Philos 111. No matter which one you register for, it’s the same class and you’ll get QLA credit (as long as you get a C or better.
    • Math/Philos 111 has the same prereq as Math 103.

    Question: I don’t meet the prerequisite of Math 103. What math course should I take?

    Question: When can I take Math 092 and 102?

    • Students must complete remedial course work (Math 9x) in their first 30 credits, basically the first year.
    • Math 092&102 will primarily shift to Spring and be offered in paired co-requisite sections. For example, Math 092 Section 57 MW 11-12:15, and Math 102 Section 57 TR 11-12:15, will meet in the same room with the same instructor, and the courses will be paired in PAWS so that students must enroll in both. Students will work in 092 on Mon (or Wed) on the background needed for the 102 material for Tues (or Thurs).
    • For students with extraordinary schedule restrictions: there will be a very limited number of freestanding 092’s each term, with enrollment by permission only. Contact Kelly Kohlmetz [email protected] for permission to enroll.
    • For students taking a paired 092+102, their 092 grade will not be lower than the grade they earn in 102.
    • Starting Spring 2018, there will be a very limited number of freestanding sections of 102 offered each fall and spring. For example, for students who take the linked 092/102 and who pass 092 but not 102, or students who were given permission to take a freestanding 092.
    • Starting Summer 2018, each of 092 and 102 will be offered in summers, not paired, likely both online, with 092 in first 6 week session and 102 in second 6 week session.
    • Starting in Fall 2018, online versions of each of 092 and 102 will be offered, not paired, with enrollment by permission only (for permission contact Kelly Kohlmetz [email protected])

    Question: Why must I take 6 credits of math in the same semester, if I’m not in a hurry to finish math since I have no more math requirements?

    • The short answer: We care about your success, and you’re more likely to pass both courses if you’re taking math every day–focusing more on math and having fewer other courses to keep track of.
    • The traditional model of taking Math 092 in fall and 102 in spring is a “leaky pipeline.” Only about half of students who enroll in Math 092 in fall finish 102 in spring. This failure negatively affects student success and retention.
    • The co-requisite model has been showed to significantly increase student success. For example, the entire Tennessee Board of Regents Universities and Community Colleges increased the percent of students who passed a credit-bearing math course in their first year from 59% to 75% by implementing co-requisite remediation!

    Question: Why must I take 6 “credits” of math courses in one semester, if I struggle with math?

    • As stated above: We care about your success, and you’re more likely to pass both courses if you’re taking math every day–focusing more on math and having fewer other courses to worry about.
    • Meeting four days per week allows the instructor to get to know your individual areas of strength and weakness – and to have the time and flexibility to address them in ways to help you succeed.
    • PLEASE NOTE: You will be taking six-credits of math! Be sure that you are allowing yourself enough time in your schedule for homework every night, weekly tutoring and weekly meetings with your instructor. This will be the equivalent of TWO courses worth of homework and studying, which means about 12 hours of time is needed outside of class. We advise you not to load your schedule with more than two other courses.

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    Polygons & Polygrams

    Triangle(3) Square(4) Pentagon(5) Hexagon(6) Heptagon(7) Octagon(8) Nonagon(9) Decagon(10) Hendecagon(11) Dodecagon(12)

    Kite, Rhombus, Parallelogram, Square, Trapezoid, Trapezium, Isosceles Trapezoid, Rectangle (Golden Ratio

    4 Equilateral Triangles (PS-12)

    Platonic Solids

    Complete set of 5 Platonic Solids ideal for Geometry, Physics, and Astronomy class

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    9" - $55

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    9" - $55

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    9" - $75

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    Misc Tetrahedral symmetry

    Abstract Tetrahedron (TS-53)

    Tetrahedron 3 Bent Squares (TS-16)

    Tetrahedron 90' Cube Corners (TS-34)

    4 Morphed Tetrahedron (TS-28)

    Tetrahedron Closed Weave (TS-26)

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    9" - $20

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    Octahedron 18 squares (OS-17)

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    9" - $25

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    9" - $75

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    9" - $45

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    Icosadodeca 6 Decagons (IS-04)

    Dodecahedron of 20 Triangles (IS-6)

    9" - $85

    Dodecahedron (IS-8) 12 great pentagons

    Icosahedron (IS-14) 12 pentagons & stars

    9" - $85

    Icosahedron (IS-16) 12 pentagons

    Icosa Beveled Pentagons (IS-49)

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    Math 410: Modern Geometry

