Mathematics: General resources

Balmond, Cecil & Jannuzzi Smith, Informal, Prestel, NA2750 .B323 2002

Burrows, Roger, 3D thinking in design and architecture: from antiquity to the future, Thames & Hudson, NA2750 .B87 2018

Choma, Joseph, Morphing: a guide to mathematical transformations for architects and designers, Laurence King Publishing, NA2760 .C476 2015

Evans, Robin, The Projective Cast: architecture and its three geometries, MIT Press, NA2760 .E93 1995

Frei, Hans & Pamela Johnson, "The mathematics of the Shinohara House," AA Files, 2016, #73, p.145-153 (PDF via JStor)

Hahn, Alexander, Mathematical excursions to the world's great buildings, Princeton University Press, NA2750 .H325 2012

Harris, James, Fractal architecture: organic design philosophy in theory and practice, University of New Mexico Press, NA2760 .H37 2012

Hesselgren, Lars, et al (editors), Advances in architectural geometry 2012, Springer, NA2760.A354 2013

Pottman, Helmut, Architectural geometry, Bentley Institute Press, NA2760 .A728 2007

Terzidis, Kostas, Algorithmic architecture, Architectural Press, NA2728 .T47 2006

Tzonis, Alexander, Classical architecture: the poetics of order, MIT Press, NA260 .T96 1986

Bafna, Sonit, “On the idea of the mandala as a governing device in Indian architectural tradition,” JSAH, March 2000, p.26-49 (PDF via Jstor)

Green, Judy Green & Paul S. Green, “Alberti's perspective: a mathematical comment,” The Art Bulletin, December 1987, p.641-645 (PDF via Jstor)

Hiscock, Nigel, The wise master builder: Platonic geometry in plans of medieval abbeys and cathedrals, Ashgate, NA4850 .H57 2000

Necipoğlu, Gülru, The Topkapi Scroll: geometry and ornament in Islamic architecture, Getty Center for the History of Art and the Humanities, NA2706.A783N45 1995 (Chapter 5)

Opaéciâc, Zoë, Diamond vaults: innovation and geometry in medieval architecture, Architectural Association, NA2880 .O63 2005 

Özdural, Alpay, “Omar Khayyam, mathematicians and conversazioni with artisans,” JSAH, March 1995, p.54-71 (PDF via Jstor)

Philbbs, John, “Projective geometry,” Garden History, Summer 2006, p.1-21 (In 18^th century English gardens; PDF via Jstor)

Robison, Elwin C., “Optics and mathematics in the domed churches of Guarino Guarini,” JSAH, December 1991, p.384-401 (PDF via Jstor)

Rossi, Corinna, Architecture and mathematics in ancient Egypt, Cambridge University Press, NA215 .R67 2004

Sunderland, Elizabeth Read, “Symbolic numbers and Romanesque church plans,” JSAH, October 1959, p.94-103 (PDF via Jstor)

Van Liefferinge, Stefaan, “The Hemicycle of Notre-Dame of Paris: Gothic design and geometrical knowledge in the twelfth century,” JSAH, December 2010, p.490-507 (PDF via Jstor)

Waddell, Gene, “The principal design methods for Greek Doric temples and their modification for the Parthenon,” Architectural History, Vol.45, 2002, p.1-31 (PDF via Jstor)

Wittkower, Rudolf, “Principles of Palladio's architecture,” Journal of the Warburg and Courtauld Institutes, Vol.7, 1944, p.102-122  (See also Part II)

Wrightman, Greg, “The Imperial Fora of Rome: some design considerations,” JSAH, March 1997, p.64-88 (PDF via Jstor)

Carpo, Mario, “Drawing with numbers: geometry and numeracy in early modern architectural design,” JSAH, December 2003, p.448-469 (PDF via Jstor)

Eisenman, Peter, “Aspects of Modernism: Maison Dom-ino and the self-referential sign,” Log, Winter 2014, p.139-151 (PDF via Jstor)

Rowe, Colin, The mathematics of the ideal villa, MIT Press, NA7110.R68

Bell, Michael, Space replaces us: essays and projects on the city, Monacelli Press, NA737.B42 A4 2004

Harris, James, Fractal Architecture: Organic Design Philosophy in Theory and Practice, University of New Mexico Press, 2012 (via Ebook Central)

Klanten, Robert & Lukas Feireiss, Strike a pose!: eccentric architecture and spectacular spaces, Die Gestalten Verlag, NA687.S77 2008

Sasaki, Mutsuro, Morphogenesis of flux structure, AA Publications, NA2728 .S27 2007

