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Review

• Review Topics for Exam I, Sample Exam

• Mathematician Pictures - for Exam I

  • Check Stamps and Stamps from Mathematica

• Review Topics for Exam II, Sample Exam

• Mathematician Pictures - for Exam II

• Mathematicians Timeline

  • More on Legal Paper

  • MacTutor Timelines

  • Mathigon Timeline

• MAA Portrait Gallery - A-M, N-Z

• MathsHistory at Twitter

• Heinz Klaus Strick's histories

• Presentation Rubric New

• Weierstrass Substitution

Breaking Math News

• Why are the formulas for the sphere so weird? (major upgrade of Archimedes' greatest discoveries) (32:22) Nov. 25 2023
• The Man Who Invented Fifteen Hundred Necktie Knots, Nov 10, 2023, The New Yorker.
• Newfound Mathematical ‘Einstein’ Shape Creates a Never-Repeating PatternApr 10, 2023, Scientific American and follow-up, Jul 1, 2023

Ancient Mathematics

• Ancient African Mathematics

  • The Lebombo Bone: The Oldest Mathematical Artifact in the World

  • Ishango bone

  • Black African Traditional Mathematics

  • On Mathematics in the History of Sub-Saharan Africa

• Egyptian Mathematics

  • Rhind Mathematical Papyrus - at The British Museum

  • Quadrature in Ancient Egypt Revisited

• Three thousand years of sexagesimal numbers in Mesopotamian mathematical texts

• Babylonian Mathematics

  • Babylonian Mathematics Handout

  • Mesopotamia

  • Sherlock Holmes in Babylon

  • Online number converter

• Pythagorean Thm - Wizard of Oz

• Mathematics Across Cultures; The History Of Non Western Mathematics

Greek Mathematics

• Videos in Canvas to make up for some of missed class - 8/31/23
• Man, Myth, Mathematician - Pythagoras of Samos - Genius Video (45:18)
• The Story of Maths, Part II: The Greeks Video (43:13)
• Greek Mathematics (Part 1) Video about Euclid's Elements (12:21)
• Some 2021 lectures on the Greeks: Lecture 4, Lecture 5, Lecture 6

• A History of Greek Mathematics - Sir Thomas Heath, 1921, Euclid pg 354

• Physics of Music

• 2,500 years of musical temperaments

• Pythagorean Tuning

• History of Music Sir John Hawkins, 1853, and stories of Pythagoras' death.

• Pythagorean Thm Tiling

• Angle Trisection

• Euclid

  • Euclid's Elements online - Useful for assignments

  • Euclid's Elements of Geometry, Heiberg's Greek text

  • Book I Dependencies Also useful for assignments

• Heptadecagon

• Constructible Polygons

• Platonic Solids

• Apollonius of Perga - Conics

• Dandelin spheres

  • https://www.geogebra.org/m/EcFmZR24 and

  • http://nonagon.org/ExLibris/dandelin-spheres-conic-sections

  • Wikipedia

• Greek Mathematician Timeline

• Archimedes Palimpsest, Video, TED Talk

• Archimede's Inventions

  • Archimedean Screw

  • List of Achivements, Top 10 list

  • Mythbuster's

• Approximations of Pi

• Hypatia

  • Hypatia: Sifting the Myths; 39:36

  • History Buffs: Agora; 28:54

  • Hypatia of Alexandria: The (REAL) history ... 20:50

• Library of Alexandria - What does the Evidence Say? 27:00

• Changes in Greece through maps

• Greek Cities link to Mathematicians

• Mathematical Methods in Ancient Astronomy - O. Neugebauer

• Tetrahedron Volume - PNG

Numbers

• Sieve of Eratosthenes - Example

• Tables of Primes - One, Two and Twin Primes

• MATLAB - Prime numbers less than or equal to input value: https://www.mathworks.com/help/matlab/ref/primes.html

• Largest Primes

• GIMPS - Great Internet Mersenne Prime Search

• The On-Line Encyclopedia of Integer Sequences

• Euclid on Numbers

  • Euclidean Algorithm - Book VII Prop 1.

