Download Symmetry and Tessellations

Introduction to Geometry and a First
Look at Tessellations
Introduction
Have you ever wondered how M.C. Escher made some of his patterns? Escher has done
many different types of artwork. We will start off with his regular divisions of the plane.
Here are some examples:
It is not as difficult as it looks. If you want to make drawings like this, all you
need is some geometry. These regular divisions of the plane go by several different
names. They are called tessellations, tilings and even wallpaper patterns.
You of course all know examples of these tilings. Think about the tile in your
bathroom. You most often have an entire pattern made up out of squares. Some older
bathrooms may actually have hexagonal (six sided) tiles. If you do, the pattern will
resemble that of chicken wire. If you look at the side of a brick building, you will see a
tessellation by rectangles.
We will see that all of Escher’s patterns are derived from just a handful of
geometric patterns. There are several different techniques that Escher used, and
sometimes he combined techniques as well. The simplest example of an Escher like
tessellation is based on a square. We will start out with a simple geometric pattern, and
then change that ever so slightly.
In this example I have made all the vertical pieces look like a lightning bolt, and the
horizontal pieces like another jagged piece. Note that all vertical pieces are changed in
the same way, and that all horizontal pieces look the same as well.
Now it becomes an inkblot test. What do you see? Some will think it looks like inter
locked men, others may see birds. By decorating each of the newly formed tiles you will
create an Escher like tessellation. This method is the simplest way to create Escher –like
tessellations, and also works well if you start with a tessellation by rectangles or
parallelograms. You should give this a try! It’s not hard, and it’s a lot of fun.
In this first section we will first discuss polygons, followed the theory of these much
simpler geometric patterns, and finally we will use the theory of isometries (sometimes
called rigid motions) to make Escher-like patterns.
Polygons
For more details, see
http://www.mathleague.com/help/geometry/polygons.htm
Definition A polygon is a closed figure made by joining line segments, where each
line segment intersects exactly two others.
The following are examples of polygons:
These are not polygons:
The left figure is not closed, and the figure in the middle is not made of line segments.
The figure on the right is not a polygon, since its sides intersect each other.
Regular Polygons
Definition: A regular polygon is a polygon whose sides are all the same lengths, and
whose angles are all congruent. The sum of the angles of a polygon with n sides,
where n is 3 or more, is 180˚ * (n - 2) degrees.
Definition: The vertex of an angle is the point where the two rays that form the angle
intersect
Definition:
The vertices of a polygon are the points where its sides intersect.
Triangles
Triangles can be classified in different ways. We may identify triangles according to the
lengths of their sides, or we may identify then according to the measure of their angles.
Identifying a Triangle using the Length of the Side
Based on the lengths of their sides, triangles are classified into three categories:
Equilateral triangle: If the lengths of all three sides of the triangle are equal, then it
is called an equilateral triangle.
Isosceles triangle: If only two sides of a triangle are equal in length, it is called an
isosceles triangle.
Scalene triangle: If all the sides of a triangle have different lengths it is called a
scalene triangle.
Identifying a Triangle using the Angles
Based on the measure of their angles, triangles are classified into three categories:
Acute triangle: A triangle in which all the angles are acute (less than 90°) is called
an acute triangle.
Obtuse triangle: A triangle in which one of the angles is obtuse (more than 90°) is
called an obtuse triangle. Since the sum of all the angles of a triangle is 180°, the
other two angles must be acute.
Right Triangle: A triangle in which one of the angles is a right angle (exactly 90°).
The other two angles must be complementary angles (they add up to 90°). In a right
triangle the side opposite to the right angle is called the hypotenuse.
A special case of an acute triangle is when all the three acute angles are equal. Such a
triangle is called an equiangular triangle. It is not hard to see than an equiangular
triangle must be equilateral. Since the sum of all the angles of a triangle is 180°, each
angle of an equiangular triangle is 60°.
For more examples, see
http://www.pinkmonkey.com/studyguides/subjects/geometry/chap2/g0202301.asp
Quadrilaterals
Definition
A quadrilateral is a four-sided polygon.
Examples:
The sum of the angles of any quadrilateral is 360°.
Types of Quadrilaterals
Rectangle
A four-sided polygon having all right angles.
Square A four-sided polygon having equal-length sides meeting at right angles.
Parallelogram
Rhombus
A four-sided polygon with two pairs of parallel sides.
A four-sided polygon having all four sides of equal length.
Trapezoid A four-sided polygon having exactly one pair of parallel sides. The two
sides that are parallel are called the bases of the trapezoid.
Examples:
Rectangles
Squares
Parallelograms
Rhombuses
Trapezoids
Other polygons worth mentioning
Pentagon
A five-sided polygon. The sum of the angles of a pentagon is 540 degrees.
Hexagon
A six-sided polygon. The sum of the angles of a hexagon is 720 degrees.
Regular pentagon.
An irregular
pentagon.
Regular Hexagon.
An irregular
hexagon.
Geometry and Basic Tessellations
We need to understand our basic definitions very well. We also need to make sure that
we can work with geometry in a fairly creative manner. In this course, you should always
try to think in the most general terms. An example would be, if you were asked to give an
example of a triangle. You might be tempted to quickly draw an example of a triangle,
without thinking about it much and then move on. This will get you into trouble later on
in the course. If you’re asked to draw a triangle, try to come up with as many diverse
types of triangles as you can.
Similarly if someone asks you to draw a parallelogram, how many significantly different
ones can you draw? Note that the following are all parallelograms:
Squares and rectangles also meet the requirements in the definition of a parallelogram.
This means that squares and rectangles are just special classes of parallelograms. Not
realizing these kinds of facts would make it difficult to recognize patterns you might be
looking for. Think about the definitions and questions; see how exotic the examples can
be. At all cost you should avoid “hit-and-run mathematics”: coming up with a quick
answer without thinking about it too deeply. This (bad) practice would keep all the really
interesting ideas and patterns hidden from you.
We often think of all the quadrilaterals as a large family, where some of the members are
related to one another. Here are some examples:
Note that every square is a rectangle (because all angles are right angles), but not every
rectangle is a square.
Every rectangle is a parallelogram (because opposite sides are equal and parallel), but not
every parallelogram is a rectangle. Some of the relationships are represented in the
following diagram. We assumed here that trapezoids have exactly one pair of parallel
sides. Note that kites can be convex or non-convex, hence we can’t connect the kites to
either one of those classes of quadrilaterals.
QUADRILATERALS
CONVEX QUADRILATERALS
TRAPEZOIDS
PARALLELOGRAMS
NON-CONVEX QUADRILATERALS
KITES
RECTANGLES RHOMBUS
SQUARES
Exercises: Geometry and Basic Tessellations
1. Carefully draw several distinct examples of the following quadrilaterals: a kite, a
parallelogram, a rhombus, a rectangle and a trapezoid.
2. The rectangle is a subclass of which other groups of quadrilaterals given in problem
1?
3. Show that if the opposite angles in a quadrilateral have equal measure, then that
quadrilateral has to be a parallelogram.
4. True or False:
a. Every square is a rectangle.
b. Every equilateral triangle is an isosceles triangle.
c. Every rhombus is a parallelogram.
d. The sum of the angles in a triangle is 180°.
e. Every diagonal bisects a quadrilateral into two triangles.
f. There exists a diagonal that bisects a quadrilateral into two triangles.
5. Draw an equilateral triangle, and draw the lines of symmetry. How many lines of
symmetry are there?
6. Draw a right triangle, an acute triangle and an obtuse triangle. Find the midpoint of
each of the sides, and connect the midpoints. (This should create 4 smaller triangles.)
