Jay Kappraff

THE GEOMETRY OF COASTLINES: A STUDY IN FRACTALS

Abstract The geometry of coastlines, based on an empirical study by Lewis Richardson, is presented as a way of introducing the subject of fractals developed by Benoit Mandelbrot. It is shown how the statistically self-similar nature of coastlines can be generalized to an interesting class of point sets, curves and surfaces with the same property. Brownian and fractional Brownian motion are introduced as ways of generating statistically self-similar curves with the appearance of coastlines and mountain ranges.

A COURSE IN THE MATHEMATICS OF DESIGN

Abstract A project-oriented course on the Mathematics of Design taught for the past six years to freshman architecture students at the New Jersey Institute of Technology is described. The course uses mathematics as the organizing force linking scientific, artistic and cultural subject areas together. The sequence of topics is graph theory with application to planning a floor plan; polyhedra applied to Platonic solids; tilings of the plane with application to lattice designs; tiling of three-dimensional space and space-filling polyhedra; similarity, proportion and the golden mean with application to architectural design; transformations; mirrors and symmetry; and vectors applied to analysis of polyhedra and ruled surfaces. The mathematical elements of each topic lead students to carry out a two- or three-dimensional construction. Students are helped to focus on the ideas behind their work by writing a series of essays.

Polygons and Chaos

Abstract An infinite number of periodic trajectories are derived for the map giving rise to the Mandelbrot set at a value of the parameter corresponding to the extreme point on the real axis of the Mandelbrot set. Beginning with the edge of a family of star n-gons as the seed, the trajectory of the logistic map cycles through a sequence of edges of other star n-gons. Each n-gon for n odd is shown to have its own characteristic cycle length. The logistic map is shown to be the first of two infinite families of maps, all exhibiting periodic trajectories, derived from two families of polynomials, the Chebyshev polynomials and another related to the Lucas sequence. Using the family of Lucas polynomials, the Mandelbrot set is generalized to an infinite family of sets with similar properties.

In Memoriam: Slavik Jablan 1952–2015

After a long and brave battle with a serious illness, our dear friend and colleague Slavik Jablan passed away on 26 February 2015. [...]

Meanders, knots, labyrinths and mazes

There are strong indications that the history of design may have begun with the concept of a meander. This paper explores the application of meanders to new classes of meander and semi-meander knots, meander friezes, labyrinths and mazes. A combinatorial system is introduced to classify meander knots and labyrinths. Mazes are analyzed with the use of graphs. Meanders are also created with the use of simple proto-tiles upon which a series of lines are etched.

Musical Proportions at the Basis of Systems of Architectural Proportion both Ancient and Modern

In architecture, systems of proportions facilitate technical and aesthetic requirements, ensure a repetition of ratios throughout the design, have additive properties that enable the whole to equal the sum of its parts, and are computationally tractable. Three systems of architectural proportion meet these requirements: a system used during Roman times, a system of musical proportions used during the Renaissance, and Le Corbusier’s Modulor. All three draw upon identical mathematical notions already present in the system of musical proportions. While the Roman system is based on the irrational numbers √2 and θ = 1 + √2, the Modulor is based on the Golden Mean, ϕ = (1 + √5)/2. Both can also be approximated arbitrarily closely (asymptotically) by integer series. Underlying the Roman system is the “law of repetition of ratios” and the geometrical construction known as the “Sacred Cut,” both of which geometric expressions of the additive properties of the Roman systems and ensure the presence of musical proportions. The discussion concludes with a system of “modular coordination” based on both musical proportions of Alberti and Fibonacci numbers.

A participatory approach to modern geometry

The Origin of Geometry in Design A Constructive Approach to the Pythagorean Theorem Lines and Pixels Compass and Straightedge Constructions Congruent Triangles and Trigonometry The Art of Proof Parallel Lines and Bracing of Frameworks Perpendicular Lines and Vornoi Domains Doing Algebra with Geometry Areas, Vectors and Geoboards From Right Triangles to Logarithmic Spirals The Golden and Silver Means Transformational Geometry and Isometries Kaleidoscope and Frieze Symmetry An Introduction to Symmetry Groups Fractals, Isometries and Matrices Thirteen Fundamental Constructions of Projective Geometry.