  • Infinite Number of Primes - Book IX, Prop 20

  • Perfect Number - Def 22 in Book VII and Prop. 36 in Book IX

  • Unique Factorization - Book VII Pro 30-32 and Book IX Prop 14

• Archimede's Cattle Problem, An Article, MathWorld, and Another solution

• Solving the Pell Equation, Cubic Pell's Equation

• History of Pell's Equaton

• Chinese Remainder Theorem

Antikythera Mechanism

• The Antikythera Mechanism: A Shocking Discovery from Ancient Greece 1:48:50

• The Antikythera Cosmos - 29:57

• The Antikythera Mechanism - 7:51

• A Model of the Cosmos in the ancient Greek Antikythera Mechanism - 12 Mar 2021

Make-up Videos

• Fall 2021 Lectures

  • Archimedes Codex C, Eratosthenes, Heron, Ptolemy, Hypatia

  • Ancient Chinese Mathematics

  • Xian's Triangle, Chinese Remainder Theorem, Indian Mathematics, Rise of Islamic Mathematics

• External Videos - Archimedes, Hypatia, Antikythera

  • Archimedes Palimpsest Site. Long Video (1:04:04), TED Talk (14:53)

  • Tricolor Stomachion (14:12)

  • History Buffs: Agora; 28:54

  • Hypatia of Alexandria: The (REAL) history ... 20:50

  • The Antikythera Mechanism - 7:51

  • The ancient 'computer' that simply shouldn't exist (6:53)

  • Library of Alexandria - What does the Evidence Say? 27:00

Chinese Mathematics

• Chinese Mathematics - The Story of Mathematics

• Brief Guide to Chinese Math - Swetz and Se

• The Evolution of Mathematics in Ancient China - JSTOR

• Mathematics in China - timeline, chronology of mathematicians and mathematics

• Chinese mathematics - by dynasty

• The Thirteen Dynasties

• The Ten Classics of Ancient Chinese Mathematics (1:33:57)

• Gougu Rule Swetz MAA article

• 九章箅术 "Jiu zhang suan shu" (Nine Chapters on the Art of Mathematics) - An Appraisal of the Text, its Editions, and Translations by Joseph W. Dauben

• The Ten Mathematical Classics - List at MacTutor

• Historical Development of Chinese Remainder Theorerm by Kansheng

• The History of The Chinese Remainder Theorem - Ing

• Solving Systems of Equations in Ancient China (9:50) Counting rods example

•  The Chinese Hexagram

•  Liu Hui and Tsu Keng-chih on the Volume of a Sphere

•  The "Piling up of squares" in Ancient China

• Chinese multiplication table

Japanese Mathematics

• Japanese Mathematicians

• Mathematical Treasures of Japan in the Edo Period

• Wasan (Japanese mathematics)

• How to Multiply Using a Soroban (6:20)

• See A History of Japanese Mathematics Book by David E. Smith and Yoshio Mikami. 1914 Edition

Korean Mathematics

• History of Korean Mathematics ICM 2014

• History of Korean Mathematics, 1657-1868

Indian Mathematics

• Aryabhata's Sine Table, Maple file: BaskaraSineFn.mw

• Indian Mathematics - MacTutor

• Indian Mathematics - The Story of Mathematics

• A Brief History of Indian Mathematics

• Pell's Equation - Solution by Continued Fractions

• The Kolam Tradition

• The Sine of a Single Degree

• Heron's Formula and Brahmagupta's Generalization

• The History and Historiography of the Discovery of Calculus in India (47:20)

• Mining the Treasure of Palm Leaf Manuscripts

• Yuktibhāṣā - by Jyesthadeva of the Kerala school of mathematics around 1530

  • Charles Matthew Whish - MacTutor

  • On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference ... 