Now repeat this subdivision for the three smaller triangles that contain one of the
original vertices. (i.e. subdivide the three “corner triangles”). What other triangles are
the small triangles congruent to? What other triangles are the small triangles similar
to? (Remember that congruent means that they should have the same size and shape.
Similar means that they should have the same shape, but can be of different size.)
7. Draw a right triangle, an acute triangle and an obtuse triangle. Pick one of the sides,
and rotate the triangle about that midpoint. What shapes do you get? Show that this
rotation always results in a parallelogram.
8. Using paper, make copies (''pattern blocks'') of each of the polygons indicated. In
each case, see if there is a tessellation. If there is, sketch it.
a. Isosceles triangles that are not equilateral.
b. Scalene triangles.
9. Which triangles can tessellate a plane?
10. In each of the following cases, see if there is a tessellation. If there is, sketch it
a. Rectangles that are not squares.
b. Parallelograms that are not rectangles or rhombuses.
c. Kites.
d. Trapezoids that are not isosceles.
e. Other quadrilaterals (make one up).
11. Which quadrilaterals can tessellate the plane?
12. Create a geometric tessellation by triangles, then make a drawing inside the triangles
to make it look more Escher-like.
13. Below are three early cubist paintings (beginning of the twentieth century).
a. What geometric shapes do you recognize in Braque’s Woman and Guitar?
b. What geometric shapes do you recognize in Picasso’s Ambrose Vollard?
c. What geometric shapes do you recognize in Metzinger’s Cubist Landscape?
Braque (1913) Woman and Guitar
Metzinger (1911) Cubist Landscape
i.
Picasso (1910) Ambrose Vollard
Explorations For Euclidean Geometry
ii.
Exploration: Quadrilaterals
Objective
(Re-) familiarize us with the different classes of quadrilaterals. Which quadrilaterals
can be defined in different ways? Which classes of quadrilaterals are subclasses of other
quadrilaterals?
Polygons, like living things, can be organized by family, genus, and species. Some
members of the quadrilateral class are subclasses of other types of quadrilaterals.
Squares, for example, are a subclass of rectangles, because a square is a rectangle whose
sides are all the same length. Rectangles, however, are not a subclass of squares, because
not all rectangles are squares.
One quadrilateral may be a subclass of more than one other kind of quadrilateral. Squares
are a subclass of both rhombuses and rectangles. But since most rhombuses are not
rectangles, and most rectangles are not rhombuses, neither is a subclass of the other.
Below, read about a dozen types of quadrilaterals by definition, all have four sides, but
each has unique qualities. Then answer the questions below.
A. Convex quadrilateral whose angles are all less than 180 degrees.
B. Cyclic quadrilateral: a quadrilateral whose corners all lie on a single circle.
C. Kite: a quadrilateral with two pairs of adjoining sides of equal length.
D. Parallelogram: a quadrilateral whose opposite sides are parallel.
E. Quadrilateral: a flat polygon with four sides.
F. Rectangle: a quadrilateral with four 90° angles.
G. Rhombus: a quadrilateral whose sides are all the same length.
H. Square: a quadrilateral with four sides of equal length and four 90° angles.
I. Trapezoid: a quadrilateral having only two parallel sides.
J. Unnamed quadrilateral 1: has opposite angles of equal measure.
K. Unnamed quadrilateral 2: has one pair of opposite angles both 90°.
L. Unnamed quadrilateral 3: has one pair of opposite sides of equal length.
Questions
1. Draw representative illustrations (several) of these types of quadrilaterals.
2. Which two definitions describe exactly the same type of quadrilateral?
3. Besides E (Quadrilateral), which two types of quadrilaterals are not necessarily
convex?
4. Quadrilaterals of class K (one pair of opposite angles both 90°) are a subclass of
which two other types of quadrilaterals, not counting E?
5. Can you find a chain of six of the quadrilaterals in which each type is a subclass of
the next? Hint: The first is a square and the last is a quadrilateral.
6. Can you find one set of four quadrilaterals in which no one quadrilateral is a subclass
of another? Hint: Try drawing pictures.
BONUS QUESTION: Can you define a fifth type of quadrilateral that is neither a
subclass nor a super-class of these four quadrilaterals?
Further information can be found at:
http://www.discover.com/jul_00/bogglers.html
http://www.mathleague.com/help/geometry/polygons.htm
© Copyright 2000 The Walt Disney Company. (With permission of Scott Kim, the
author.)
Exploration: Triangles and Quadrilaterals
Objective:
Triangles can be combined to create other shapes. This fact will be used later in the
course.
1. Start this activity with four congruent 30°-60°-90° triangles. These should be the
white triangles.
iii.
Measure, in centimeters, the three sides of your triangle.
iv.
Try to find some relationship between your angle measures and the measures of
the sides.
2. Put two triangles together with no gaps and no overlapping to obtain an equilateral
triangle. How many lines of symmetry does an equilateral triangle have?
3. Put two of the triangles together to obtain an isosceles triangle. (That is not also
equilateral.)
v.
What does the ''line down the middle'' do to the angle of the big triangle?
vi.
What does it do to the side it intersects?
vii.
How many lines of symmetry does an isosceles triangle have if it's not
equilateral?
4. Use two of the triangles to form a quadrilateral that is not a parallelogram.
5. Use two of the triangles to form a parallelogram that is not a rectangle and not a
rhombus. Write down all the things you observe about this new figure. (i.e. angles,
lengths of sides, diagonals, symmetry etc.)
6. Use two of the triangles to form a rectangle. Record your observations.
7. Use four of the triangles to form a rhombus. Record your observations.
8. Use all four of the triangles to form another triangle. How does its size and shape
compare with the size and shape of the original triangle? What other observations can
you make about the new figure?
9. Use all four of the triangles to form a square with a square hole in it. This can be done
in two ways; sketch them both.
10. Which of the questions 2-9 could you do with four congruent triangles that are right
triangles, but do not have a 30˚ angle? Try and see.
11. Which of the questions 2-9 could you do with four congruent triangles that are not
right triangles? Try and see.
Exploration: First look at Tessellations
Objective:
We are introduced to some basic ideas about tessellations. Manipulatives are used to
help develop our intuition. Explore how many tessellations we can make using just one
type of geometric shape.
When polygons are fitted together to fill a plane with no gaps or overlaps, the pattern is
called a tessellation. You have seen them in floor tilings, quilts, art designs, etc.
Tessellation patterns can be made from one shape or from more than one shape; here our
investigation will use one shape at a time.
We will start with a hands-on approach, and use pattern blocks to explore tessellations.
1. Use the green tiles to create a tessellation by equilateral triangles. Sketch one
tessellation. How many possibilities are there? How many possibilities are there of
vertices are only allowed to meet other vertices?
2. Use the blue tiles to create a tessellation by rhombuses. Sketch one tessellation. How
many possibilities are there? How many possibilities are there of vertices are only
allowed to meet other vertices?
3. Use the red tiles to create a tessellation by isosceles trapezoids. Sketch one
tessellation. How many possibilities are there? How many possibilities are there of
vertices are only allowed to meet other vertices?
4. Use the yellow tiles to create a tessellation by hexagons. Sketch one tessellation. How
many possibilities are there? How many possibilities are there of vertices are only
allowed to meet other vertices?
5. Use grid paper to show that the following shapes tessellate:
A
B
6. Look at Escher prints 1-21 in Visions of Symmetry [VOS, pg 116-132]. Make a list
of the geometric shapes that are used to tessellate the plane. For example: in print 1,
Escher bases his print on a tessellation by parallelograms.
C
D
7. Find a pentagon that will tessellate. Sketch the tessellation.
8. Find a pentagon that will not tessellate. Explain why not (i.e. try to explain what goes
wrong when one tries to tessellate the plane using this pentagon).
9.