Sakamoto, Tomolo & Albert Ferré, From control to design: parametric/algorithmic architecture, Actar-D, NA2728 .F665 2008

Terzidis, Kostas, Algorithmic architecture, Architectural Press, NA2728 .T47 2006

Watts, Andrew, Modern structural design : constructing complex forms, Birkhauser, NA687 .W38 2022

Ashton, Anthony, Harmonograph: a visual guide to the mathematics of music, Walker, QC228.3 .A78 2003

Ginzburg, M. I. A., Rhythm in architecture, Artifice books on architecture, NA2760 .G513 2016

Loy, D. Gareth, Musimathics: a guided tour of music, MIT Press, ML3805 .L78 2006

Maor, Eli, Music by the numbers: from Pythagoras to Schoenberg, Princeton University Press, ML3805 .M3 2018

Rahn, John, “The Swerve and the Flow: Music's Relationship to Mathematics,” Perspectives on New Music, Winter 2004, p.130-148 (PDF via Jstor)

Rosen, Charles, The classical style, W. W. Norton, ML195.R68 1997

Society for Mathematics & Computation in Music

“Architectural proportion,” and  “Human proportion,” Oxford Art Online

Cohen, Preston Scott, Contested symmetries and other predicaments in architecture, Princeton Architecture Press, 2001 (via Ebook Central)

Elam, Kimberly, Geometry of design: studies in proportion and composition, Princeton Architectural Press, N7431.5 .E44 2001

Huntley, H. E., The divine proportion: a study in mathematical beauty, Dover Publications, QA466 .H85 1970

Kepes, Gyorgy, Module, Proportion, Symmetry, Rhythm, Braziller, N76.K4

Leyton, Michael, Shape as memory: a geometric theory of architecture, Birkhäuser, NA2760.L49 2006

Moffitt, John F., The Islamic design module in Latin America: proportionality and the techniques of neo-Mudéjar architecture, McFarland & Co., NA702.2 .M64 2004

Panofsky, “The History of the Theory of Human Proportions,” Meaning in the Visual Arts, N7445.2.P35 1982

ANALYSIS in mathematics studies sets of numbers, points and functions that are infinitely large, small, near and divisible by means of the FUNCTION--a rule or equation that identifies for each independent value a corresponding dependent value. The rate at which the dependent value changes relative to the independent value is the DERIVATIVE. REAL ANALYSIS is comprised of DIFFERENTIAL CALCULUS, which studies the difference between consecutive values of continuously varying quantities and their rate of change; and INTEGRAL CALCULUS, which studies the sums of infinitesimal functions of the variable. VECTOR ANALYSIS applies real analysis to quantities having direction and magnitude. COMPLEX ANALYSIS studies functions which include variables that are complex (include the square root of -1). DIFFERENTIAL EQUATIONS are a type of equation in which the function is related to its derivative, which is useful for describing the behavior of rates of change. For example, differential equations describing behavior within a mathematical space--a manifold--can describe the occurrence of random behavior within a deterministic system: CHAOS. FOURIER ANALYSIS studies functions which include variables that are trigonometric (include sine & cosine). The CALCULUS OF VARIATIONS studies the length and path of arcs connecting two points. FUNCTIONAL ANALYSIS studies sequences of functions with the techniques used to describe numbers and points.

“Calculus,” and “Differential equations,” Kahn Academy

Kojima, Hiroyuki, The manga guide to calculus, Ohmsha, QA300 .K57513 2009

Novak, Miroslav M., Complexus mundi: emergent patterns in nature, World Scientific, 2006 (via Ebook Central)

Novak, Miroslav M., Thinking in patterns: fractals and related phenomena in nature, World Scientific, 2004 (via Ebook Central)

ARITHMETIC includes elementary number systems, measurement and the art of computation with addition, subtraction, multiplication, division, powers, roots and logarithms.

“Arithmetic and pre-algebra,” Kahn Academy

Lam, Lay Yong & Tian Se Ang, Fleeting footsteps: tracing the conception of arithmetic and algebra in ancient China, World Scientific, 2004 (via Ebook Central)

Potter, Michael D., Reason's nearest kin: philosophies of arithmetic from Kant to Carnap, Oxford University Press, 2000 (via Ebook Central)

Serre, Jean-Pierre, A course in arithmetic, Springer, QA243.S4713

COMBINATORICS studies arrangements, operations, and selections within a finite system. One of the basic problems is to determine the number of possible configurations, e.g., graphs, designs, arrays. COMBINATIONAL GEOMETRY studies relations among members of finite systems of geometric figures subject to various conditions and restrictions.

