Exploring the

Topograph

Trevor Hyde

Abstract for 14 February 2019

Binary quadratic forms have a long history passing through algebra,geometry, and number theory. Their study attracted some of thebiggest names in classical mathematics including Fermat, Euler,Legendre, and Gauss. But to the modern eye much of this worklooks like cryptic computations and algebraic manipulations. Sowhat’s the big deal? In the 1990’s John H. Conway introduced agreat way to visualize binary quadratic forms through a tool hecalls the topograph. In this talk we explain how the topographworks and demonstrate how it sheds new light on some beautifulclassical mathematics. Along the way we will make connectionswith continued fractions and hyperbolic geometry.

Abstract for 08 March 2018

Phil Tosteson

Normally, we think of [0, 1) and [0, 1] as having the same size, eventhough [0, 1] has one more point. If we keep track of this extrapoint when we measure shapes, it leads to surprising math involvinggeometry, topology, and their interaction.

Let RS (resp., RA) denote the average number of runs scored (resp.,allowed) in a baseball game by a team. It was numerically observedyears ago that a good predictor of a team’s win-loss percentage isRS2/(RS2 +RA2), though no one knew WHY the formula worked.We review elementary concepts of probability and statistics anddiscuss how one can build and solve a model for this problem. Inthe course of investigating this problem we discuss how one attacksproblems like this in general (what are the features of a good model,how to solve it, and so on). The only prerequisite is simple calculus(no baseball knowledge is required, though Red Sox knowledge isalways a plus).

Steven Miller earned his BS in mathematics and physics from Yaleand his PhD in mathematics from Princeton. He has taught at nu-merous colleges and universities, including Brown, Mount Holyoke,NYU, The Ohio State University, Princeton, Smith and Williams.He is the author of over 100 papers in accounting, computer science,economics, geology, marketing, mathematics, physics, sabermetrics,and statistics, as well as five books. He has taught continuingeducation classes at the Teachers As Scholars program for years,and been supported by multiple NSF grants for both research and

1

Exploring the

Topograph

Trevor Hyde

Abstract for 14 February 2019

Binary quadratic forms have a long history passing through algebra, geometry,and number theory. Their study attracted some of the biggest names in classicalmathematics including Fermat, Euler, Legendre, and Gauss. But to the moderneye much of this work looks like cryptic computations and algebraic manipulations.So what’s the big deal? In the 1990’s John H. Conway introduced a great wayto visualize binary quadratic forms through a tool he calls the topograph. In thistalk we explain how the topograph works and demonstrate how it sheds new lighton some beautiful classical mathematics. Along the way we will make connectionswith continued fractions and hyperbolic geometry.

Abstract for 08 March 2018

Phil Tosteson

Normally, we think of [0, 1) and [0, 1] as having the same size, even though [0, 1]has one more point. If we keep track of this extra point when we measure shapes, itleads to surprising math involving geometry, topology, and their interaction.

Let RS (resp., RA) denote the average number of runs scored (resp., allowed) ina baseball game by a team. It was numerically observed years ago that a goodpredictor of a team’s win-loss percentage is RS2/(RS2 +RA2), though no one knewWHY the formula worked. We review elementary concepts of probability and sta-tistics and discuss how one can build and solve a model for this problem. In thecourse of investigating this problem we discuss how one attacks problems like thisin general (what are the features of a good model, how to solve it, and so on). Theonly prerequisite is simple calculus (no baseball knowledge is required, though RedSox knowledge is always a plus).

Steven Miller earned his BS in mathematics and physics from Yale and his PhD inmathematics from Princeton. He has taught at numerous colleges and universities,including Brown, Mount Holyoke, NYU, The Ohio State University, Princeton,Smith and Williams. He is the author of over 100 papers in accounting, computerscience, economics, geology, marketing, mathematics, physics, sabermetrics, andstatistics, as well as five books. He has taught continuing education classes atthe Teachers As Scholars program for years, and been supported by multiple NSFgrants for both research and expository writing. He has also worked with numerousMichigan students (undergraduate and graduate) over the years, and hopes to workwith more this summer at the Williams College SMALL REU.

Cookie Monster meets

the Fibonacci Numbers

mmmmmm – theorems!

1

Exploring the

Topograph

Trevor Hyde

Abstract for 14 February 2019

Binary quadratic forms have a long history passing through algebra,geometry, and number theory. Their study attracted some of thebiggest names in classical mathematics including Fermat, Euler,Legendre, and Gauss. But to the modern eye much of this worklooks like cryptic computations and algebraic manipulations. Sowhat’s the big deal? In the 1990’s John H. Conway introduced agreat way to visualize binary quadratic forms through a tool hecalls the topograph. In this talk we explain how the topographworks and demonstrate how it sheds new light on some beautifulclassical mathematics. Along the way we will make connectionswith continued fractions and hyperbolic geometry.

Abstract for 08 March 2018

Phil Tosteson

Normally, we think of [0, 1) and [0, 1] as having the same size, eventhough [0, 1] has one more point. If we keep track of this extrapoint when we measure shapes, it leads to surprising math involvinggeometry, topology, and their interaction.

