''Project Mathematics!'' (stylized as ''Project MATHEMATICS!''), is a series of educational video modules and accompanying workbooks for teachers, developed at the

California Institute of Technology The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...

to help teach basic principles of mathematics to high school students. In 2017, the entire series of videos was made available on

YouTube YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by ...

Overview

The ''Project Mathematics!'' series of videos is a teaching aid for teachers to help students understand the basics of

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

. The series was developed by

Tom M. Apostol Tom Mike Apostol (August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks. Life and career Apostol was bor ...

and James F. Blinn, both from the

. Apostol led the production of the series, while Blinn provided the

computer animation Computer animation is the process used for digitally generating animations. The more general term computer-generated imagery (CGI) encompasses both static scenes (still images) and dynamic images (moving images), while computer animation refe ...

used to depict the ideas beings discussed. Blinn mentioned that part of his inspiration was the Bell science series of films from the 1950s. The material was designed for teachers to use in their curriculums and was aimed at grades 8 through 13. Workbooks are also available to accompany the videos and to assist teachers in presenting the material to their students. The videos are distributed as either 9 VHS videotapes or 3 DVDs, and include a history of mathematics and examples of how math is used in real world applications.

Video module descriptions

A total of nine educational video modules were created between 1988 and 2000. Another two modules, ''Teachers Workshop'' and ''Project MATHEMATICS! Contest'', were created in 1991 for teachers and are only available on videotape. The content of the nine educational modules follows below.

''The Theorem of Pythagoras''

In 1988, ''The Theorem of Pythagoras'' was the first video produced by the series and reviews the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

. For all

right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...

s, the square of the

hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...

is equal to the sum of the squares of the other two sides (a² + b² = c²). The

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

is named after

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...

of ancient Greece.

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

s occur when all three sides of a right triangle are

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s such as a = 3, b = 4 and c = 5. A

clay tablet In the Ancient Near East, clay tablets (Akkadian ) were used as a writing medium, especially for writing in cuneiform, throughout the Bronze Age and well into the Iron Age. Cuneiform characters were imprinted on a wet clay tablet with a stylu ...

shows that the

Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...

ns knew of Pythagorean triples 1200 years before Pythagoras, but nobody knows if they knew the more-general Pythagorean theorem. The Chinese proof uses four similar triangles to prove the theorem. Today, we know of the Pythagorean theorem because of

Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...

, a set of 13 books on mathematics—from around 300

BCE Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...

—and the knowledge it contained has been used for more than 2000 years. Euclid's proof is described in book 1, proposition 47 and uses the idea of equal areas along with

shearing Sheep shearing is the process by which the woollen fleece of a sheep is cut off. The person who removes the sheep's wool is called a '' shearer''. Typically each adult sheep is shorn once each year (a sheep may be said to have been "shorn" or ...

and

rotating Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

triangles. In the ''dissection proof'', the square of the hypotenuse is cut into pieces to fit into the other two squares. Proposition 31 in book 6 of Euclid's Elements describes the similarity proof, which states that the squares of each side can be replaced by shapes that are similar to each other and the proof still works.

''The Story of Pi''

The second module created was ''The Story of Pi'', in 1989, and describes the mathematical constant pi and its history. The first letter in the Greek word for "perimeter" (περίμετρος) is , known in English as "pi". Pi is the

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...

to its

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

and is roughly equal to 3.14159. The circumference of a circle is

2 \pi r

and its area is

\pi r^2

. The

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

and

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

of a

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

and

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

are calculated using pi. Pi is also used in calculating planetary orbit times,

gaussian curve In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is ...

s and alternating current. In

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, there are

infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

that involve pi and pi is used in

. Ancient cultures used different approximations for pi. The Babylonian's used

\tfrac

and the

Egyptians Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian ...

used

\tfrac

. Pi is a fundamental constant of nature.

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

discovered that the area of the circle equals the square of its

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

times pi. Archimedes was the first to accurately calculate pi by using

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

s with 96 sides both inside and outside a circle then measuring the line segments and finding that pi was between

\tfrac

and

\tfrac

. A Chinese calculation used polygons with 3,000 sides and calculated pi accurately to five

decimal places Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expre ...

. The Chinese also found that

\tfrac

was an accurate estimate of pi to within 6 decimal places and was the most accurate estimate for 1,000 years until

arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...

were used for

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

. By the end of the 19th century,

formulas In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

were discovered to calculate pi without the need for geometric diagrams. These formulas used infinite series and

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

to calculate pi to hundreds of decimal places. Computers were used in the 20th century to calculate pi and its value was known to one billion decimals places by 1989. One reason to accurately calculate pi is to test the performance of computers. Another reason is to determine if pi is a specific

fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

, which is a ratio of two

integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

called a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

that has a repeating pattern of digits when expressed in decimal form. In the 18th century,

Johann Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French language, French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Switzerland, Swiss or France, French, who made i ...

found that pi cannot be a ratio and is therefore an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...