  • Charles Whish At Calicut, and the Madhava School - blog

  • South Indian Sanskrit Manuscripts - palm leaf manuscripts

Islamic Mathematics

• Islamic Mathematics - The Story of Mathematics

• Arabic Mathematics - MacTutor

• al Kwarizmi

• Ibn al-Haytham (Alhazen) - Father of Optics

• Kamal Al-Din Al-Farisi's Explanation of the Rainbow

• The Algebra of Abu Kamil

• The House of Wisdom is a Myth - Is it?

Other Civilizations

• Mayan Mathmematics - MacTutor

• Mathematics of the Incas - MacTutor

• The Maya Mathematical System

• Living Maya Time, Calendar System

• Maya Astronomy and Mathematics—Yesterday and Today (1:04:21)

• When a Number System Loses Uniqueness: The Case of the Maya - Number Bases

• Aztec Arithmetic: Positional Notation and Area Calculation

• Mathematics of the Hnahnu: the Otomies

• Graphs in Cultures

• Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic

• Relacion de las cosas de Yucatan by Diego de Landa

Renaissance Mathematicians

• Timeline via Artists

• Painting During The Early Renaissance

• Piero della Francesca
• Piero della Francesca's Polyhedra
• Rediscovering the Archimedean Polyhedra
• Luca Pacioli, Book on Chess

• List of Renaissance Mathematicians

  • Fibonacci (Leonardo Pisano), Liber Abaci, 1202 AD (updated in 1254 AD)

  • Leonardo of Pisa and his Liber Quadratorum
  • Michael of Rhodes (ca.1380-1445) and Abacus Schools (42:44), Journal, The Pamiers Manuscript

  • Luca Pacioli's Summa de Arithmetica, Geometria, Proportioni et Proportionalita published in Venice, 1494 AD.

  • Scipione del Ferro, Niccolò Fontana Tartaglia, Gerolamo Cardano (Ars Magna, 1545 AD), Lodovico Ferrari.

  • Rafael Bombelli published L'Algebra with imaginary quantities, 1572 AD.

  • Simon Stevin's De Thiende (The Art of Tenths), published in Dutch, 1585 AD.

  • Regiomontanus (Johannes Müller von Königsberg) - table of sines and cosines, 1533 AD.

  • Bartholomaeus Pitiscus, first to use ''trigonometry'' Trigonometria, 1595 AD.

• The teaching of mathematics in the Renaissance

• Universities during Renaissance, Another map

• Harvard College Curriculum, 1640-1880
• Dispute between abacists and algorists
• Abacists Versus Algorists - by W. E. Stone in JSTOR

European Reawakening

• Omar Khayyam's Solutions for a Cubic

• Omar Khayyam:Geometric Algebra and Cubic Equations

• Cubics Prior to Cardano (and the others)

• The quest to Solve Cubics

  • The Cubic Equation as Poetry - Tartaglia's Poem

  • The Cardano-Tartaglia Dispute - PDF

  • The Sordid Past of the Cubic Formula - June 30, 2022, Quanta Magazine

  • Review of The Secret Formula, July 20, 2022

  • Tartaglia, Cardano, Ferrari

  • Sidelights on the Cardan-Tartaglia Controversy - M. A. Nordgaard in JSTOR, Includes some of the questions from Fior challenge.

• A Modern Vision of the Work of Cardano and Ferrari on Quartics - Introduction

• 500 years of NOT teaching THE CUBIC FORMULA - 36:57

• Solution of Cubic and here

• Solving the Cubic with Cardano

• Why you can't solve quintic equations

• Nested Radicals

• van Roomen's 45th Degree Equation, Modern Explanation

• Solution of van Roomen Problem

• Vieta's Formulas for cubics at this page

• Models illustrating a classification of plane cubic curves

• Origins of Analytic Geometry

• Zero: The Symbol, the Concept, the Number

• Mathematical Symbols
• A History of Mathematical Notations
• Table of mathematical symbols

Projective Geometry

• Oresme picture

• History of Perspective

• A (Brief) History of Perspective in European Art - Video (3:47)

• Art Before Perspective - Google Image Search

• Linear Perspective: Brunelleschi's Experiment - Video (4:15)

• Projective Geometry - Britannica

• perspective.png

• ProjectiveGeom.mw - Maple File

• Mathematics of Perspective Drawing

• One Point Perspective Drawing

• Pappas' Theorem

• Desargues' Theorem - with applet

• Infinity: does it exist - a debate - Video (42:57)

• Piero della Francesca. A Mathematician’s Art, The Geometry ...