Let RS (resp., RA) denote the average number of runs scored (resp.,allowed) in a baseball game by a team. It was numerically observedyears ago that a good predictor of a team’s win-loss percentage isRS2/(RS2 +RA2), though no one knew WHY the formula worked.We review elementary concepts of probability and statistics anddiscuss how one can build and solve a model for this problem. Inthe course of investigating this problem we discuss how one attacksproblems like this in general (what are the features of a good model,how to solve it, and so on). The only prerequisite is simple calculus(no baseball knowledge is required, though Red Sox knowledge isalways a plus).

Steven Miller earned his BS in mathematics and physics from Yaleand his PhD in mathematics from Princeton. He has taught at nu-merous colleges and universities, including Brown, Mount Holyoke,NYU, The Ohio State University, Princeton, Smith and Williams.He is the author of over 100 papers in accounting, computer science,economics, geology, marketing, mathematics, physics, sabermetrics,and statistics, as well as five books. He has taught continuingeducation classes at the Teachers As Scholars program for years,and been supported by multiple NSF grants for both research and

1

Exploring the

Topograph

Trevor Hyde

Abstract for 14 February 2019

Binary quadratic forms have a long history passing through algebra, geometry,and number theory. Their study attracted some of the biggest names in classicalmathematics including Fermat, Euler, Legendre, and Gauss. But to the moderneye much of this work looks like cryptic computations and algebraic manipulations.So what’s the big deal? In the 1990’s John H. Conway introduced a great wayto visualize binary quadratic forms through a tool he calls the topograph. In thistalk we explain how the topograph works and demonstrate how it sheds new lighton some beautiful classical mathematics. Along the way we will make connectionswith continued fractions and hyperbolic geometry.

Abstract for 08 March 2018

Phil Tosteson

Normally, we think of [0, 1) and [0, 1] as having the same size, even though [0, 1]has one more point. If we keep track of this extra point when we measure shapes, itleads to surprising math involving geometry, topology, and their interaction.

Let RS (resp., RA) denote the average number of runs scored (resp., allowed) ina baseball game by a team. It was numerically observed years ago that a goodpredictor of a team’s win-loss percentage is RS2/(RS2 +RA2), though no one knewWHY the formula worked. We review elementary concepts of probability and sta-tistics and discuss how one can build and solve a model for this problem. In thecourse of investigating this problem we discuss how one attacks problems like thisin general (what are the features of a good model, how to solve it, and so on). Theonly prerequisite is simple calculus (no baseball knowledge is required, though RedSox knowledge is always a plus).

Steven Miller earned his BS in mathematics and physics from Yale and his PhD inmathematics from Princeton. He has taught at numerous colleges and universities,including Brown, Mount Holyoke, NYU, The Ohio State University, Princeton,Smith and Williams. He is the author of over 100 papers in accounting, computerscience, economics, geology, marketing, mathematics, physics, sabermetrics, andstatistics, as well as five books. He has taught continuing education classes atthe Teachers As Scholars program for years, and been supported by multiple NSFgrants for both research and expository writing. He has also worked with numerousMichigan students (undergraduate and graduate) over the years, and hopes to workwith more this summer at the Williams College SMALL REU.

Cookie Monster meets

the Fibonacci Numbers

mmmmmm – theorems!

1

Exploring the

Topograph

Trevor Hyde

Abstract for 14 February 2019

Binary quadratic forms have a long history passing through algebra, geometry,and number theory. Their study attracted some of the biggest names in classicalmathematics including Fermat, Euler, Legendre, and Gauss. But to the moderneye much of this work looks like cryptic computations and algebraic manipulations.So what’s the big deal? In the 1990’s John H. Conway introduced a great wayto visualize binary quadratic forms through a tool he calls the topograph. In thistalk we explain how the topograph works and demonstrate how it sheds new lighton some beautiful classical mathematics. Along the way we will make connectionswith continued fractions and hyperbolic geometry.

Abstract for 08 March 2018

Phil Tosteson

Normally, we think of [0, 1) and [0, 1] as having the same size, even though [0, 1]has one more point. If we keep track of this extra point when we measure shapes, itleads to surprising math involving geometry, topology, and their interaction.

Let RS (resp., RA) denote the average number of runs scored (resp., allowed) ina baseball game by a team. It was numerically observed years ago that a goodpredictor of a team’s win-loss percentage is RS2/(RS2 +RA2), though no one knewWHY the formula worked. We review elementary concepts of probability and sta-tistics and discuss how one can build and solve a model for this problem. In thecourse of investigating this problem we discuss how one attacks problems like thisin general (what are the features of a good model, how to solve it, and so on). Theonly prerequisite is simple calculus (no baseball knowledge is required, though RedSox knowledge is always a plus).

Steven Miller earned his BS in mathematics and physics from Yale and his PhD inmathematics from Princeton. He has taught at numerous colleges and universities,including Brown, Mount Holyoke, NYU, The Ohio State University, Princeton,Smith and Williams. He is the author of over 100 papers in accounting, computerscience, economics, geology, marketing, mathematics, physics, sabermetrics, andstatistics, as well as five books. He has taught continuing education classes atthe Teachers As Scholars program for years, and been supported by multiple NSFgrants for both research and expository writing. He has also worked with numerousMichigan students (undergraduate and graduate) over the years, and hopes to workwith more this summer at the Williams College SMALL REU.

Cookie Monster meets

the Fibonacci Numbers

mmmmmm – theorems!

1