Find a hexagon that is not regular, but which will tessellate. Do you think any
hexagon will tessellate? Explain your answer.
Symmetry
Symmetry
Introduction
Examples in Visions of Symmetry (Schattschneider):
viii. Plane-filling Motifs on page 7
ix.
Wall mosaic from the Alhambra Pg 9
x.
Development I Pg 30
xi.
Plane Filling Motif with Reptiles Pg 100
xii.
Shield of the Swiss Canton of Valais Pg 238
xiii. Letter puzzles on Pg 240
xiv. Trees and Animals on Pg 301
xv.
Six sided rosette with fish on Pg 309
Symmetry is something all human beings look for and seem to intuitively understand.
The dictionary gives different definitions. One way to describe symmetry is to say that it
is harmony or beauty of form that results from balanced proportions. A little bit more
technical description is to say that symmetry is a correspondence in the position of pairs
of points of a geometric object that are equally positioned about a point, line or plane that
bisects the object.
If points of an object are equally positioned about a line, then we say the object has
reflectional or bilateral symmetry. A good example is for instance a hart or a smiley face.
If you draw a vertical line though these figures, then you will notice that for every point
on the left of the line there is an equally spaced point to the right of the line. The vertical
line is called the mirror line or the line of symmetry. Note that the cross on the right has
four mirror lines. (A vertical line, a horizontal line, and the two diagonals.)
If points on an object are equally positioned about a point, then we say the object has
rotational symmetry.
In the figures above all the points on the object are equally spaced about the centre. In the
figure on the left you see eight copies of a triangle evenly spaced around a circle. This
figure has 8-fold rotational symmetry. The figure in the centre is a 5-pointed star and has
5-fold rotational symmetry. The hourglass on the right consists of 2 triangles equally
spaced around the centre. The hourglass has 2-fold rotational symmetry.
Sometimes the rotational symmetry is expressed in degrees. If a figure has n-fold
360 o
rotational symmetry then we compute
.
n
360 o
 45 o rotational symmetry. This means that if you
8
rotate the figure 45o about thecentre, it will line up exactly with the original. Similarly
360 o
 72 o rotational symmetry and the hourglass has
the 5-pointed star is said to have
5

360 o
 180 o rotational symmetry.
2
The figure on the left then has


We can label a shape based on what symmetries it has. If a shape has reflectional
symmetry, then we say it has a dihedral symmetry group, and it gets the label D. If a
shape does not have reflectional symmetry then we say it has a cyclic symmetry group,
and gets the label C. We also record what the highest degree of rotational symmetry is.
Some examples:
The hexagon has reflectional symmetry and the highest degree rotational
symmetry is 6. This means it will be assigned the symmetry group D6.
The smiley face has reflectional symmetry, and the highest degree
rotational symmetry is 1 (you would have to rotate it through 360 degrees
to get it to match up again. This means it has symmetry group D1.
This shape does not have reflectional symmetry, and hence gets a C. The
highest degree rotational symmetry is 2, so this shape has symmetric group
C2.
We will see these symmetry groups again in a later section.
Exercises: Symmetry
1. Draw the reflected image of the F for each of the four mirrors shown below.
a.
c.
F
F
b.
F
F
d.
2. For each of the polygons below, draw all of the reflection lines. What will the
reflection lines look like for a 13-sided polygon? A 20 sided polygon? What is the
general pattern?
3. In Escher's Castrovalva [VOS pg 6;MOE] there are no obvious mathematical
symmetries, but Escher's eye for patterns is already evident. Find at least four areas
of this print, which have repetition of pattern. Describe the area and the pattern that
repeats.
4. For the twenty-six capital letters, determine all symmetries of each letter then group
the letters into symmetry classes. All members of the same class should have the
same symmetries.
5. Determine the symmetry group for each of the pictures below:
a.
b.
c.
d.
e.
6. Discuss the symmetries of the following Escher works. What is the symmetry group
of each? Do the colors affect the symmetries? Does the picture have symmetries that
are "almost" there?
a. The Scapegoat [VOS pg 5]
b. Alhambra wall mosaic [VOS pg 9]
c. Development I [VOS pg 30]
d. "Amazing Images" [VOS pg 240]
e. Circle Limit I [MM pg 21]
f. Drawing Hands [MM pg 26]
g. Moebius Strip I [MM pg 99]
7. Choose one asymmetrical motif. Draw your motif, and then use it to create Rosettes
with symmetry groups C2, C3, C4, C5, C6 and D2, D3, D4, D5, D6. (You can use
Kali to help out, if you like).
8. This problem is to show that a finite picture cannot have two 90° rotation symmetries
with different centers. Take a full sheet of graph paper, and draw two 90° rotation
centers near the middle of the sheet. Now outline one of the squares of the graph
paper, near the two centers. This is your first square, so put a 1 in it:
1
Now rotate the square around one of the
centers. Outline the new square, label it 2.
Rotate again, around the other center, and
label with a 3. Repeat until the square
leaves the page.
9. On page 7 [VOS] you will find two motifs. Locate the center of each figure. Answer
the following questions: (Short answers suffice, no explanation needed.)
a. Top figure:
i. Does the motif have reflectional symmetry? (Look VERY closely!)
ii. Is there rotational symmetry? If so, what is the degree? (2-fold, 3-fold, 4-fold,
6-fold?)
iii. What is the symmetry group based on your previous answers?
b. Bottom figure:
i. Does the motif have reflectional symmetry?
ii. Is there rotational symmetry? If so, what is the degree?
iii. What is the symmetry group, based on your previous answers?
10. What is the symmetry group for the wall mosaic in the Alhambra, shown on page 9
[VOS]?
11. Below you see a copy of Dali’s Swans Reflecting Elephants.
a. Explain the title of the painting
b. What would be different if Dali had decided to paint Swans Rotating Elephants?
c. Give a (rough) sketch of what the painting paint Swans Rotating Elephants would
have looked like. (Simple line drawing is sufficient).
Dali Swans Reflecting Elephants.
12. Below are two of Escher’s prints. Answer the following questions for (i) and (ii):
a. Do they have rotational symmetry?
b. Do they have reflectional symmetry?
(i)
(ii)
Explorations For Symmetry
Exploration: Reflectional and Rotational Symmetry
Objective
Define reflectional and rotational symmetry. See some examples restricted to
geometric shapes.
Definition A figure has reflectional symmetry when there is a line of symmetry that
separates the figure into two parts, one of which is a mirror image of the other part.
Some figures have more than one line of reflectional symmetry.
1. Here are two of the mirror lines for the square below. Find the other 2.
2. Find all the mirror lines for the pentagon (5-gon), the hexagon (6-gon) and the star
below.
3. Given the two examples in question 2, how many mirror lines do you think there are
in an n-gon? Draw some more examples if you need to.
Note that there are plenty of shapes that do not have reflectional
symmetry. For example, this lightning bolt:
Definition A figure has rotational symmetry if there is a point
called the center of rotation around which the figure can be
rotated so that it aligns with itself in less than one complete rotation of 360°. The
angle of rotational symmetry is the smallest angle through which the figure must be
rotated so that it aligns with itself. The order of rotational symmetry is the number
of small rotations that must be made to return to the original orientation.
For instance, the square above has 4-fold rotational symmetry. The smallest angle
through which we can rotate so that it aligns with itself is 90. The center of rotation
is located in the center of the square. The easiest way to see that the angle of rotation
is 90 is to note that 360/4 = 90. (360 divided by the order of the rotation gives you
the smallest angle of rotation.)
Remark: It is true that we can also rotate the square through 180 and cause the
rotated image to align with the original. The point is that we are looking for the
smallest possible angle!
