THE GEOMETRY AND ART OF TESSELLATION
Exercises Set 5.1
1.
A plane tessellation is a pattern made up of one or more shapes completely
covering a surface without any gaps or overlaps.
2.
The word tessellation originates from the Latin word tessella, meaning the small
square stone or tile used in ancient Roman mosaics.
3.
The Moors were ancient people of North Africa who invaded and occupied Spain
for roughly seven hundred years. Moorish artists created abstract geometric
mosaic designs without people, animals or real-world objects.
4.
In general, any triangles and quadrilaterals can tessellate a plane because 360° is a
multiple of the sum of the interior angles of these polygons.
5.a.
Construct a scalene triangle ABC. Translate triangle ABC from point A to point
B. Continue translating in a similar fashion until you have reached the edge of the
plane. Reflect the image repeatedly until the plane is filled.
5.b.
Construct the parallelogram ABCD. Translate parallelogram ABCD from point A
to point B. Continue translating in a similar fashion until you have reached the
edge of the plane. Reflect the image repeatedly until the plane is filled.
5.c.
Construct the right trapezoid ABCD. Translate side AB from point A to point C.
Translate side DC from point D to point B. Translate side AD to complete the
second right trapezoid. Reflect the image repeatedly both vertically and
horizontally until you have filled the plane.
5.d.
Refer to figure 5.1.3. Construct a convex quadrilateral ABCD. Translate side AB
from point A to point C. Translate side DC from point D to point B. Translate
side AD to complete the second convex quadrilateral. Translate side AD from
point a to point C. Translate side BC from point B to point D. Translate side AB
to complete the third convex quadrilateral. Translate sides DC and BC to
complete covering the plane around point C. Translate the image repeatedly both
vertically and horizontally until you have filled the plane.
5.e.
Construct a nonconvex quadrilateral. Translate three of the sides with the fourth
side being common to the first and second nonconvex quadrilaterals. Translate
image repeatedly both vertically and horizontally until you have filled the plane.
the
6.
Construct the pentagon as described. Label the 60° vertex point E. Mark one of
the parallel sides as mirror and reflect the entire pentagon. Translate the parallel
sides to points E and the reflection of E. Close the pentagon by translating the
third side of the square. At this point the image should resemble a zipper or how
upper and lower teeth meet. Translate the image repeatedly along the axis of the
“zipper” until you have reached the edge of the plane. Reflect the image
repeatedly until the plane is filled.
7.a.
Construct a convex quadrilateral ABCD as shown in Figure 5.1.9. Translate side
BC from point C to point A. Translate side AD from point D to point B. Close
the nonregular hexagon by translating side CD.
7.b.
Translate the nonregular hexagon from point A to point C. Translate the two
nonregular hexagons ffrom point D to point B. Repeat both vertically and
horizontally until you have filled the plane.
Exercises Set 5.2
1.
Equilateral triangles, squares, and regular hexagons are the only regular polygons
that can generate monohedral tessellations .
2.
The dual of a tessellation is the tessellation formed by joining the centers of
rotation of each tile. A regular triangle tessellation {3, 6} has a regular hexagon
tessellation {6, 3} as its dual tessellation. A regular hexagon tessellation {6, 3}
has a regular triangle tessellation {3, 6} as its dual tessellation. A square
tessellation {4, 4} has a square tessellation {4, 4} as its dual tessellation.
3.
A semi-regular tessellations are tessellations that are formed by combinations of
two or more regular polygons and they have identical arrangements of polygons at
their vertex points. Only eight semi-regular tessellations exist.
4.
The minimum number of regular polygons surrounding a vertex point is three,
while the maximum number of polygons surrounding a vertex point is six. The
360° surrounding a point must be partitioned into pieces composed of vertices of
regular polygons leaving neither a gap nor overlap.
5.
The Archimedean Tilings are the three regular and eight semi-regular
tessellations.
6.
