Mathematics of Islamic Art & Geometric Design رياضيات الفن الإسلامي والتصميم الهندسي The prominent historian, de Vaux said “(the Muslims) were indisputably the founders of plane and spherical geometry. Islamic art is prized for its beauty, complexity, harmony and intricacy. There are many good reasons for exploring Islamic art in schools. They provide a genuine cross-curricular focus, offering scope for coordinated work in mathematics and art. Here is an opportunity to show that we value contributions from our host culture, both in terms of artistic expression and mathematical knowledge.

Learning Objectives: To raise awareness of Islamic contribution to Mathematics To inspire Mathematics through the study of Islamic Geometric Designs To have fun with Islamic Geometric Designs To create works of art

Formative Assessment to Determine Knowledge Base to Track Progress After which Muslim Mathematician is the term “algorithm” named? A. al-Karaji B. al Khawrizmi C. al Biruni D. al Jazari Which Muslim Mathematician is regarded as the “Father of Optics”? A. al-Mawsili B. al Jurjani C. ibn Haytham D. al Firnas Which Muslim Polymath is regarded as the first aviator? A. ibn Firnas B. ibn Battuta C. Hasan Selebi D. al Kindi Which Muslim Mathematician, philosopher, musician and physicist has been described as one of the “Twelve great minds of history”? A. ibn Haytham B. al Wafa C. al Rammah D. al Kindi Which Italian Mathematician played a major role in promoting the use of Arabic numbers in Europe? A. Galileo B. de Vinci C. Fibonacci D. di Capprio

أشهر علماء الرياضيات المسلمين Famous Muslim mathematicians ibn al-Haytham Algebra, Geometry Thabit bin Qurra: Greatest Muslim geometer Kamal al-Din al-Farisi’s work أشهر علماء الرياضيات المسلمين Famous Muslim mathematicians Muslims see the importance of seeking knowledge from the time of the Holy Prophet (PBUH): “Seek knowledge even if you have to go to China”. There is hardly a single entity, be it business, industry, education or architecture without the Arabic numerals, the decimal point, the sine and cosine or the compass amongst others which are Islamic inventions Omar Khayyam: Poet, Mathematician, Astronomer Al-Khwarizmi The “Father of Algebra” Muhammad Al-Karaji’s work

Al-Khwarizmi The “Father of Algebra” The best known of the Islamic Mathematicians Considered one of the greatest Mathematicians of all times His books were studied long into the Renaissance To him we owe the words: Algebra and Algorithm Numbers as used in Algebra fascinated Al- Khwarizmi and where his originality and depth is most clearly evident. His treatise classifies the solution of quadratic equations and gives geometric methods for completing squares.

Al-Karaji Al-Karaji was the first to use mathematical induction to prove the binomial theorem He proved that if the first statement in an infinite sequence of statements is true, then so is the next one. He proved that since (13 + 23)= (1 +2)2, thus: (13 + 23 + … + 103) = (1 +2 + … + 10)2 He proved that: If (13 + 23) = (1 +2)2 Then (13 + 23 + 33) = (1 +2 + 3)2 and so on and so on

Omar Al Khayyam Famous poet and the writer of the “Rubaiyat”, but an important mathematician and astronomer in his own right The Moving Finger writes, and, having written, Moves on: nor all thy Piety nor Wit Shall lure it back to cancel half a Line, Nor all thy Tears wash out a Word of it. His famous short poem from the Rubaiyat: The Moving Finger writes, and, having writ, Moves on: nor all thy Piety nor Wit Shall lure it back to cancel half a Line, Nor all thy Tears wash out a Word of it.

a⁄(sin A) = b⁄(sin B) =c⁄(sin C) Nasir Al-Din Al-Tusi The first to treat trigonometry as a separate math discipline, distinct from astronomy Gave the first extensive account of spherical trigonometry One of his major mathematical contributions was the formulation of the famous law of sine for plane triangles: a⁄(sin A) = b⁄(sin B) =c⁄(sin C)

ibn Al-Haytham Systemized conic sections and number theory on analytic geometry Worked on the beginnings of the link between algebra and geometry This in turn had an influence on the development of René Descartes' geometric analysis and Isaac Newton's calculus.

Kamal Al-Din al-Farisi Applied the theory of conic sections to solve optical problems Pursued work in number theory such as on amicable numbers (e.g., 220 & 284) Factorization of an integer into powers of prime numbers

Thabit bin Qurra Greatest Muslim geometer Played an important role in preparing the way for mathematical discoveries: extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic and non- Euclidean geometry Was one of the first to create a new proof for the Pythagorean Theorem This is only the tip of the iceberg! Islamic mathematicians cover huge area of mathematics, one which reveals much more than the mere sketch presented here.