4. What are the order of rotation and the angle of rotation for the following shapes?
Exploration: Quadrilaterals and Symmetry
Objective
On this exercise sheet we want to record the common features of various quadrilaterals.
Some are alike and some are different with regard to these features.
1. In the following table, indicate by Y (yes) or N (no) whether each of the figures
always has the indicated property.
All sides congruent
Opposite sides
Opposite sides
congruent
parallel
Rectangle
Parallelogram
Trapezoid
Rhombus
Kite
Square
Any quadrilateral
All angles congruent
Opposite angles
congruent
Has right angles
Rectangle
Parallelogram
Trapezoid
Rhombus
Kite
Square
Any quadrilateral
2. Draw the two diagonals of each of the figures. In the table indicate by Y and N
whether the figure always has the indicated property of the diagonals.
Bisect each other
Are congruent
Meet at right angles
Rectangle
Parallelogram
Trapezoid
Rhombus
Kite
Square
Any quadrilateral
3. In how many ways can you fold a square so that one half matches the other half? (If
you get a fold in the paper that you already had, don't count it again.) How many
folds can you get in any rectangle so that one half matches the other half? The folds
you are getting are called lines of symmetry. In the table record your results.
Number of lines of symmetry
Rectangle
Parallelogram
Trapezoid
Rhombus
Kite
Square
Any quadrilateral
4. Does there seem to be any connection between the congruence properties from
problem 1 and the lines of symmetry?
Exploration: Symmetries in Celtic Art
Objective
Symmetry in non-geometric shapes is quite interesting. Celtic art is quite symmetrical
in nature, but note the lack of reflectional symmetry.
For a finite shape, we may classify it by its symmetry group.
We first check if the figure has reflectional symmetry or not.
xvi. If it does have reflectional symmetry, then it is denoted as a ‘D’ group.
xvii. If it does not have any reflectional symmetry, then it is denoted as a ‘C’ group.
xviii. Finally, we check what the largest degree of rotational symmetry is for our figure.
This number is the added to the ‘D’ or ‘C’ we assigned before.
With Celtic art one should be careful! The under- and overcrossings have a tendency to destroy reflectivity.
Example: The figure to the left has no reflectional symmetry, hence
we get a ‘C’. The figure exhibits 8-fold rotational symmetry (the
centre of rotation is located in the centre), and hence this figure has
symmetry group C8
1. Determine the symmetry groups of the following shapes:
a.
d.
b.
c.
e.
Exploration: Rosette Symmetry
Objective
Explore cyclic and dihedral groups using computer software. Develop intuition about
the nature of these symmetry groups.
This exploration requires the computer program Kali
Kali opens with three windows, a drawing window, a colour window, and the wallpaper
window. You can close the wallpaper window, since you won’t need it today. On the
Palettes menu, choose Rosette.
Play around with the different Rosette symmetry groups. In class, we defined cyclic and
dihedral groups. Kali’s groups are numbered, and half have *.
1. Which Kali groups correspond to our cyclic and dihedral groups?
2. Choose one of the cyclic groups, and draw a large shape. Print a copy of the shape.
Label the rotation angles. How many symmetries does your shape have?
3. Choose one of the dihedral groups, and draw a large shape. Print a copy of the shape.
Label the lines of reflection and the rotation angles. How many symmetries does your
shape have?
Hand in your marked up shapes with answers.
Exploration: Rotational and Reflectional Symmetry in Escher’s Prints
Objective
Look at symmetry in larger decorations. Learn to recognize rotations and reflections in
wallpaper patterns.
The plane filling patterns (also called wallpaper patterns) shown on pages 116 - 233 also
have symmetry groups associated to them. There are infinitely many rosette symmetry
groups, but the surprising fact is that there are only 17 wallpaper symmetry groups. These
symmetry groups depend on similar ideas. We will look at possible reflectional
symmetry, and we will look for possible rotational symmetry. (There are other “things”
we need to look for, but more about that later.)
Look at (3) on page 117:
There is 2-fold rotational symmetry where the heads meet. There is 4-fold rotational
symmetry where the elbows meet. There are lines of rotational reflection running through
the centres of the bodies. (These lines of reflection form the diagonals of the underlying
tessellation by squares.)
Next, look at (5) on page 119:
There are no lines of reflection for this pattern. We do find rotational symmetry. Note
that there is 2-fold rotational symmetry. Also note that there are two different rotational
symmetries at work here. There is a 2-fold rotation around a point on the hand of the
figure, and there is a 2-fold rotation about a point on the upper arm of the figure. These
are considered different centres of rotation!
1. Complete the following table:
reflectional 2-fold rot.
symmetry
symmetry
Number 3
1: at heads
Yes
Number 4
Number 5
No
2: at heads &
arms
3-fold rot.
symmetry
-------
-------
4-fold rot.
symmetry
1: at elbows
-------
6-fold rot.
symmetry
------
-------
Number 12
Number 13
Number 25
Number 35
Number 45
Number 85
2. Print 70 has 6-fold rotational symmetry. Find at least two more prints with 6-fold
rotational symmetry.
3. What are the underlying geometric tessellations for the prints you investigated?
4. Where are the centers of rotation with respect to the underlying geometric
tessellations? I.e. do they lie on the vertices? In the middle of the polygon? Is there
any pattern you can discern?
Geometric Tessellations
Tessellations
Suggested Reading in Visions of Symmetry:
xix. The fascination of regular division of the Plane, Pg 9-15
xx.
Other Experiments in Regular Division, Pg 44-52
In Visions of Symmetry (Schattschneider):
xxi. Lion and bat patterns on Pg 11
xxii. Escher’s experiments with tessellations Pg 27-29
xxiii. Examples of tessellations used by Escher on Pg 32
xxiv. The “fundamental” forms of regular divisions of the plane on Pg 33
xxv. Escher’s experiments with block tessellations on Pg 44-52
xxvi. Regular Division Drawings Pg 116-234
In The Magic Of M.C. Escher:
xxvii. Metamorphosis II Pg 1-13
xxviii. Eight Heads, Pg 22
xxix. Regular Division Drawings, Pg 52-53, 60-63, 74-78
xxx. Escher’s Posters Explaining Theory, Pg 58-59
Introduction
Definition A tessellation or tiling of the plane is a complete covering of the plane with
shapes so that there are no holes and no overlapping.
We want to understand how to tessellate the plane with shapes that look like fish,
lizard, horses or other things. Before we get to the more fancy tessellations we need to
first understand how to tessellate with general geometric shapes. We will look at
tessellations by triangles, tessellations by quadrilaterals, and tessellations by regular
polygons.
Triangles always tessellate the plane. On
the left is an example of a part of such a
tessellation. Tessellations are meant to cover
the whole plane, so imagine that this pattern extends in all directions.
This is just one example of a triangle tessellating the plane. It is a pretty strong
statement to make that all triangles will tessellate the plane, but this is true. We will show
later why this is true.
On the left you see a small part of a tessellation by squares. Recall from the section
on polygons that the “corner” of a polygon is called a vertex. We require in all of our
work that vertices only meet vertices. It is not necessary to demand this in the general
theory of tessellations, but for our purposes, these are the kinds of tessellations we will
later on want to work with. In other words, the tessellations by squares on the right in a
perfectly legal tessellation, but for the purpose of this course we will ignore such patterns.
It is fairly easy to see that squares tessellate. You can take squares and create
horizontal strips with them.
This collection of squares tessellates the area between two parallel lines. We know
this because the angles in a square all measure 90 degrees. So, if you line up the squares
as indicated then all the bottom segments combine into one straight line, and the tops
segments all combine into a straight line. The distance between the line on the top and the
line on the bottom is always the same everywhere, hence the two lines are parallel. It is
easy to see that stacking these tessellated strips will result in a tessellation of the plane.