There are five cases where the space around a point is filled out by an equilateral
triangle and two other regular polygons. The two regular polygons must be: two
12-gons, or a 10-gon and 15-gon, or a 9-gon and 18-gon, or an 8-gon and 24-gon,
or a 7-gon and 42-gon.
7.
There are three cases where the space around a point is filled out by a square and
two other regular polygons. The two regular polygons must be: two 8-gons, or a
6-gon and 12-gon, or a 5-gon and 20-gon.
8.
All five tetrominoes can tessellate the plane.
9.a.
The twelve pentominoes are:
9.b.
All twelve pentominoes tessellate the plane.
10.a. The four pentiamonds are:
10.b. All four pentiamonds can tessellate the plane since two of each pentiamond can
be arranged to form a non-regular hexagon with sides that are pairwise parallel.
11.
The construction of a regular hexagon and its dual should resemble this:
12.
A semi-regular tessellation 3.3.3.3.6 and its dual should resemble this:
13.
A semi-regular tessellation 3.3.4.3.4 and its dual should resemble this:
14.
A semi-regular tessellation 4.8.8 and its dual should resemble this:
15.
A semi-regular tessellation 3.6.3.6 and its dual should resemble this:
16.a. The center of a square is also the center of four-fold rotational symmetry for the
tessellation.
The center of a hexagon is also the center of six-fold rotational symmetry for the
tessellation.
16.b. Each vertex of a square is a center of four-fold rotational symmetry for the
tessellation.
Each vertex of a hexagon is a center of three-fold rotational symmetry for the
tessellation.
16.c.
Each midpoint of a side of a square is a center of two-fold rotational symmetry for
the tessellation.
Each midpoint of a side of a hexagon is a center of two-fold rotational symmetry
for the tessellation.
Exercises Set 5.3
1.
A rigid transformation is the relocation of a two dimensional figure within the
same plane without altering its size or shape.
2.
A translation (or slide) is a rigid transformation that moves every point of a figure
a given distance along an indicated line of direction.
A rotation (or turn) is a rigid transformation that moves a figure in its entirety to a
new location by rotating it about a fixed point.
a
A reflection (or flip) is a rigid transformation that moves a figure in its entirety to
new location by flipping it about a line of reflection.
A glide reflection (or simply glide) is a rigid transformation accomplished by
combining a translation and a reflection.
3.
the
4.
Construct three dashed lines parallel to line segment EF and passing through each
of the vertices of triangle ABC. Construct three dashed circles with radius EF and
center being each of the vertices of triangle ABC. The points of intersection of
corresponding dashed circles and dashed lines are the vertices of the translated
triangle.
Construct a dashed line perpendicular to line l and passing through point A.
Construct a dashed circle centered at point H and passing through point A. Label
the intersection of the dashed line and dashed circle point A’.
Line segment AA’ and line l are perpendicular.
Line segments AH and A’H are congruent.
The reflection of point B under line l is point B itself since it falls on the line of
reflection.
5.a.
Repeat the constructions to reflect point A as described in Problem 4 to reflect
points B and C under line l. Join points A’, B’, and C’ to complete the reflection
of triangle ABC under line l.
5.b.
Repeat the constructions to reflect point A as described in Problem 4 to reflect the
other vertex of the triangle not found on line l. Join the vertex on line l with the
two points reflected under line l to complete the reflection of triangle ABC under
line l.
5.c.
It is only necessary to perform the constructions necessary to reflect point A as
described in Problem 4 since the other vertices fall on line l. Join points A’, B,
C to complete the reflection of triangle ABC under line l.
and
5.d.
Repeat the constructions to reflect point A as described in Problem 4 to reflect
points B and C under line l. Note, points B’ and C’ are on one side of line l while
point A’ is on the other side of line l. Join points A’, B’, and C’ to complete the
reflection of triangle ABC under line l.
6.