    Euclid wrote the first preserved Geometry book which has traditionally been held up as a role model for logical reasoning inside and outside mathematics for thousands of years. However, Euclid has several subtle logical omissions, and in the late 1800s it was necessary to revise the foundations of Euclidean geometry. The need for such a revision was partly due to advances in mathematical logic and changes in the conception of an axiom system. In this course we will review the traditional approach, and then a modern approach based on Hilbert's axioms developed around 1900. The famous mathematician David Hilbert, building on work of several other mathematicians, was able to develop axioms that allow one to develop geometry without any overt or covert appeals to intuition. His idea was that, although intuitions are important in discovering, motivating, communicating and appreciating the theorems, rigorous proofs should not appeal to them. With the more modern approach to the axiomatic method that is not logically dependent on intuition, mathematicians are free to develop more types of geometries than the traditional Euclidean geometry. We will discuss different types and models of geometry that are used today. These include finite geometries with applications in discrete mathematics and number theory, spaces of more than three dimensions, geometries whose coordinates are not real numbers, and geometries where a line can pass through a circle without actually intersecting the circle. Many of these geometries are useful, and not just curious examples.

    A second major theme of the course will be the history and role of the parallel postulate. The parallel postulate makes the assumption that anytime you have a point P and a line l not going through P, there is one and only one line m going through P that is parallel to l (this is closely related to Euclid's original fifth postulate). Modern geometry began in the 1800s with the realization that there are interesting consistent geometries for which the parallel postulate is false. For example, hyperbolic and elliptic geometry do not satisfy the parallel postulate.

    Since this postulate is less intuitively obvious than the other axioms of geometry, many mathematicians, especially medieval Arab mathematicians and later several European mathematicians of the 1700s, tried to make the parallel postulate a theorem and not an axiom. This goes along with the traditional idea that axioms should be restricted to a few simple, self-evident propositions, and the rest of the subject should be built upon these using proof. However, no mathematician was able to show that the parallel postulate followed as a theorem from the other axioms. Several prominent mathematicians thought that they had a proof of the parallel postulate, but subtle flaws were later discovered in their proofs. Finally mathematicians such as Lobachevsky and Bolyai started to believe that it is possible for there to be geometries where the parallel postulate fails, and they proved theorems about such non-Euclidean geometries. After the discovery of (Euclidean) models of non-Euclidean geometries in the late 1800s, no one was able to doubt the existence and consistency of non-Euclidean geometry. Also, these models show that the parallel postulate is independent of the other axioms of geometry: you cannot prove the parallel postulate from the other axioms.

    Required Texts (2)

    1. Euclidean and Non-Euclidean Geometries : Development and History
    by Marvin Jay Greenberg
    ISBN: 0716799480
    Publisher: W. H. Freeman and Company (4th edition 2008)
    See Amazon or Barnes and Noble for descriptions of this textbook.

    2. Book I of the Elements of Euclid. You can use the recent translation by Richard Fitzpatrick or the classic translation by Thomas Heath.

    You have at least several options for obtaining Book 1 of Euclid.

    A. I recommend downloading Fitzpatrick's translation which is free, contains all of Euclid, and as a bonus has the original Greek. Instead of printing this out, you might want to buy the Hard bound edition.

    B. Alternatively, there is an inexpensive complete edition published by Green Lion Press.
    Click here for the first few pages of the Elements from this edition.

    C. If instead you want an edition with extensive commentary, you can get the Dover edition of Euclid instead (same translation, but with Heath's commentary): Volume 1 has all the text we need, so Volume 2 and 3 are optional.
    See Amazon or Barnes and Noble for descriptions. For the optional volumes: Amazon (vol 2), Barnes and Noble (vol 2), Amazon (Volume 3), Barnes and Noble. (Volume 3)

    D. There is an online version designed by David E. Joyce. The translation is basically Heath's but slightly less literal in order to make it more readable.

    Optional Texts (3)

    1. Geometry: Euclid and Beyond
    by Robin Hartshorne
    ISBN: 0387986502
    Publisher: Springer (2000)
    See Amazon or Barnes and Noble for descriptions of this textbook.

    2. Foundations of Geometry
    by David Hilbert
    ISBN: 0875481647
    Publisher: Open Court (1971 translation of an early 20th century classic)
    See Amazon or Barnes and Noble for descriptions of this book.
    Get an older (1902) edition for free from Project Gutenberg.

    3. Non-Eulcidean Geometry
    by Roberto Bonola
    ISBN: 0486600270
    Publisher: Dover (reprint of the 1912 edition).
    This book includes translations of articles by Lobachevski and Bolyai, two originators of non-Euclidean geometry.
    See Amazon or Barnes and Noble for descriptions of this book.

    Watch the video: find unknown measures (December 2021).