You should be able to convince yourself that this argument also works to show that
rectangles, trapezoids, rhombi and parallelograms tessellate the plane. Because of their
special properties, these special quadrilaterals all tessellate these stackable horizontal
strips, and hence they tessellate the plane. We get the following theorem:
Theorem 1a: All squares, rectangles, rhombi, trapezoids and parallelograms
tessellate the plane.
This begs the question: which the other quadrilaterals tessellate the plane? What
about other shapes? If you look at a kite or a non-convex quadrilateral, then it’s clear that
they do not tessellate these nice horizontal strips. Our argument does not generalize.
There are other ways to show that in fact all quadrilaterals tessellate the plane. We have
the following (stronger) theorem:
Theorem 1: All quadrilaterals tessellate the plane.
The proof of this fact uses rotations about the midpoint of a side. We need another result
that we will assume to be true for now. The proof of this theorem is given in a later
section.
Theorem 2: The sum of the angles in a triangle is 180 degrees.
There is an immediate consequence of this result, which tells us the angle sum in a
quadrilateral:
Theorem 3: The sum of the angles in a quadrilateral is 360 degrees.
This is quite obvious if you consider the fact that we can always subdivide any
quadrilateral into two triangles.
Let’s show why any arbitrary 4-gon tessellates the plane. If we start with an arbitrary 4gon ABCD, and find the midpoint of a side (I chose the midpoint of BC in this example)
we can rotate through 180 degree about the midpoint of a side. We can repeat this process
about the midpoints of other sides. The rotation about the midpoint of a line segment
implies that the sides will match up exactly. If we keep track of the angles around a
vertex, we will see that they are the angles of the original quadrilateral. But the angle sum
for a quadrilateral is exactly 360 degrees. Hence the angles close up. This means we have
created a pattern that doesn’t have any gaps or overlaps, and hence is a tessellation.
D’
C’
B
A’
B
A
A
C
C
B’
D
D
D’
C’
D’
A
B
A
A’
B
C’
A’
C
B’
B’
C
D
D
It may be worth showing what this looks like for a non-convex quadrilateral, as these are
the hardest to work with. Quite a few people initially think that the non-convex
quadrilaterals do not tessellate. The proof however shows not only that they do tessellate,
but also shows you how to construct the tessellation. (Just keep rotating about midpoints
of sides.)
We can now quite easily show that all triangles tessellate as well.
Theorem 4: All triangles tessellate the plane.
Suppose we take an arbitrary triangle, and rotate it about the midpoint of one of its sides.
We obtain a 4-gon that has the property that the opposite sides are equal. This implies
that the 4-gon must be a parallelogram. (In special cases it may be a square or rectangle.)
But parallelograms tessellate the plane by theorem 1a. And hence the triangles must
tessellate the plane.
In pictures we get the following:
X
Y
Z
Z
Y
X
X
Y
Z
The opposite angles are equal. This is one
of the alternative definitions of a
parallelogram. We know from before that
parallelograms tessellate horizontal strips.
These strips stack up to form a tessellation of the plane:
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
It should be noted that we could also tessellate the plane with triangles using only
rotations. This idea is explored in one of the GSP explorations.
Regular and Semi-Regular Tessellations
We will take a closer look at two very specific types of tessellation, namely the regular
tessellations and the semi-regular tessellations. Recall the following definitions:
Definition: A regular polygon has the property that all of its sides are equal and all of its
angles are equal.
For example, a regular triangle is an equilateral triangle, and a regular 4-gon is a square.
Definition: A regular tessellation is a tessellation made from exactly one type of regular
polygon.
Definition: A semi-regular tessellation is a tessellation made from more than one type
of regular polygon, and the additional property that the tessellation looks the same at
every vertex.
On the plane, the number of regular and semi-regular tessellations is very restricted. We
have the following theorems.
Theorem 5: There are exactly three regular tessellations of the plane. Only the
equilateral triangle, the square and the regular hexagon tessellate the plane.
A sketch of the proof is as follows:
Step 1: We may divide any n-gon into (n-2) triangles in such a way that the angle sum of
all the triangles corresponds to the angle sum of the polygon. This implies that the sum of
the interior angles of an n-gon is (n - 2) 180° (There are (n-2 triangles and each has an
angle sum 180°)
The sum of the interior angles of an n-gon = (n - 2) 180°
Step 2: For a regular polygon all the interior angles are equal, hence we can compute the
measure of one angle:
The measure of an interior angle in a regular n-gon =
(n – 2)180°
n
Step 3: If a regular polygon tessellates the plane, then some number of polygons has to fit
exactly around a vertex. This means that the angle measure must be an exact divisor of
360°. There are only a finite number of exact divisors of 360°:
2,3,4,5,6,8,9,10,12,15,24,30,36,40,45,60,72,90,120,180.
Step 4: A little experimentation and computation quickly shows that all regular n-gons
have angle measures greater than or equal to 60, and strictly less than 180. Examining a
list of angle measures of regular n-gons show that the only divisors of 360 that
correspond to an angle measure are 60 (triangle), 90 (square) and 120 (hexagon).
End of Sketch
Theorem 6: There are exactly 8 semi-regular tessellations of the plane.
The proof of this theorem is a bit more involved. We will not go through it in detail, but
the idea is that of we analyze the vertex configuration; we see that there are 23 different
ways in which to arrange regular polygons of two or more types around a vertex, so that
there are no gaps or overlaps at the vertex. Careful examination of each of the 23
examples then shows that only 8 of these extend to a semi-regular tessellation of the
plane.
The other vertex configurations can sometimes be extended to a tessellation, but it’s not
possible to arrange the exact same configuration at every vertex, and hence the
tessellation is not semi-regular.
Some Theory and Logic
Objective
Demonstrate the difference between examples, empirical evidence, and mathematical
proofs.
There is a big difference between showing some examples and rigorously arguing that a
statement must be true. This distinction also plays an important role outside of
mathematics. Passing from an example to a general statement is a tricky business, and
mistakes can result in erroneous conclusions. An example from mathematics is the
q  22 1
n
following formula:
For n = 0 we obtain q = 3. This is definitely a prime number.
For n = 1 we obtain q = 5. This is definitely a prime number.
For n = 2 we obtain q = 17. This is definitely a prime number.

Does this mean we have a formula that generates prime numbers? No, the next value of n
actually fails.
This example shows how examples can be deceiving. The examples that don’t work may
just not have been tried yet.
You will be amazed how often people make similar logical errors. Listen to a political
speech, and you might find such errors. Documentaries claiming to prove the existence of
UFO’s and/or ghosts are also full of logical flaws.
Axioms for Plane Geometry
In geometry we develop all our theory from a collection of undefined terms, and general
truths called axioms or postulates.
Undefined terms are concepts such as “point”, ‘line”, and “between”. The axioms for
planar geometry are as follows:
Axiom 1: We can draw a unique line segment between any two distinct points.
Axiom 2: Any line segment can be continued indefinitely.
Axiom 3: A circle of any radius and with any center can be drawn.
Axiom 4: Any two right angles are congruent.
Axiom 5: Given a line L and a point P not on L, there exists a unique line L2 through P,
which does not intersect L.
The fifth axiom can be restated as follows:
Axiom 5.v2: Suppose a line meets two other lines so that the sum of the angles on one
side is less than two right angles. Then the other lines meet at a point on that side.
Line 1
b
a
Line 2
In the diagram, a+b <180°, and Lines 1 and 2 will meet on that side of the intersecting
line.
Theorems of Plane Geometry
Theorem 1 If a line meets two parallel lines, then opposite interior angles are congruent.
Proof:
C
A B
D
We know from axiom 5.v2 that D+B = 180°. But A + B = 180°, and hence angle A and
angle D are congruent. •
Theorem 2 The sum of the angles in a triangle is 180°.