Balakrishnan, V. K., Shaum’s outline of theory and problems of combinatorics, McGraw-Hill, QA164 .B35 1995

Mazur, David, Combinatorics: A Guided Tour, American Mathematical Society, 2010 (via Ebook Central)

Koh, Khee Meng & Tay Eng Guan, Counting, World Scientific, 2002 (via Ebook Central)

COMPUTER SCIENCE. The development of software and hardware to be informed by a combination of many branches of mathematics, such as INFORMATION THEORY, performance studies of systems and the analysis of ALGORITHMS--systematic procedures for determining answers to mathematical problems.

Denning, Peter, Great principles of computing, MIT Press, QA76 .D3483 2015

Goldstine, Herman H., Computers from Pascal to Von Neumann, Princeton University Press, TK885.A5G64 2001

Harel, David, Algorithmics: the spirit of computing, Pearson Education, QA76.9 .A43 H37 2004

Louidas, Panos, Real-world algorithms: a beginner’s guide, MIT Press, QA76.9.A43 L67 2017

McEliece, Robert, Theory of information and coding, Cambridge University Press, 2001 (via Ebook Central)

Shannon, C. E., “A mathematical theory of communication,” Bell System Technical Journal, v. 27, #3, 1948 (Established information theory as a discipline)

Wiener, Norbert, Cybernetics; or control and communication in the animal and the machine, 2^nd edition, MIT Press, Q175.W6516 1961

GAME THEORY analyzes situations in which there is interplay between parties that may have similar, opposed, or mixed interests. In a typical game, players, who each have their own goals, try to outsmart one another by anticipating each other's decisions; the game is finally resolved as a consequence of the players' decisions. A solution to a game prescribes the decisions the players should make and describes the game's appropriate outcome.

Binmore, Ken, Game theory: a very short introduction, Oxford University Press, 2007

Fisher, Len, Rock, Paper, scissors: game theory in everyday life, Basic Books, 2008 (via Ebook Central)

Packel, Edward W., The mathematics of games and gambling, Mathematical Association of America, 2006 (via Ebook Central) 

GEOMETRY studies the properties of space and of objects in space. EUCLIDEAN geometry studies two- and three-dimensional figures by means of a deductive system in which assertions about the properties of figures--theorems--are derived sequentially from previous theorems, which are ultimately derived from a set of axioms. ANALYTIC geometry studies points in space, especially curves, in the language of numerical coordinates. In ALGEBRAIC geometry, figures are studied in the language of algebra.  Other kinds of geometries derive from beginning with sets of axioms different from Euclid’s: the HYPERBOLIC, in which “parallel” lines diverge; ELIPTIC, in which they meet. RIEMANNIAN geometry provides the language of the general theory of relativity. PROJECTIVE geometry studies the representation or mapping of a line or plane onto another line or plane, from a point not lying in either. DIFFERENTIAL geometry studies curves and surfaces in space in the language of calculus. TOPOLOGY studies the properties of geometric figures that remain unchanged even when under distortion.

Archimedes, The works of Archimedes, Dover Publications, QA31 .A692 2002

Bass, Alan, Geometry: fundamental concepts and applications, Pearson, QA445 .G475 2008

Casse, Rey, Projective geometry: an introduction, Oxford University Press, 2006 (via Ebook Central)

Chavel, Riemannian geometry: a modern introduction, 2002 (via Ebook Central)

De Morgan, Augustus, On the study and difficulties of mathematics, Dover Publications, QA11.2 .D44 2005

Descartes, Rene, The geometry of Rene Descartes, Dover Publications, QA33 .D5 1954

Frame, Michael, Fractal worlds: grown, built, and imagined, Yale University Press, QA614.86 .F795 2016

“Geometry,” Kahn Academy

Gray, Jeremy, János Bolyai, non-Euclidean geometry, and the nature of space, MIT Press, QA685 .G73 2004

Heath, Thomas L., The Thirteen Books of Euclid’s Elements, Dover Publications, QA31.E875 1956 vols 1-3 (Standard English edition)

Liu, Qing, Algebraic geometry and arithmetic curves, Oxford University Press, 2006 (via Ebook Central)

Radzevich, S. P., Geometry of surfaces, Wiley, TA418.7 .R33 2013

Todd, J.A., Projective and analytical geometry, Sir Isaac Pittman & Sons, QA471 .T6

Weeks, Jeffrey R., The shape of space, Marcel Dekker, QA612.2 .W44 2002

Yan, Min, et al, Introduction to topology, De Gruyter, Inc., 2016 (via Ebook Central)

NUMBER THEORY is concerned with the properties of numbers.  Branches include the study of PRIME numbers, the study of ALGEBRAIC numbers--complex numbers that are roots of polynomial equations with integer coefficients--and the opposite of algebraic numbers, TRANSCENDENTAL numbers.

