# Theorems & Constructions

**Dragon Curve in Red**

Acrylic on canvas, 2013

30" x 26"

**Green Dragon**

Acrylic on canvas, 2013

24" x 36"

**City Dragons**

Acrylics on canvas, 2013

36" x 26"

**Ice-cream on the Beach**

(Abu Wafa Buzjani)

Acrylic on canvas, 2011

24" x 20"

Abul Wafa lived in Buzjan, modern day Iran between 900 and 998. He wrote several treatises on Euclid, Diophantos and Al-Khwarizmi. He is best remembered for the puzzles he created on dissection with the use of only a straight edge and a pair of rusty (fixed opening) compasses. In this puzzle, you have to construct the largest possible equilateral triangle in a given square.

Bubbles in May

(Three Circles through a Point)

Acrylic on canvas, May 2010

24" x 24"

If three circles of equal radii pass through a point, then another circle of the same radius can be drawn through the points of intersection of the three circles.

Sharing a Square

15" x 30" Acrylic on canvas, 2008

How would you share a square pizza between any number of people (3 or 5 shown here) so that everyone gets an equal share of the crust?

Golden Ratio

15" x 30" Acrylic on Canvas

2009

Pascal's Theorem & The Pascal Line

24" x 24" Acrylic on canvas,

February 2010

At the age of 16, Blaise Pascal discovered and published his famous theorem entitled Essai pour les coniques. The theorem states that if a hexagon is inscribed in a conic then the three points in which the opposite sides meet are collinear. The line is The Pascal Line. The above work shows The Pascal Line in a zig-zag inscribed hexagon.

A Form

24" x 24" Acrylic on canvas, 2010

The red rectangle is in the A Form ratio. This construction shows how it can be constructed from a square.

Golden Petals

(The construction of golden sections and the golden rectangle)

20" x 24" Acrylic on Canvas

2009

Abu' Wafa Buzjani's construction of an equilateral triangle in a square.

Abu'l Wafa Buzjani

24" x 30" Acrylic on canvas

2008

Abu'l Wafa Buzjani's problem in Blue and Green

Fermat Point

Acrylic on canvas

20" x 24"

To Divide a Line

Acrylic on canvas

20" x 30"

Pythagoras

Acrylic on canvas

24" x 30"

Monge's Theorem

Acrylic on canvas

18" x 24"

Double or Nothing

Acrylic on canvas, 2009

30"x 30"

Holditch

Acrylic on canvas, 2008

18" x 24.5"

The Holditch curve is generated by a fixed point on a chord of a curve as the chord moves around a convex curve.

Sacred Cut

Acrylic on canvas, 2010

24" x 30"

The Sacred Cut was perhaps historically used to find a method to double the area of a given square. For example, in order to double the altar they could not simply double the sides. The Sacred Cut gave a means to do it. It produces the Silver Rectangle with ratio of sides 1:√2 which is used in A Form paper. This work illustrates how to construct the Silver Rectangle or the Sacred Cut and also gives an impression of doubling both the rectangles and the squares.