Proof: L
A
 
B
bb
C
Let  ABC be a triangle. Construct the line L through A that is parallel to the line
segment BC. (The existence of the line follows from Axiom 5.) By the previous theorem
ABC =  and ACB = . Note that  +  + BAC =180°. This implies that
ABC + ACB + BAC =180°. •
A result that is a consequence of a theorem is called a corollary. From Theorem 2 we can
easily show the following:
Corollary The sum of the angles in a quadrilateral is 360°.
Proof: Any quadrilateral can be divided into two triangles. The sum of the angles of the
quadrilateral is equal to the sum of the angles of the two triangles, and the result follows.
•
Sometimes we will give a sketch of a proof. We will outline the general argument, but
omit some details. We will illustrate that for the following theorem:
Theorem 3 All triangles tessellate the plane.
Sketch of proof:
xxxi. Any triangle can be rotated about the midpoint of one of its sides, and this will
construct a parallelogram for us.
xxxii. Parallelograms tessellate the area between two parallel lines. We’ll refer to these
areas as “strips”.
xxxiii. These strips will tessellate the plane by stacking them. •
We would need to prove part one and two of the sketch to turn this into a real proof.
Theorem 4 All quadrilaterals tessellate the plane.
Sketch of proof:
xxxiv. Rotate the quadrilateral about the midpoints of its sides.
xxxv. Show that this gives a tessellation about every vertex of the quadrilateral.
xxxvi. Argue that this method gives an algorithm for tessellating the plane. •
Theorem 5 There are exactly three regular tessellations of the plane.
Sketch of proof:
xxxvii. The sum of the interior angles in an N-gon is (N-2) 180°.
xxxviii. It follows that the interior angle of a regular polygon is (N-2) 180°/N.
xxxix. To tessellate we need to arrange an integer number of tiles around a vertex.
xl.
Construct a table giving the interior angle for a given N-gon.
xli.
Use the table to argue that there are only three regular N-gons that will tile the
plane. •
Theorem 6 There are exactly eight semi-regular tessellations of the plane.
Sketch of proof:
xlii. There are exactly 23 ways to configure multiple types of regular polygons around
a vertex.
xliii. Only eight of these vertex configurations extend to a tessellation of the plane. •
xliv.
xlv.
xlvi.
xlvii.
Exercises: More on Tessellations
1. Find a print of Metamorphosis III [MOE inside cover sleeve, VOS pg 258-262]. Find
all the areas that have purely geometric tessellations. Describe each such tessellation,
and (briefly) describe what comes before and after in the print.
2. Find a print of Verbum [MOE pg 80, VOS pg 259]. Describe what’s going on in this
picture. What’s happening with black and white? Where are the “final” animals that
aren’t just in-betweens?
3. Using a piece of graph paper, draw a triangle with the same dimensions as triangle A
shown below. Now draw more of them to tessellate at least 1/4 of the page.
4. Tessellate using triangle B.
5. Tessellate using quadrilateral C.
6. Tessellate using quadrilateral D.
A
B
D
C
7. Which of these shapes are convex?
a.
b.
c.
d.
e.
8. One pleasing method for tessellating with pentagons was the basis for Escher's
sketches on pages 224-227 [VOS]. It is shown as the red lines on page 106 and there
is a discussion of how to make the pattern on that page. You'll see the "simple grid of
squares" if you cut each pentagon in half down its axis of symmetry.
Make a good-sized (half page at least) sketch of this pentagon tessellation on graph
paper. If you want, use two colors to emphasize the "overlapped hexagons" as Escher
does on page 225-226.
9. Tetrominoes are tiles made of four squares. There are only five kinds possible:
Using graph paper, show a way to tile the plane with each tetromino.
10. Pentominoes are like tetrominoes but made of five squares. There are 12 of them:
Using graph paper, show how each one of these can tile the plane (make 12
sketches!). All of the twelve can tessellate in more than one way, except for one.
Which is the pentomino that can only tile the plane in one way?
11. A septomino is made of seven squares. Find a septomino, which cannot tile the
plane.
Regular Tiles
12. Recall that a regular polygon has all sides and all angles equal. If we just require all
sides to be equal, is the polygon regular? What about if we require all the angles to
be equal?
13. What are the angles of a regular 11-gon? What is its angle sum?
14. Explain why you cannot tile the plane with a regular n-gon, for n larger than 6.
15. Draw a tessellation that uses regular octagons and squares.
16. From the back of the reader tear out a page of triangular dot paper. Draw three
tessellations using hexagons and equilateral triangles. Make all three have different
symmetry groups (not all p6m!).
17. Explain, by discussing angles, why you might hope to tile with regular pentagons and
regular decagons (10 sides). Explain why it can’t be done.
Tessellations in modern Art
18.
This is a famous print by Salvador Dali
(“Cinquenta...tigre Real”). What are the geometric
shapes that make up the tessellation? Is the tessellation
regular? Is it semi-regular? Explain.
19.
This painting by Mondriaan (“Composition A:
Composition with Black, Red, Gray, Yellow, and
Blue 1920”) is also a tessellation. What are the
geometric shapes that make up the tessellation? Is this
tessellation periodic or aperiodic? Explain
REMARK: you should look at these prints in color! (You should be able to find them on
the Internet.)
20.
This painting is by Kandinsky.
a. What geometric shape is the tessellation based
on?
b. Is this tessellation periodic or aperiodic?
Carefully explain your answer.
Explorations For Geometric Tessellations
Exploration: Basic Tessellations
Objective
We find a method to tessellates the plane with any triangle. We introduce regular and
semi-regular tessellations.
Tessellations by triangles and quadrilaterals
1. Convince yourself that the parallelogram below will tessellate the plane. Draw your
tessellation on a separate piece of paper. It should cover 1/4 of your page.
2. Show that the square and the rectangle below each tessellate the plane. Your
tessellation should cover 1/4 of your page.
3. Draw an acute, an obtuse and a right triangle. Now convince yourself of the fact that
two congruent copies of the same triangle fit together to form a parallelogram or a
rectangle.
4. Why does this imply that all triangles will tessellate the plane?
Tessellations of the plane by regular polygons
Definition A regular tessellation is a tessellation made of regular, congruent, convex
polygons.
Definition A semi-regular tessellation is a tessellation made of regular polygons of
two or more types so that the arrangement of polygons at each vertex is the same.
5. What is the common name for a regular 3-gon?
6. What is the common name for a regular 4-gon?
A tessellation is a regular tessellation if it is constructed from regular convex polygons of
one size and one shape. There are exactly three regular polygons that tessellate the plane.
Above, we see the tessellations of the plane by an equilateral triangle, a square, and a
regular hexagon.
The regular pentagon does not tessellate the plane! The figure below shows that when
we try to arrange the pentagons around a vertex, then we will always have a gap or an
overlap.
7. There are non-regular pentagons that tessellate the plane. Sketch the tessellation for a
pentagon that looks like the outline of a house (see below) to illustrate this point.
Semi-Regular Tessellations
We can also look at tessellations formed by a combination of several regular polygons.
Definition: A tessellation is a semi-regular tessellation if it is composed of regular
polygons of two or more types so that the arrangement of polygons at all the vertices
is the same.
8. Extend the following tessellation. You should draw at least 2 more layers of polygons
on each side. Is this tessellation semi-regular? Explain.
9. Extend the following tessellation. You should show at least 2 more layers of polygons
on each side. Is this tessellation semi-regular? Explain.
10. Are the following tessellations semi-regular? Explain.
Exploration: Why There Exist Exactly Three Regular Tessellations
Objective
There are only three regular tessellations. We will show through some exercises why
this is true.