  • Piero della Francesca's Polyhedra

  • Polyhedra & Plagiarism In The Renaissance

  • Alberti and Piero

  • Diagonal Construction

• The Intersection of Visual Science and Art in Renaissance Italy - 2020

• Luca Pacioli

• Leonardo da Vinci

• Mathematics Meets Art: Early Renaissance Art and the Need for Perspective (34:02)

Development of Calculus

• A History of Calculus - MacTutor

• The History of Calculus - Rosenthal, pdf

• Archimedes' Quadrature of the Parabola Revisited

• Archimedes’ quadrature of the parabola and the method of exhaustion

• The Method of Fluxions

• The Calculus Wars - A book review

• CALCULUS WARS The Integral Awakens (8:29)
• Calculus Rhapsody (6:03)
• The Calculus War: The Ultimate Clash of Genius - A Role-Playing Game

• A Brief History of Infinitesimals

• Arithmetica Infinitorum by John Wallis
• Oxford's Savilian Professors of Geometry: 400 Years On (49:40)

• The War against Disorder: The Jesuit Victory over Indivisibles

• Fermat's dilemma: Why did he keep mum on infinitesimals?

• On the history of epsilontics and in journal article
• History of Stokes' Theorem

The Lost Calculus (1637–1670): Tangency and Optimization without Limits

• Jan Hudde's Second Letter: On Maxima and Minima

• Pascal and the Problem of Points - History of Probability, Devlin Talk (1:08:18)

• Geometriae Pars Universalis by James Gregory - 1668
• Isochronous curve of Huygens , isochronous curve of Leibniz i
• Kissing the cheeks of Huygens
• The Brachistochrone Problem at MacTutor
• The Cycloid

Infinite Series

• Origin of Taylor Series (12:18)

• The Discovery of the Series Formula for p by Leinbniz, Gregory and Nilakantha

• Certain Mathematical Achievements of James Gregory

• Bernoulli Family

  • Bernoulli Brothers - pdf

  • The Remarkable Bernoulli Family

  • Bernoulli Family Tree

  • Who Solved the Bernoulli Differential Equation

  • Bernoulli Numbers

• How Euler Did It - articles on specific works of Euler

  • Basel Problem, with Integrals

  • Arc length of ellipse

  • Euler and Pell

  • Bernoulli Numbers

  • Amicable Numbers

  • Euler's Greatest Hits

  • Euler Identity

  • Gamma the function and Gamma the constant

• More Euler

  • Elementary Works of Euler

  • The Euler Archive

    • Rectification of Ellipse

  • Counterexample to Euler's Conjecture

  • Euler's Polyhedron Formula

  • Euler's Amicable Numbers

• Lemniscate Curve and others

• The circumference of an ellipse blog

• Construction of Equations

• Maria Agnesi and the Witch of Agnesi (and here)

Elliptic Integrals

• The elastica: a mathematical history

• Euler's Elastica and Beyond PDF

• In Search of the Birthday of Elliptic Functions

• Abel on Elliptic Integrals: A Translation

• AGM [arithmetic-geometric mean of Gauss and Legendre]

• The Borwein brothers, Pi and the AGM

• A Formula for the Perimeter of an Ellipse

• A Brief History of Elliptic Integral Addition Theorems

• agm.m - MATLAB Code from Class (modified)