Astrolabe The Astrolabe was highly developed in the Islamic World by 9th Century It was introduced to Europe from Islamic Spain (Al Andalus) in the early 12th Century It was the most popular astronomical instrument until about 1650 The Astrolabe is the original GPS – it allows the user to find its position. The astrolabe was inherently valuable in Islam because of its ability to determine the time of day and, therefore, prayer times and as an aid in finding the direction to Mecca.

What is Taught and What Should be Taught? Francois Vieta was the first to utilize algebraic symbols. In 1591, he wrote an algebra book describing equations with letters. What Should be Taught: Muslim mathematicians invented algebra. In early 9th century, they introduced the concept of using letters for unknown variables in equations.

What is Taught and What Should be Taught? In 1614, John Napier invented logarithms and logarithmic tables. What Should be Taught: Islamic Mathematicians invented logarithms and produced logarithmic tables. These were common in the Islamic world as early as the 13th Century.

What is Taught and What Should be Taught? The use of decimal fractions in mathematics was first developed by a Dutchman, Simon Stevin, in 1589. What Should be Taught: Al-Kashi's book, Key to Arithmetic, was the stimulus for the application of decimals to whole numbers and fractions. It was written at the beginning of the 15th century.

What is Taught and What Should be Taught? The concept that numbers could be less than zero was unknown until 1545 when Cardano introduced the idea. What Should he Taught: Muslim mathematicians introduced negative numbers for use in arithmetic functions at least 400 years prior to Cardano. There many such examples you can find in literature!

Mathematics of Islamic Art & Geometric Design Islamic art explores the geometric systems of the regular division of the circle Islamic Art increases appreciation and understanding of geometry Working only with a ruler and compass, students can discover how to create and study many of the geometric designs Among the most important aspects of Islamic geometry is SYMMETRY, REPETITION & VARIATION. Symmetry plays an important part in most Islamic geometry. There may be single line of reflective geometry, or there may be 3 or 4 lines of symmetry.

Circles, Squares & Octagons The eight-points star, made of two overlapping squares in a circle, is the basis of many Islamic patters

Seven overlapping circles

Discovering Patterns with Triangle Grid

Discovering Patterns with Five Overlapping Circle Grids

Discovering Patterns with the Diagonal Grid

Activities based on geometric Islamic patterns can support learning about shapes, spaces and measures In Primary, students can learn to draw and recognise circles, triangles, squares, hexagons and octagons Create pictures using 2-D shapes Learn to identify lines of symmetry Recognise reflective and rotational symmetry Upper Primary and Middle school students can study symmetric patterns to produce tessellations. High school students can look at molecular & crystal shapes and calculate spaces occupied A regular tessellation is a pattern made by repeating a regular polygon. There are only 3 regular tessellations: triangles, squares and hexagons A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!

Work of AAM Students

http://cmcuworkshops.net/?page_id=13

http://www.dynamicgeometry.com/

Review of Math topics using Jigsaws and Islamic Geometric Designs Formulator Tarsia is designed for Teachers of Mathematics to create activities in a form of jigsaws for use in a class. It includes the powerful equation editor for building the math-expressions for the activities.  http://www.mmlsoft.com/index.php?option=com_content&task=view&id=9&Itemid=10

3-D Applications (G&T Projects) ABA pattern: Mg

A few References on Islamic Art & Math Islamic Art and Geometric Design: Activities for Learning Copyright ©2004 by The Metropolitan Museum of Art, New York: http://www.metmuseum.org/~/media/Files/Learn/For%20Educators/Publications%20for%20Educators/Islamic_Art_and_Geometric_Design.pdf Mathematics, Geometry and the Arts Resources (under: Islamic Art and the Sciences) http://cmcuworkshops.net/?page_id=13 ; http://www.dynamicgeometry.com/ Islamic Art & Culture: A resource for Teachers http://ahmadladhani.files.wordpress.com/2009/10/islamic-tp1.pdf The connection between Islamic art and Mathematics http://www.dartmouth.edu/~matc/math5.pattern/lesson5A&M.connection.html Using Technology to investigate mathematics in Islamic Art: http://cmcuworkshops.net/?page_id=13 Formulator Tarsia known earlier as Formulator Jigsaw is an editor designed for Teachers of Mathematics creating the activities in a form of jigsaws or dominos etc for later use in a class. It includes the powerful equation editor for building the math-expressions for the activities.  http://www.mmlsoft.com/index.php?option=com_content&task=view&id=9&Itemid=10 Book of Curiosities of the Sciences and Marvels of the Eyes http://cosmos.bodley.ox.ac.uk/store/Teacher_s-Pack-Inside-pages.pdf Islamic Geometric Patterns by Eric Broug, published by Thames & Hudson Geometric Concepts in Islamic Arts by El-Said Islamic Design: A Genius for Geometry (Wooden Books) by Daud Sutton

Learning Objectives: LOLOLO To raise awareness of Islamic contribution to Mathematics To inspire Mathematics through the study of Islamic Geometric Designs To have fun with Islamic Geometric Designs To create works of art LOLOLO