Alaca, Saban & Kenneth S. Williams, Introductory algebraic number theory, Cambridge University Press, 2004 (via Ebook Central)

Dudley, Underwood, A guide to elementary number theory, The Mathematical Association of America, 2009 (via Ebook Central)

Maor, Eli, To Infinity and beyond; a cultural history of infinity, Princeton University Press, QA9.M316 1991 (Interesting cultural history)

Number Theory Web

Ribenboim, Paulo, Little Big Book of Primes, Springer-Verlag, QA246.R472 1991

Shidlovskii, A.B., Transcendental numbers, De Gruyter, Inc., 1989 (via Ebook Central)

NUMERICAL ANALYSIS is “concerned with obtaining numerical answers by approximations, rather than by analytic solution,” (Oxford Dictionary of Physics), useful for describing and modeling the behavior of real-world systems.

Bultheel, Adhemar, & Ronald Cools (editors), The birth of numerical analysis, World Scientific, 2009 (via Ebook Central)

Eisley, Joe G. & Anthony Waas, Analysis of structures: an introduction including numerical methods, John Wiley & Sons, 2011 (via Ebook Central)

Keinitz, Joerg & Daniel Wetterau, Financial modelling: theory, implementation and practice, Wiley, 2012 (via Ebook Central)

Pardoux, E., Markov processes and applications: algorithms, networks, genome and finance, Wiley/Dunod, 2008 (via Ebook Central)

Rao, G. Shanker, Numerical analysis, New Age International, 2006 (via Ebook Central)

OPTIMIZATION is a technique for improving or increasing the value of some numerical quantity that in practice may take the form of temperature, air flow, speed, pay-off in a game, political appeal, destructive power, information, monetary profit, and the like. Also known as OPERATIONS RESEARCH/MANAGEMENT SCIENCE (OR/MS).

Allaire, Grégoire, Numerical analysis and optimization: an introduction to mathematical modeling and numerical simulation, Oxford University Press, 2007 (via Ebook Central)

Institute of Operations Research and the Management Sciences (INFORMS)

PROBABILITY & STATISTICS are concerned with activities in which the outcome of a given trial cannot be predicted with certainty, although the collective results of a large number of trials display some regularity. DESCRIPTIVE STATISTICS concerns methods of presentation of data, via tables, graphs and summaries.

Bertin, Jacques, Semiology of graphics: diagrams, networks, maps, ESRI Press, QA90 .B47513 2010

Everitt, Brian, Chance rules: an informal guide to probability, risk and statistics, Springer, 2008 (via Ebook Central)

Gonick, Larry & Woollcott Smith, The Cartoon Guide to Statistics, Harper Perennial, QA276.12.G67 1993

Hand, D. J., Statistics: a very short introduction, Oxford University Press, QA276 .H3183 2008

Huff, Darrell, How to lie with statistics, Norton, HQ29.H82 1993

Mlodinow, Leonard, The drunkard’s walk: how randomness rules our lives, Pantheon Books, QA273 .M63 2009

“Probability & statistics,” Kahn Academy

Woolfson, Michael M., Everyday probability and statistics: health, elections, gambling and war, Imperial College Press, 2008 (via Ebook Central)

SET THEORY concerns collections of definite, distinguishable objects of perception or thought conceived as a whole, especially as it permits the study of the infinite as a mathematical object. Work in formalizing the axioms and principles of set theory has rendered the field almost equivalent to the study of the foundations of all mathematics.

Grattan-Guinness, I., Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Godel, Princeton University Press, 2000 (via Ebook Central)

O’Leary, Michael, A first course in mathematical logic and set theory, John Wiley & Sons, 2015 (via Ebook Central)

Stoll, Robert R., Set Theory and Logic, Dover Publications, QA248.S7985

Van Heijenoort, Jean, Frege and Gödel; Two Fundamental Texts in Mathematical Logic, Harvard University Press, QA9.V28

TRIGONOMETRY is the branch of mathematics concerned with specific functions of angles and their application to calculations in geometry.

Lial, Margaret, Trigonometry, Pearson/Addison-Wesley, QA531 .L5 2009

Morrow, H. W. & Robert P. Kokernak. Statics and strength of materials, Pearson Education, TA405 .M877 2007

“Trigonometry and pre-calculus,” Kahn Academy