M.C.Escher: Mathematics and Visual Arts

Diane Hanson, Saskatchewan Education (O.M.L.O.)

(A more detailed version of this unit, containing supplementary ideas and activities, worksheets, evaluation instruments, a magazine article and a bibliography will be inserted in the mathematics curriculum guide for middle years which will be available in June 1996.)


Integration is a term that is appearing more and more frequently in the professional teaching literature. In its simplest form, integration means making connections or links.

Integrated teaching brings the following advantages:

For teaching according to the new mathematics curriculum, several ways of integrating can be identified: In this unit, we see integration of two subject-languages (mathematics and visual arts) with the concrete (the art of M.C. Escher).

This unit is intended for use at the grade 7 level, but it could easily be modified and adapted so as to respond to the needs of students at other levels.

There is a large variety of levels and ways of achieving integration.It is recommended that teachers experiment with a number of methods for carrying out integration.

Summary of the Unit

After they've studied examples of tiling encountered in everyday life, the students explore tiling from the perspective of mathematics and visual arts through studying the works of M.C. Escher.

Instructional Approaches They are:

Resources and Materials Vocabulary and Structures

In the course of this unit, the student will understand and use:


Present the students with examples of paving, tiling or tessellations.

Examine a number of these paved surfaces and point out which shapes are used in creating them.

(The following examples can be taken from magazines or books or be found in the environment: patterns used in kitchen or bathroom floors, patios, honeycombs, wallpaper, fabrics, quilts, stained glass windows, mosaics, ornamentation from different cultures, works of art.)

By studying these techniques you can develop a definition of paving, tiling or tessellation.

(Tiling, paving or tessellation is defined as covering a surface or an area by means of polygons that are placed in such a way as to leave no space between them and to have no overlap of the polygons.)

(Another way to develop a definition of paving is to use the learning strategy called concept attainment, and to show students examples and non examples of paving. See the pamphlet That's a Yes! Concept Attainment, which appears in the collection titled Instructional Strategies Series.)

Give each student a pattern block (the yellow hexagon, the red trapezoid, the green triangle, the orange square or the blue rhombus) and ask the students to cover a surface or pave an area (a rectangle measuring 20 cm by 25 cm) so as to produce a tessellation. The teacher can demonstrate at the same time, using the overhead projector, by placing the tile in the middle of the area, tracing its outline, and placing it in another spot, and continue to work in this way (from the middle out to the sides) until the whole area has been covered (without leaving any space between the tiles).


Is it possible to develop criteria for determining whether a shape can be used to tessellate or not?


Give the students (who are working in pairs or in groups) a large number of two dimensional shapes and ask them to divide these shapes into two groups according to the following criteria:

Students can be asked to produce a hypothesis regarding this classification (criteria of classification), to verify it, and to compare the results with their hypotheses.

These results can be shared with the rest of the groups.

You can pin up the examples of paving and tiling in the classroom and ask the students to add further examples.

You can keep the criteria and the results of this experiment posted on the bulletin board in order to be able to come back to them as the unit goes on.

Here is a sample of the shapes that can be used: the circle, the oval, a variety of polygons (triangles, quadrilaterals, pentagons, hexagons, octagons, etc.). Irregular polygons should also be presented. The different forms can be numbered to make discussion easier.

It's an appropriate juncture at which to encourage teachers to begin drawing up a list of mathematical and other terms that are specific to this unit. The list can be pinned up and added to throughout the unit. The students can refer to it. Making an illustrated dictionary can be an activity they carry out along with their projects.

Activity: Students select one or more shapes and cover the area (completely cover the surface).

Students use their imagination to draw and colour the design.


Present to the class a number of works by the artist M.C. Escher and make mention of the fact that he was fascinated by regularity and mathematical structure, by continuity and the infinite, and by the latent conflict in each image. Discuss with them the sources of his inspiration and what had influenced him.

Escher used geometrical shapes which he would transform so as to generate other interesting shapes. He would then draw and/or colour his piece. Show the students how to transform a square using translation and rotation.

The students can then cover the surface of the area using the work of M.C. Escher as a model.

For further details on how to proceed, refer to the article titled The Art of Mosaics in the Arithmetic Teacher of March 1990.


Ask the students to write down their personal impressions of the activities in this unit, the relationship that exists between mathematics and visual arts, and such creative strategies as feeling the excitement of discovery, looking for the various possibilities afforded by the representation of an idea, and bringing out the contradictions.


Present a scenario for the whole class or ask the students to choose between the following options:

Scenario 1:

Ask the students to perform transformations starting with shapes other than the square, such as the triangle, the hexagon, etc.

Draw and/or colour them.

Scenario 2:

Use the software program Tessellmania to explore different transformations and ways to cover a surface.

Scenario 3:

Make a kaleidoscopic construction. Find three square mirrors measuring approximately 30 cm per side. Glue them together to form a corner. Cut out shapes that have been traced on wood, plastic, stiff cardboard or some other material. Arrange a composition on the mirror's surface. Draw attention to the fact that the figures are completed by their reflection in the mirror. For example, a semicircle becomes a circle. Glue the arrangement to the surface with a suitable type of glue.

Scenario 4:

Mount a project for a Math Fair. You can plan to invite parents and other classes from the school.

Possible projects:

Other Suggestions for Adapting the Unit

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