• Arithmetic-geometric mean to calculate Pi

• Mathematical Analysis Of Fly Fishing Rod Static And Dynamic Response

Complex Analysis

• Short History

• Augustin Louis Cauchy

• Euler and the FTA - Dunham

• The Fundamental Theorem of Algebra (31:58)

• The Fundamental Theorem of Algebra - MacTutor

• Quaternions

  • William Rowan Hamilton (10:06)
  • Quaternions and 3d rotation

  • The Quaternion Debate

  • The great vectors-versus-quaternions debate

  • A History Of Complex Noncommutative Rotation Groups

  • Quaternions and Clifford

  • Elements of Quaternions - Hamilton

  • Peter Tait's An Elementary Treatise on Quaternions

  • Kelland and Tait Introduction to Quaternions

  • UNCW Theses

    • Modeling Twisted Tori, Trinity White, 2017.

    • Geometric Approaches To The Characterization Of Qubits And Entanglement, Gregory Johnson, 2017.

    • Quaternion Approach To The Spinning Top With Simulations In Matlab, Daniel Hancock, 2014.

  • Charting the 3-Sphere--An Exposition for Undergraduates

  • A derivation of the quaternion Maxwell's equations using geometric algebra

• Visualizing Complex Functions

• History of Mathematics - Complex Analysis - Oxford

  • Part 1: complex numbers. (21:15)

  • Part 2:  functions of a complex variable. (22:43)

• The Fundamental Theorem of Fourier Series (47:09)

Select Women Mathematicians (Mostly of the 1800's)

• Timeline of women in mathematics

• Hypatia See also History Buffs: Agora

• In 1700's

  • Maria Agnesi - First Math Appointment, Math Book

  • Laura Bassi - 1st Physics Professorship

  • Emilie du Chatelet - First woman to have paper published by Paris Academy

  • The 18th-Century Lady Mathematician Who Loved Calculus and God

• In 1800's

  • Marie Sophie Germain

  • Ellen Amanda Hayes

  • Sophia Kovalevskaya

  • Christine Ladd-Franklin

  • Ada Lovelace

  • Florence Nightingale

  • Elizaveta Fedorovna Litvinova

  • Ada Isabel Maddison

  • Helen Abbot Merrill and a story

  • Mary Frances Winston Newson

  • Emmy Noether

  • Mary Emily Sinclair

  • Mary Fairfax Greig Somerville

  • Pauline Sperry

  • Alicia Boole Stott

  • Anna Johnson Pell Wheeler

  • Grace Chisholm Young

  • Philippa Fawcett First woman to obtain the top score in the Cambridge Mathematical Tripos
  • The Woman Who Bested the Men at Math
  • A history of the study of mathematics at Cambridge by W. W. Rouse Ball, 1889
  • Wikipedia on Mathematical Tripos and Senior Wrangler

• Alphabetical Index and Biographies

• Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem

• Sophie Germain Scientific American
• Chladni Plates (3:35) and Paper on theory

• A Few of My Favorite Spaces: The Kovalevskaya Top

• Kovalevskaya Top (25:08)

• A History of Women in Higher Education
• Hard-earned gains for women at Harvard
• Seven Sisters Colleges
• Women at Oxford 1878-1920

Non-Euclidean Geometry

• Non-Euclidean Geometry & the Shape of Space (49:57)

• A Brief History

• MacTutor History

• Comment by D.M.Y. Sommerville on early history

• Bolyai and Lobachevsky

• Beltrami-Klein model

• Beltrami-Poincare' model

• Illuminating Hyperbolic Geometry (4:25)

• Hyperbolic Geometry and the Pseudosphere

• Hyperbolic World

• Tiling the Plane

  • Kepler and the regular polygon tilings, The Harmony of the World

  • The Trouble with 5

  • 5 and Penrose Tiling (7:00)

• Hyperbolic Tiling

Tiling in Hyperbolic Plane

• Tesselations of Hyperbolic Plane and M. C. Escher

• Circle Limit IV (Heaven and Hell)

