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# New book on participatory approach to modern geometry caters to non-math majors

World Scientific

World Scientific has published a book on the Participatory approach to Modern Geometry by Jay Kappraff.

Written to meet a specific need as there is no book on rigorous Euclidean geometry available on a level appropriate for non-Math majors, this book is based on geometric constructions throughout. Many of these constructions are carried out with compass and straightedge and so the student is asked from the beginning to carry a toolkit to class consisting of a compass, protractor, straightedge and ruler, glue stick, markers, scissors, and graph paper. Each topic is introduced through applications and constructive activities.

As a result the book is driven by context rather than technique. Whenever possible the student is urged to carry out the geometrical ideas using drawing programs such as Corel draw, Photoshop, or Adobe Illustrator. In this manner, lecturers are able to meet their design interests halfway.

The book begins with the study of geometry with the construction of triangular and square arrays of circles. Some anthropologists feel that this was the origin of geometry and the prime ingredient of what is called sacred geometry. It is the basis of a construction known as "the tree of life". This term is found to grab the attention of the students as it begins the creation of geometrical complexity from simplicity. It also suggests relationships found in the natural world and leads to the creation of 6, 8, 12 and 16 pointed stars. Stars are then explored and found to contain much mathematical power. These two systems of circles are represented by the geometry based on the two great systems of ancient geometry: the 30, 60, 90 and 45, 45, 90 triangles. The reader is also introduced to the Pythagorean Theorem. Of the more than seven hundred proofs of the Pythagorean Theorem, the book illustrates this important mathematical theorem through several geometrical proofs. The Brunes star is then introduced as a way to create a star design exhibiting 3, 4, 5-triangles at four different scales.

Furthermore, lines are studied in the context of pixels whereas pixels are shown to violate the axioms of Euclidean geometry. The student is then informed about the strategy of calculus to approximate complex smooth curves by lines at least locally and show how this differs from fractals which are nowhere smooth. The student is then introduced to the idea of fundamental compass and straightedge constructions. This is then applied to studying "the world within a triangle" in which the meeting points of angle bisectors, altitudes, medians, and perpendicular bisectors meet at points associated with the triangle.

Congruent triangles enter by giving the student partial information about a triangle and having them construct that triangle with compass, straightedge, and protractor and finding the remaining information about the triangle by measurement of sides and angles. These problems are often like little puzzles. If the triangle can be constructed then it can also be solved by trigonometry. First the sides and angles are determined by measurement and then by trigonometry. To this end, an introduction to trigonometry is included. In that particular exercise, the most important concept to get across to the students is the idea of measurement based on an arbitrarily selected unit. Congruence comes in because given partial information about a triangle, sometimes more than one triangle can be constructed. Congruence tells us when only one triangle is constructible. It also gives a mechanism by which corresponding elements of a pair of triangles are equal.

After the notion of perpendicular lines is introduced, it is applied to constructing Voronoi domains. The technical difficulties of this topic are bypassed and simple domains are constructed. The notion of Voronoi domains are also then applied to problems of pattern recognition. After parallel lines are introduced this frees us to study how the notion of parallel lines is key to studying the rigidity of frameworks which also gives us an opportunity to introduce the rudiments of graph theory. Parallel lines are also applied to Eratstathenes' measurement of the circumference of the Earth. Similarity is introduced and then applied to showing how a right triangle leads to the construction of a logarithmic spiral using only compass and straightedge. Similarity is then used to introduce Jay Hambridge's notion of Dynamic Symmetry which is then applied to introducing the golden and silver means with all their applications to design. The golden and silver means are shown to give novel methods of constructing pentagons, decagons, and octagons with compass and straightedge.

Compass and straightedge construction are revisited to show how algebra and arithmetic can be carried out by geometry. Construction of square roots can then be used for the construction of logarithmic spirals. Areas are introduced in a playful way using old fashioned geoboards. These geoboards are then used to study geometrical vectors which are then applied to determining areas.

After these topics are introduced the course makes a sharp turn to studying transformational geometry where Euclidean geometry is now studied through the isometries of the plane: translations, rotations, reflections and glide reflections. Transformations are then carried out by compass, straightedge, and protractors.

In the last chapter of the book these transformations are carried out by matrices. It is further shown that the fundamental transformations lie at the basis of symmetry and the idea of a symmetry group, kaleidoscopes, dihedral symmetry, and frieze symmetry is introduced. Frieze patterns are then examined and constructed using a stamp pad and gum erasers. Fractals are also studied and the symmetries of a square that lead to the construction of a family of fractals referred to as the Iterative Function System are shown. These ideas lead to the construction of a fractal wallhanging.

Apart from the comprehensive content, there is also an extensive appendix describing thirteen constructions from projective geometry without going into the theory of that subject.

One thing is guaranteed for this book, that unsophisticated and even math-phobic students will be well-served with sophisticated mathematical ideas, and will be able to sharpen their algebraic skills which are often greatly lacking. When students see mathematics reflecting on design, they will begin to feel a sense of ownership of the geometry and are more willing to work at improving their mathematical deficiencies. On the other, there is sufficient novel material to pique the interest of stronger students.

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The book retails for US\$65 / £43 at all major bookstores. More information on the book can be found at: http://www.worldscientific.com/worldscibooks/10.1142/8952.