1. How many different shapes are there if you are color blind?
2. Count the vertices where 3 (or more) lines joint together.
3. What is the smallest cluster that can cover the whole surface ?
4. Do you see rotational symmetry and where are the pivot points ?
5. Are the shapes staggered (different x-shifts) for alternate rows ?
6. What is the easiest way to draw a thick line ?
7. How many points to define the basic polygon shape ?
8. Should we use a different color to outline the shapes ?
9. Where to place the origin to save computation?
10. Define the shift vectors (directions and distance)?
11. Do you see mirror symmetry and where is the mirror plane ?
12. Are there internal lines or shading inside of the shapes ?
13. Is there a way to draw all the shapes using no rotation ?
14. How many different shapes are there if you count also the colors ?
15. Can the co-ordinates of this polygon shape be integer numbers ?
16. How many colors are needed for adjacent shapes to be different ?
17. How to minimize or eliminate "jaggies" on slope lines ?
18. How to generalize a program to cover several similar designs ?
19. Can the differences between designs be isolated into a parameter ?
Words of wisdom from a Grade 3 class
From the book "Creative Mathematics: Exploring Children's Understanding", Rena Upitis and
Eileen Phillips both in Canada. Book is published by Routledge, London & New York, 1997.
(1) Keep it simple;
(2) Use only one or two shapes;
(3) Make sure the pattern repeats, and do not make a maze;
(4) Cover the paper;
(5) No overlapping of shapes is allowed.
Just how exactly do you learn to do interlocking shapes ? In my experience,
a combination of hand sketching and mathematical understanding work the best.
Escher first hand sketched all the Islamic tiling he saw in Spain.
Later, upon the advise of his brother, he did extensive reading in the mathematics
of crystal packing.
Here are several teaching method and tools.
(a) Translucent tracing paper (or foil) and felt pens that are
colorfast on the sheet.
(b) Photocopies of Escher's work or computer print out from the WEB.
Print 10 sheets of the same thing in extra light impression.
(c) Print a design on slightly thicker paper and cut out the silhouette
with a pair of scissors.
(d) Use a clean pad of paper with quarter-inch grids. The grid
points are used to locate the vertices for free hand sketching.
Exercise #1
Use a light color marker or a "high light marker" color the shapes that
are in one orientation. Pick a different color for the shape that is
transposed by rotation or mirror. You will need as many as 6 colors.
Exercise #2
Superimpose the paper cutout silhouette on one image. Then find the
rotation vertex and rotate into the next position. In some drawings, you
will have to flip over (mirror) the shape. This practice help you see the
action that is required to get from one position to the next.
Exercise #3
Put the translucent tracing paper on an Escher drawing. Mark out the
vertices where 3 shapes meet together. You can also do this with the extra
light photocopies. Focus on one shape and mark out all the vertices.
Connect these vertices with straight lines. The shape with
straight line will most likely be a square, rectangle, a parallelogram or a
hexagon. In this sense, the interlocking shape is just a hexagon with the
boundary severely bend.
Exercise #4
Focus on the line segments from the last exercise. See how the line
segments are repeated all over the page by mostly translation. Some lines
will rotate at the vertices by a systemic half turn, quarter turn, 1/3 turn
or 1/6 turn. Some vertices are just junctions of dissimilar lines.
Exercise #5
Sketch a design on a blank piece of paper with square grid.
Start by marking out the vertices. Connect the vertices so that the whole
page is divided into hexagons or parallelograms. Replace the straight line
with a curved line starting and ending at the vertices. Now move this
curved line into other positions (2 to 6 positions) by rotation or
mirror action. Now repeat this transposed line into all positions on the page.
The results may amaze you.
*
Periodic Drawings with JAVA programs
*
*
Mathematical Basis of Interlocking Shapes
*
Math/ART definitions