Interior Angles of Polygons
Warm up: We know that the sum of the angles in a triangle is 180°
1.
Demonstrate that the sum of the angles in a triangle is 180°?
2. What is the sum of the angles in a quadrilateral? How do we
show that this is true based on the fact that the sum of the angles in a
triangle is 180°?
In general, we can divide a polygon up in triangles. (We do have to
do this in a clever way.)
3.
How many triangles make up a 4-gon?
How many triangles make up a 5-gon?
How many triangles make up a 6-gon?
How many triangles would you say there are in an n-gon based on
the pattern described above?
4. Using the information from question 3 argue that:
The sum of the interior angles of an n-gon = (n - 2) 180°
5. We claimed above that we have to divide up our polygon in a clever way. Why is the
following subdivision not appropriate when trying to find the sum of the interior
angles?
6. By definition, all the angles in a regular polygon have the same angle measure. This
implies that we get the following formula for the measure of an angle in a regular
polygon:
The measure of an interior angle in a regular n-gon =
Would this formula work for just any n-gon? Why or why not?
7. Complete the following table:
n
Angle = (n – 2) 180° / n
3
60˚
4
90˚
5
6
7
8
(n – 2)180°
n
9
10
11
12
15
20
50
100
If regular polygons are going to fit around a vertex, then their angle measures have to be
an exact divisor of 360˚.
8. Which of the angle measures in the table are exact divisors of 360˚?
9. How do you know that there are no other regular polygons with angles that are exact
divisors of 360˚ besides the ones mentioned on the list?
10. (Easy) Which regular n-gons are the only ones that can tessellate the plane using just
one type of tile?
Exploration: Polyominoes
Objective
Tetrominoes are made up out of four squares, combined in different configurations (as
seem in Tetris). These tiles give good examples of oddly shaped polygons that tessellate.
Definition: A polyomino is a polygon made from squares of the same size, connected
only along complete edges.
Triominoes
Definition:
A triomino is a polyomino made from three squares.
1. There are exactly two triominoes. Draw both of them. (Two triominoes are
considered the same if one can be obtained from the other by rotation or reflection).
2. What kind of symmetry does each of the triominoes have? (Rotational? What angle?
Reflectional? How many lines of symmetry?)
3. Both triominoes tessellate the plane. Show a tessellation for each of the triominoes.
Tetrominoes
Definition A tetromino is a polyomino made from four squares. Another name for a
tetromino is a quadromino. These are the shapes that are used in the popular
computer game Tetris.
4. There are five different tetrominoes. Draw them. (Two tetrominoes are considered the
same if one can be obtained from the other by rotation or reflection).
5. What kinds of symmetry does each of the tetrominoes have? (Rotational? What
angle? Reflectional? How many lines of symmetry?)
6. All tetrominoes tessellate the plane. Show a tessellation for each of the tetrominoes.
Pentominoes
Definition:
A pentomino is a polyomino made from five squares.
7. There are twelve different pentominoes. Draw them. (Two pentominoes are
considered the same if one can be obtained from the other by rotation or reflection).
8. Pick five of the pentominoes that you found and for each individual pentomino draw
at least one tiling pattern that can be developed with it.
9. How many different tilings can you create using just the "long pentomino"?
This text is based a webpage created by Mike Riedy:
http://forum.swarthmore.edu/sketchpad/intro/gsp.introlab1.html
Exploration: The Geometer's Sketchpad
Objective
Learn basic operations of The Geometer’s Sketchpad.
You will need to find the icon in the `'Alias Palette'' that looks like a circle with a triangle
and another circle in it. Once you find the icon, double-click on it. This will get the
program up and running. Click anywhere to start in the program.
Making Mistakes
You are bound to make some mistakes while working with the GSP. Don't panic! There
is a very simple way to correct a mistake. Simply press and hold down the command key
(that's the one with the funny little apple on it), then press the "Z" key. This combination,
referred to as "Apple Z," will undo, in reverse order, any previous steps you made. There
is one catch: Once you save your work you can only "Apple Z" back to the point at which
you saved. "Apple Z" can also be accomplished by choosing "Undo" under the EDIT
menu.
The Toolbox
select tool point tool
line tool
circle tool
label tool
As GSP starts up, you will see a picture with the words "The Geometer's Sketchpad" in
the centre of the screen. Click once anywhere to clear the picture. Now you are looking
at the basic GSP layout, called a sketch. To the left on the sketch you will see six
different squares with different objects in them. These objects are called the tools. The
five most commonly used tools and their names are at the top of this page. The area to
the right of the tools is called the window. We are going to examine how to use each of
the five tools. Note that the software is very particular. If you are, for instance, in the
circle tool mode, then you can only draw circles. If you want to draw line segments or
select part of your drawing, then you will have to select that tool first.
The Circle Tool:
Select the circle tool by using the mouse to position the arrow over the circle tool and
click once. To draw a circle, move the mouse to the right onto the sketch window. Press
and hold the mouse button while moving the mouse. A circle should appear. Draw
several circles of different size. (Five or six should suffice.) Now practice erasing them
by using "Apple Z" or "Undo." One by one the circles should disappear. Now draw
three circles of different sizes that fit on the screen.
The Select Tool:
Choose the select tool (sometimes called the translate tool). When you press and hold the
select tool three other pictures appear.
1. Sketch each of the other tools.
For now, make sure you are using the first one with the arrow only. Move onto the
window and highlight a circle by moving the arrow close to a circle until the arrow
becomes horizontal. When the arrow is horizontal it means the arrow is pointing to
something. Try to avoid the point on the circle; you'll need that shortly. With the arrow
horizontal, hold down the mouse button and move the mouse around. What happens?
Arrange the circles from smallest to largest so that they look like a snowman.
2. By using the select tool, how can you change the size of the circle you have already
drawn? (Hint: paragraph above.)
The Point Tool:
Select the point tool - it looks like a dot. Using this tool is very simple: wherever you
click, a point appears. Use the point tool to put eyes, nose, mouth, and buttons on your
snowman. (You can also put snowflakes all around your snowman.)
The Segment Tool:
The segment tool allows you to draw straight objects. When you press and hold the
segment tool three other pictures appear.
3. Sketch each of the other tools and give their names.
Off to the side of your snowman you are going to practice drawing one of each of the
three types of straight objects. To draw a straight object, click and hold the mouse button
while dragging in the direction you want your object to go. Try drawing one
horizontally. What happens when you hold down the SHIFT key while drawing a
segment? (Hint: move the endpoint around in a circle.)
Now use the segment tool to put arms and hands on your snowman. (You could also
draw the ground and a put a hat on the snowman.)
The Labeling Tool:
When you draw anything GSP gives it a name called a label. Sometimes, however, the
label is invisible. Every point, circle, and segment you have drawn on your snowman has
a label - even if you can't see it. Let's see what the program labelled your snowman's
nose. Choose the labelling tool from the toolbox on the left of the screen (it looks like a
little hand). Move the hand close to the nose of your snowman. When you are pointing
at the nose the hand will turn black. Click once and a letter should appear.
4. What is the label for your snowman's nose?
Click on the nose again to make the label disappear.
The labelling tool actually serves two purposes: labelling tool and text tool. You have
already seen what the labelling does. The text tool allows you to write whatever you
want anywhere on the sketch. To put your name on the snowman, find an empty place on
your picture and move the labelling tool there. Click and hold the mouse button while
dragging diagonally. A box should appear. This is called a text box. Type some text of
your choice in the text box. (Your name would do.)
Optional: Printing and Saving:
Let's print the snowman to see how he looks. Before you print, however, make sure your
picture is complete with all the necessary parts. Your name and the date should also be
on the sketch. Once all this is complete choose "Print" under the FILE menu. Your
snowman will pop out at one of the printers in the room.
