The Ryoanji Suite
I was in Kyoto in July 2013 and was very moved by the rock gardens of the Ryoanji Temple. The rock garden is enclosed in a rectangular courtyard surrounded by lush Japanese gardens. The rock garden within is an austere arrangement of rocks on neatly raked gravel. Their proportions and positions defy symmetry yet they have an aesthetic balance.
I have tried to describe that in these works:
Ryoanji is constructed in a series of 20 horizontal and vertical strokes. The strokes were made so that their intersections formed the consecutive numbers 1 through 9 in that order. The artwork was the process of producing the intersections in a particular order and the result of the process is the calligraphy, Ryoanji.
Problems of dissection were common challenges posed to mathematicians in the past. Abu Wa’fa was famous for some of his constructions and dissections with straight edge and a pair of compasses. Henry Dudeney is remembered for his very elegant dissection of a square into four different parts which swings around in a hinged manner to form an equilateral triangle.
The three paintings in this series show how four pieces, all different can be arranged together in different ways to form enlargements of each of the four shapes. With the restriction of 4 pieces, I do not know if there are finite or an infinite number of solutions. Of course, if there are a finite number of solutions, then I would like to record them by painting them all. If you should find solutions other that these three, please let me know as I would be very happy to paint them and name them after you.
These three paintings are at the Saxion University,the Netherlands.
Midnight in Mongkok
Acrylic on canvas, 2013
36” x 24”
This game of leapfrog shows the minimum number of moves needed for a game with 3 counters each.
(Gauss’ Staircase of the Primes)
Acrylic on canvas, 2013
26" x 18", Acrylic on canvas, 2013
This work shows a tiling of a rectangle with seven Incomparable Rectangles. Two rectangles are called incomparable if neither of them will fit inside the other with their sides parallel. That is, one of the rectangles is both longer and narrower than the other. At least 7 and at most 8 incomparable rectangles are required to tile any rectangle (Wells, 1991).
All in Squares
24" x 24", Acrylic on canvas, 2011
This is a puzzle which hides the digits 1 through 9 in the 4 squares. No digit is used twice. The squares are correct relative to each other by size. Work out their areas to find all the digits.
For the answer go to the bottom of the page.
Acrylic on Canvas, 2011.
30" x 20"
The Klein bottle is one-sided, no-edged, with an outside but no inside. In theory this transformation is possible but no such physical bottle has been made without self-intersection. It seems a fair reflection of my emotional state at the moment.
20" x 16", Acrylic on canvas, 2011
20" x 18", Acrylic on canvas, 2011
20" x 18", Acrylic on canvas, 2011
24" x 26", Acrylic on canvas, 2011
For details go to Number Sequences
26" x 18", Acrylic on canvas, 2010
for details go to Number Sequences
30" x 24", Acrylic on canvas, 2010
Kites shows how the ratio 1:√2 was obtained in early times. This ratio is also called the Sacred Cut and was used in construction, especially to double the area of a square plinth. I have fond memories of making and flying kites with my father and hence this work.
30" x 19.5" Acrylic on canvas, 2010
This painting is a homage to the most delectable of mangoes - the Alphonso mango which grows in Ratnagiri in Western India. The slices I have eaten describe the Sacred Cut.
Journey in 4 x 6
20" x 30" Acrylic on canvas, 2010
I would not like to explain too much and spoil your pleasure. Suffice it to say that there are many routes one can take but depicted above is one that I have enjoyed taking.
24" x 24" Acrylic on canvas, 2010.
This describes the only journey of a cube around a 4x4 grid that I could find which starts and ends with the same orientation.
Journey 4 in 6x6
24"x 24" Acrylic on canvas, 2010
This is a journey in 6x6 with an interesting route. And so is the one below - a different route around the same 6x6.
Journey 3 in 6 x 6
24"x24" Acrylic on canvas, 2010.
Out of a Box - Cluster
14" x 18" Acrylic on canvas, completed 2012
The 11 possible nets of cubes.
Chord that Halves!
36" x 36" Acrylic on canvas, 2010
Chords that run through the centre of rectangles and parallelograms divide the perimeter in half just as they do in a circle. In this shape made of two small semicircles and a big semicircle too a chord that passes through the centre divides the perimeter in half.