• Coxeter discusses the math behind Escher's circle limit (7:58)

• Visualizing the sphere and the hyperbolic plane: five projections of each (12:46)

• Bird, Fish - Escher, 1941
• Polyominoes

• Tilings of the 6xl0 Rectangle

• Tiling with Polyonimoes - Golomb

• Tiling with polyonimoes and combinatorial group theory -Conway

• Ideas for research - Tile Invariants for Tackling Tiling Questions

• Aperiodic Tiling
• Penrose Tiles and Aperiodic Tesselations
• The Infinite Pattern That Never Repeats (21:11)
• Mysterium Cosmographicum
• Aperiodic tiling
• An Aperiodic Monotile The hat PDF Webpage
• A Chiral Aperiodic MonotileThe spectre PDF Webpage
• Aperiodic Monotile - Mad as a Hat (14:34)
• Ein Stein Revisited - The Spectre Tile (53:07)
• A Hobbyist Just Solved a 50-Year-Old Math Problem (17:58)

• Elliptic Geometry

• Riemannian Geometry

• Tom Lehrer - Lobachevsky

  • Also, New Math, The Elements, Poisoning Pigeons In The Park

  • While you're at it - I Will Derive!

Group theory

• Evariste Galois

• The Last Mathematical Testament of Galois

• Evolution of Group Theory

• Group Theory - MacTutor

• History of Group - Theory WikiVisually

• Galois Theory Explained Simply (14:44)

Topology and Knots

• Vortex Dynamics: The Legacy of Helmholtz and Kelvin

• Topology, matter, and space, I: Topological notions in 19thcentury natural philosophy

Eversions - Sphere and Torus

• Sphere Eversions Notes from Topology Class

• Turning a Sphere Inside Out Documentary (Nelson L. Max, 1976) (22:57)

• Outside In (21:24)

• The Optiverse (8:43)

• Torus Eversion

• Smale's Paradox (13:47)

• Boy's Surface (2:14), How to Draw (10:15)

Fourier Analysis via Vibrations

• Gauss and the FFT

• The Vibrating String Controversy

  • Unpublished resolution

  • Controversy in science
  • Bernoulli on Vibrating Strings.

• Can one hear the shape of a drum? Mark Kac; and Cleve's Corner

• Drummed up for spectra in geometry, 1922

• You Can't Always Hear the Shape of a Drum, Article

• Chladni Figures and Vibrating Plates;

  • GIF - Vibrating Plates

• Sophie Germain,

  • https://hackaday.com/2018/03/20/sophie-germain-the-mathematics-of-elasticity/

  • from PBS: http://www.pbs.org/wgbh/nova/physics/sophie-germain.html

Turkeys

• How to cook a turkey. Musing of an Energy Nerd

• The Panofsky Turkey Constant

• Wolframalpha - how long should you cook a turkey

• "Spherical Turkeys and Vibrating Balloons," MAA-SE 2015, Dr. Herman's talk

Set Theory and Other Topics

• Mathematical paradoxes

• MacTutor History of Set Theory

Other Sites

• MacTutor History of Mathematics archive

  • History Topics Index

  • Biography Index

  • Chronology

• The Story of Mathematics

• Betrand Russell, The Teaching of Euclid

• Tartaglia versus Cardan

• https://cs.slu.edu/~clair/history/

• Wikipedia Timeline of Mathematics

• Richardson's timeline - http://www.math.wichita.edu/~richardson/timeline.html

• A Timeline of Mathematics - https://www.scaruffi.com/science/mathemat.html

• http://www.timelineindex.com/content/select.php/1495/1023,820,1495

• History of Mathematics - MAA

• Who was Charles Dodgson?

• Catholic Scientists and Mathematicians

• Papers of Riemann

• Papers of Hamilton

• Brougham Bridge story

• A History of Mathematics in Cambridge

• Crossword Puzzles

Why?

• Why Teach History of Mathematics? Torkil Heiede