A New Sketch:
You now need a new, blank sketch to work with. Choose "New Sketch" under the FILE
menu. What is the keyboard shortcut for starting a new sketch?
Highlighting:
The most important thing in GSP is how to highlight objects. This is your way of telling
the computer, "I'm talking about this object." To highlight you must use the select tool.
The computer puts little black squares on the object to highlight a straight object or a
circle. A point is highlighted when it looks like a bulls eye.
xlviii. On your new sketch, draw several points. Notice that the last point you drew is
highlighted.
xlix. Draw a circle and two segments that start outside the circle and pass through it.
l.
Use the select tool to move the circle just a little bit but still intersecting the
segments.
What if you want to move more than one thing at a time? Version 4.0 will allow you to
select multiple objects by clicking on them in turn. Make sure you do not click outside
the figures. This will deselect everything. Now highlight the circle and one of the
segments. Move them away from the second segment so that the second segment and the
circle no longer intersect.
Hiding Objects:
Very seldom when using GSP will you delete objects. Instead you will hide them.
Delete means to completely erase something, while hide means that it's still there but you
just can't see it. Look through the menus to find the "Hide" command.
5. What menu was it under? What is the keyboard shortcut?
Highlight the circle and hide it using either the menu or the keyboard shortcut. The circle
should disappear. Suppose you really didn't want the circle to be gone because you need
it after all. To bring it back you need to use the "Show All Hidden" command under the
DISPLAY menu. This will bring back the circle.
Transformations
Later we will look at properties of transformations, but for now let's look at the basics.
Start with a new sketch. Draw a triangle and place a single point somewhere outside the
triangle.
6. Describe how you constructed the triangle.
7. There is another way to construct a triangle. How?
Highlight only the single point and choose "Mark Centre" under the TRANSFORM
menu. What is the keyboard shortcut for marking centre?
You can also accomplish this by double-clicking on the point (a little bulls eye should
appear around the point for a fraction of a second). Next, change the select tool to the
rotation tool. Highlight all parts of the triangle. Drag the triangle.
8. What happens?
Change the rotation tool to the dilation tool. Now try dragging the triangle.
9. What happens when you try to move the triangle closer to and farther away from the
point?
The select tool is sometimes called the translate tool.
10. Define the three words translate, rotate, and dilate.
Start with a new sketch. On the left half of the sketch draw a small smiley face and a
vertical line just to the right of it. Feel free to add some colour and make your smiley
face one solid colour. Highlight the line and choose "Mark Mirror" under the
TRANSFORM menu. Highlight your entire smiley face and choose "Reflect" under the
TRANSFORM menu.
11. Now try moving one of the eyes on the left smiley face. What happens to the eye on
the smiley face at the right?
Exploration: Triangle Tessellations
Objective
In this project, you will use Geometer's Sketchpad to create tessellations with triangles.
Doing the construction on the computer allows you to change the shape of your original
triangle and see how the tessellation changes.
Make A Triangle
1. Run Geometer's Sketchpad (GSP). You should have an Untitled window for
sketching. If you don't, go to the file menu and choose "New Sketch".
2. Use the line segment tool to draw a segment. Then draw two more segments to make
a triangle.
3. Construct the interior of the triangle:
First, use the select tool (the arrow) to select all three corners of the triangle. Selected
things are outlined in red, or have red checkers. To select nothing, click on any
empty area of the sketchpad.
Choose "Triangle interior" from the Construct menu to construct the interior.
With the interior selected, you can change its colour and shading (using the Display
menu) if you want.
A Triangle Tessellation
1. Construct the midpoint of one side of the triangle. To do this, select a side of the
triangle. Then choose "Midpoint" from the Construct menu.
2. Rotate the triangle around the midpoint you just made. To do this, select the
midpoint and choose "Mark Center" from the Transform menu. The rotation center
will flash. Now select the triangle and choose "Rotate" from the Transform menu.
You want to rotate by 180°. Make the new triangle a different color than the first.
Your picture should look something like this (but in colour, and probably with a different
shape of triangle):
3. Try moving the original triangle's corners. Notice that the new triangle changes
shape too, and the picture is always a parallelogram. You probably want to save your
work at this point.
4. Use translations to finish the tessellation.
To do a translation, you first need to specify the length and direction, which is called a
"vector". Select one corner of your triangle, and then select a second corner. Now
choose "Mark Vector" on the transform menu. The translation direction and length
should flash.
Select both triangles and choose "Translate" from the Transform menu. You should get
another parallelogram, translated over from the first.
Keep making copies (use different translation vectors when you need to), until you fill
out a good-sized piece of the plane:
Move the corners of the original triangle. Everything else should change shape with it.
5. Add your name to the sketch using the Text palette. Print out your tessellation.
6. Write answers to the following questions on your printed sheet.
a. For a random triangle, what symmetries (reflectional and rotational) does the
picture have?
b. Make the triangle a 45°-45°-90° triangle. Now what symmetries does the picture
have?
c. Make an equilateral triangle. . Now what symmetries does the picture have?
45°-45°-90° triangle:
Equilateral triangle:
Exploration: Quadrilateral Tessellation
Objective
In this project, you will use Geometer's Sketchpad to create a tessellation with
quadrilaterals. Doing the construction on the computer allows you to change the shape of
your original quadrilateral and see how the tessellation changes.
Directions
1. Run Geometer's Sketchpad (GSP). You should have an untitled window for
sketching. If you don't, go to the file menu and choose "New Sketch".
2. Use the line segment tool to draw a segment. Then draw three more segments to
make a quadrilateral.
3. Construct the interior of the quadrilateral. Choose a color.
4. Construct the midpoints of all four sides of the quadrilateral.
5. Rotate the quadrilateral 180° around each midpoint. Color all four new copies with
one color.
Your picture should look something like this:
6. Use translations to finish the tessellation.
7. Move the corners of the original quadrilateral. Examine what happens when it is not
convex.
8. Add your name to the sketch using the Text palette. Print out your tessellation.
9. What symmetries does the picture have??
Exploration: Square Block Patterns
Objective
Explore, as Escher did, the tessellations one can create using square blocks as stamps.
Read Pg 46 and up from Visions of Symmetry.
1. The following three patterns were made with the blocks at the top of page 46. For
each pattern, give a two-by-two number array that generates it.
A
B
C
2. For any number array, there are other arrays that generate the same overall pattern,
after translation or rotation. Give five other number arrays that generate pattern C
above.
3. Write down a new two-by-two number array using 1,2,3, and 4, then (on graph paper)
sketch the pattern created by these blocks. Escher found 23 distinct patterns - find
three more and sketch them.
4. A block with some symmetry has fewer possible positions. For example, the block
below has 180° rotational symmetry, so it has only two positions, labeled as 1 and 2:
1
2
There are 16 possible two-by-two arrays of numbers using only 1s and 2s (one with
four 1s, one with four 2s, six with two of each, and eight more). Find them all. Now,
notice that not all arrays produce different patterns. For example, the four 1 array,
and the four 2 array produce the same picture, just rotated. In fact, only four distinct
patterns are produced. Sketch all four, and classify which arrays produce which
patterns.
5. How many patterns can be made with block (a) below?
How many patterns can be made with block (b) below?
Do you get more patterns if you use block (a) and its mirror image?
Do you get more patterns if you use block (b) and its mirror image?
(a)
(b)
6. Using one square asymmetrical stamp (such as Escher's stamp at the top of page 46),
which of the 17 wallpaper groups can you make? (Hint: Can you get reflections?)
7. Use this block
to make a pattern with symmetry group pgg, p4, or cmm. (You pick).
Hint: you can put a few blocks together to make a more complicated unit, and use the
unit to build your pattern.