. Pi shows up in many areas having no obvious connection to circles. For example; the fraction of points on a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornam ...

viewable from an origin point is equal to

\tfrac

''Similarity''

Discusses how scaling objects does not change their shape and how angles stay the same. Also shows how ratios change for perimeters, areas and volumes.

''Sines and Cosines, Part I'' (Waves)

Visually depicts how

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

s and cosines are related to waves and a

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

. Also reviews their relationship to the ratios of side lengths of

''Sines and Cosines, Part II'' (Trigonometry)

Explains the

law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...

and cosines how they relate to sides and angles of a triangle. The module also gives some real life examples of their use.

''Sines and Cosines, Part III'' (Addition formulas)

Describes the addition formulas of sines and cosines and discusses the history of

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...

. It also goes into details of

Ptolemy's Theorem In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician ...

. Animation shows how sines and cosines relate to harmonic motion.

''Polynomials''

How

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s can approximate sines and cosines. Includes information about

cubic spline In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding ...

s in design engineering.

''The Tunnel of Samos''

How did the ancients dig the Tunnel of Samos from two opposite sides of a mountain in 500

? And how were they able to meet under the mountain? Maybe they used geometry and trigonometry.

''Early History of Mathematics''

Reviews some of the major developments in mathematical history.

Production

The ''Project Mathematics!'' series was created and directed by Tom M. Apostol and James F. Blinn, both from the California Institute of Technology. The project was originally titled ''Mathematica'' but was changed to avoid confusion with the mathematics software package. A total of four full-time employees and four part-time employees produce the episodes with help from several volunteers. Each episode took between four and five months to produce. Blinn headed the creation of the computer animation used in each episode, which was done on a network of computers donated by Hewlett-Packard.

Funding

The majority of the funding came from two grants from the

National Science Foundation The National Science Foundation (NSF) is an independent agency of the United States government that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National I ...

totaling $3.1 million. Free distribution of some of the modules was provided by a grant from Intel.

Distribution

''Project Mathematics!'' video tapes, DVDs and workbooks are primarily distributed to teachers through the California Institute of Technology bookstore and were popular enough that the bookstore hired an extra person just for processing orders of the series. An estimated 140,000 of the tapes and DVDs were sent to educational institutions around the world, and have been viewed by approximately 10 million people over the last 20 years. The series is also distributed through the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

and NASA's Central Operation of Resources for Educators (CORE). In addition, over half of the states in the US have received master copies of the videotapes so they can produce and distribute copies to their various educational institutions. The videotapes may be freely copied for educational purposes with a few restrictions, but the DVD version is not freely reproducible. The video segments for the first 3 modules can be viewed for free at the ''Project Mathematics!'' website as streaming video. Selected video segments of the remaining 6 modules are also available for free viewing. In 2017, Caltech made the entirety of the series, as well as three

SIGGRAPH SIGGRAPH (Special Interest Group on Computer Graphics and Interactive Techniques) is an annual conference on computer graphics (CG) organized by the ACM SIGGRAPH, starting in 1974. The main conference is held in North America; SIGGRAPH Asia ...

demo videos, available on

Availability in different languages and formats

The videos have been translated into Hebrew, Portuguese, French, and Spanish with the DVD version being both English and Spanish. PAL versions of the videos are available as well and efforts are underway to translate the materials into Korean.

Releases

All of the following were published by the California Institute of Technology: *''Project Mathematics!'', workbooks (1990), *''Project Mathematics!'', 9 videotapes (VHS, 30 minutes each, 1994), *''Project Mathematics!, DVD 1'', videodisk (DVD, 68 minutes, 2005), *''Project Mathematics!, DVD 2'', videodisk (DVD, 81 minutes, 2005), *''Project Mathematics!, DVD 3'', videodisk (DVD, 82 minutes, 2005),

Awards

''Project Mathematics!'' has received numerous awards including the Gold Apple award in 1989 from the National Educational Film and Video Festival. *1988 International Film and TV Festival of New York

''Interactive Project Mathematics!''

A web-based version of the materials was funded by a third grant from the National Science Foundation and was in phase 1, .

References

Sources

{{cite book , last=Borwein , first=Jonathan M. , author-link=Jonathan Borwein , editor=Jonathan M. Borwein , title=Multimedia tools for communicating Mathematics, Volume 1 , url=https://books.google.com/books?id=V4_NSXjDNu0C&q=%22Project+Mathematics%22&pg=PA1 , access-date=20 August 2010 , edition=illustrated , volume=1 , orig-year=2002 , year=2002 , publisher=Springer , isbn=978-3-540-42450-5 , oclc=50598138 , page=1

External links

Project Mathematics! websiteInteractive Project Mathematics! websiteYouTube playlist of original Project Mathematics! videos
1988 American television series debuts 2000 American television series endings 1980s American documentary television series 1990s American documentary television series 2000s American documentary television series Mathematics education television series American educational television series