Garden of Charms
12" x 24" Acrylic on canvas, 2010
In the Garden are four distinct pieces; taken in pairs, they form two equal charms, all together they form a whole. What are the charms?
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.
Pascal's Theorem & The Pascal Line
24" x 24" Acrylic on canvas,
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.
Acrylic on Canvas, 2010
18" x 24"
This was inspired by the puzzle MAKE IT SQUARE which runs like this: "This design contains exactly sixty-four little squares, and the puzzle consists in showing how it may be cut into the least possible number of pieces to make a large eight by eight square, with the pattern preserved." The original puzzle is most probably one of Sam Loyd's puzzles.
Three Sisters - (The Four Colour Conjecture)
15" x 30" Acrylic on canvas, 2010
A plane map can be coloured in at most four colours. The first panel shows that if a map is drawn in one continuous curve starting and ending at the same point without lifting the pen off the paper, then it can be coloured in only two colours. If the continuous curve is drawn so that it starts and end at different points, then at most three colours will be needed to paint that map - as shown in the second panel. If more than one continuous curve is needed to draw the map (the pen is lifted off the paper), then at most four colours will be needed to colour the map - the third panel.
24" x 24" Acrylic on canvas, 2010
The A Form ratio is got from the Sacred Cut. This work is another slant on the Sacred Cut and shows my current pre-occupation with this structure. For details see Sacred Cut above.
The Pedal Triangle
20" x 24" Acrytic on canvas, 2009
The so called pedal triangle is also the billiard ball path (on a triangular billiards table!). It is the shortest complete repeating route that touches the three sides of the triangle. This work and the one below illustrate how you can find the pedal triangle by joining the feet of the altitudes of a triangle.
Pedal Triangle in Red
24"x 24" Acrylic on canvas, 2010.
24"x24" Acrylic on canvas, 2010
The common chords of three intersecting circles go through a common point.
4x4 Dice Route
20" x 20" Acrylilc on canvas, 2009
This work Illustrates the path of a die around all squares of a 4x4 grid so that the die returns to the starting point in the original orientation.
Faultfree Stepping Stones
20" x 24" Acrylic on canvas, 2009
Meditating on Holditch
20"x 24" Acrylic on canvas
The Holditch curve is traced by a fixed point on a chord that slides around a convex curve. If the point on the chord divides the chord into segments p and q, then the area between the Holditch Curve and the original curve is Pi *p*q
20" x 30" Acrylic on Canvas 2009
If two triangles (in red) are in perspective, then pairs of corresponding sides meet at three points which are collinear. Two triangels are said to be in perspective if lines joining corresponding vertices meet at a point.
4x4 Dice Roll
(Inspired by Erik & Marty Demaine's Dice Roll Demonstration)
30" x 30" Acrylic on canvas
Cheeky One Cut
24" x 30" Acrylic on Canvas 2009
(An interesting configuration produced by one fold of a sheet and one curved cut)
15"x30" Acrylic on Canvas 2009
If there are two towns (top right and bottom left of the "dress") along a railway line, where should the station be located so that the least total roadway is needed to serve both towns?
Light Under the Door
18" x 24" Acrylic on Canvas 2009
Fold the edges and corners of a sheet to touch a point on it.
Acrylic on Canvas 2009
How would you cut and rearrange the squares using the fewest cuts possible so as to form a chess board? The solution is in the background.
Holditch on a Towel
Acrylic on Cotton 2009
The Sacred Cut
24" x 30" Acrylic on Canvas 2009
Illustrates the Sacred Cut or A Form cut or the ratio one to square root two.
Perfect Dice Roll (II)
20"x30" Acrylic on Canvas 2009
Illustrates the path of a die around all the squares of a 4x6 grid so that it returns to the starting with the original orientation, perhaps.
Light Under Tut's Door
24"x30" Acrylic on Canvas 2009
Illustrates folding a rectangular sheet of paper so as to touch a point on it.
All in Squares Puzzle
Hint: Take the Yellow Square to be 9.
Answer: Yellow is 3 squared = 9, Red is 9 squared = 81, Blue is 18 squared = 324 and the whole is 24 squared = 576.
9,81,324,576 all nine digits in the squares.