Escher Exhibition: Rigid Motion (4º ESO)

8/3/2019 Escher Exhibition: Rigid Motion (4 ESO)

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Activity: Escher Exhibition (4 ESO) Mathematics and History Departments

IES Albayzn (Granada) Page 1

REGULAR DIVISION OF THE PLANE1

In addition to being wonderfully engaging art, the work of Maurits Cornelius

Escher also displays some of the more beautiful and intricate aspects of

mathematics. In 1936 Escher, became obsessed with tessellations, that is, with

creating art that used objects to cover a plane so as to leave no gaps.

Symmetry became a cornerstone of Eschers famous

tessellations.

Escher kept a notebook in which he kept background

information for his artwork. In this notebook, Escher

characterized all possible combinations of shapes, colors

and symmetrical properties of polygons in the plane. By

doing so, Escher had unwittingly developed areas of a

branch of mathematics known as crystallography yearsbefore any mathematician had done so!!

These pictures have beencreated by Escher using the

rules of transformationalgeometry. One of the targetsof this activity will be todiscover that rules.

1This activity has been developed using different materials selected from the following bibliographic sources:AAVV, A Survey of Mathematics with Applications, Eighth Edition (Pearson Education, 2009). Recursos digitalesde la Editorial ANAYA (digital.com).


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Transformational Geometry

We will now introduce a type of geometry called transformational geometry. In

transformational geometry we study various ways to move a geometric figure

without altering the shape or size of the figure. When discussing

transformational geometry, we often use the term rigid motion.

The act of moving a geometric figure from some starting position to some

ending position without altering its shape or size is called a rigid motion (or

transformation).

When discussing rigid motion of two-dimensional figures, there are four basic

types of rigid motions: Reflections, Rotations, Translations, and Glide

Reflections. We call these four types of rigid motions the basic rigid motions in a

plane.

A reflection is a rigid motion that moves a geometric figure to a new position

such that (tal que) figure in the new position is a mirror image of the figure in the

starting position. In two dimensions, the figure and its mirror image are

equidistant from a line called the reflection line or the axis of reflection.

A translation (or glide) is a rigid motion that moves a geometric figure by

sliding (deslizar) it along a straight line segment in the plane. The direction and

length of the line segment completely determine the translation. A concise way

to indicate the direction and the distance that a figure is moved during thetranslation is with a translation vector.


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A rotation is a rigid motion performed by rotating a geometric figure in the

plane about a specific point, called the rotation point or the center of rotation.

The angle through which the object is rotated is called the angle of rotation.

We will measures angles of rotation using degrees. In mathematics, generally,

counterclockwise angles have positive degree measures and clockwise angles

have negative degree measures.

A glide reflection is a rigid motion formed by performing a translation(or glide)

followed by a reflection.

As a summary of the basic rigid motion in a plane we can bear in mind the

following image:


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Activities

1) Construct the reflection of polygon ABCDE, shown in Figure a, about line

eand the reflection of polygon ABCD, shown in Figure b, about point O.

2) Given the shapes shown in Figure a and b, and translation vector ,

construct the translated shapes (ABCDE).

3) Use the given figure and rotation point Oto construct the indicated

rotations

a) A 30 rotation of point A about point O.

b) A 90 rotation of segment ABabout point O.


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c) A 30 rotation of trapezoid ABCDabout point O.

If you compare this rigid motion to the one in the exercise 1b, what

would you say about it?

4) Construct the reflection of polygon shown below about line e.

5) Construct the reflection of polygon shown below about point O.


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6) Given triangle OAB, where 0,0,1,3 y 4,1:

a) Plot all the points and draw the triangle.

b) Construct the translation of triangle OAB using = 5,1 as a

translation vector.c) Determine the coordinates of the three vertices of triangle OAB.

7) John has to study this composition that he has found out in an art

exhibition, could you help him to answer the following questions?

What rigid motion would you use to transform tile 1 into tile 2? And tile

1 into tile 3? And tile 1 into tile 6?

8) Construct a glide reflection of square ABCD using vector and reflection

line e.


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9) Determine whether the following Eschers work have been created using

reflection, translation, rotation or glide reflection.

You can use a tracing paper (a transparent sheet placed over the

original) to answer this question.


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Escher Exhibition: Rigid Motion (4º ESO)