Construct a circle centered at point E and passing through point F. Construct
another circle centered at point F and passing through point E. The intersection of
these two circles is a point F’. Since triangle EFF’ is an equilateral triangle point
F’ is point F rotated 60° about point E.
is
Construct a circle centered at point E and passing through point F. Construct a
line perpendicular to line segment EF and passing through point E. The
intersection of the circle and perpendicular line is a point F’. Since triangle EFF’
a right triangle point F’ is point F rotated 90° about point E.
Construct a dashed circle centered at point E and passing through point F.
Construct a thin circle centered at point F and passing through point E. The
intersection of
these two circles is a point G. Bisect angle GEF. Label the
intersection of the bisector and the dashed circle point F’. Point F’ is point F
rotated 30° about point E.
7.
Construct a dashed circle centered at point D passing through point A. Construct
another dashed circle centered at point A passing through point D. Label the
intersection of these two dashed circles point A’. Repeat this process for points B
and C. Join points A’, B’, and C’ to complete the 60° rotation of triangle ABC
about point D.
Construct a dashed circle centered at point D passing through point A. Construct
ray AD. Label the intersection of ray AD and the dashed circle point G.
Construct another dashed circle centered at point G passing through point D.
Label the intersection of these two dashed circles point A’’. Construct a thin
circle centered at point D passing through point B. Construct another thin circle
centered at point A’’ with radius AB. Label the intersection of these two thin
circles point B’’. Repeat this process for point C’’. Join points A’’, B’’, and C’’
to complete the 120° rotation of triangle ABC about point D.
8.
Mark the midpoint of segment EF point G. Mark the midpoint of segment EM
last
9.
point M. Construct a line, p, parallel to segment EF passing through point C.
Construct a circle centered at point C with radius EM. Label the intersection of
the last two constructions point E’. Construct a line, r, perpendicular to line p
passing through point E’. Reflect triangle ABC about line r. Construct a circle
centered at point C’ (the reflection of point C) passing through point E’. Label the
intersection of this circle with line p point E’’. Construct a line, r’, perpendicular
to line p passing through point E’’. Reflect triangle A’B’C’ about line r’. This
construction, triangle A’’B’’C’’, is the translation of triangle ABC in the direction
of ray EF with a magnitude EF.
Construct a line, m, passing through point O and a vertex of triangle ABC.
Construct a line, n, perpendicular to line m passing through point O. Bisect, with
line p, the angle formed by lines m and n. Reflect triangle ABC about line p.
Reflect this figure about line n. This last construction, triangle A’’B’’C’’, is the
rotation of triangle ABC 90° about point O.
Construct a line, q, passing through point O and a vertex of triangle ABC.
Construct a line, r, perpendicular to line q passing through point O. Reflect
triangle ABC about line r. Reflect this figure about line q. This last construction,
triangle A’’B’’C’’, is the rotation of triangle ABC 180° about point O.
10.
Horizontal Line Flip: Reflect the given figures about the dashed horizontal line.
Vertical Line Flip: Reflect the given figure about the dashed vertical line.
¼ Turn (90°): First, reflect the given figure about a line at 45° to the line between
the point and the original figure. Then, reflect the intermediate figure about a line
perpendicular to the line between the point and the original figure.
½ Turn (180°): First, reflect the given figure about a line perpendicular to the line
between the point and the original figure. Then, reflect the intermediate figure
about a line between the point and the original figure.
¾ Turn (270°): First, reflect the given figure about a line at 135° to the line
between the point and the original figure. Then, reflect the intermediate figure
about a line perpendicular to the line between the point and the original figure.
Slide: Move all aspects of the figure the same distance along the same line.
Vertical Glide: First, slide the figure a given distance along the vertical line.
Then, reflect the figure about the dashed vertical line.
Horizontal Glide: First, slide the figure a given distance along the horizontal line.
Then, reflect the figure about the dashed horizontal line.
11.a. Consider a shaded tile first. The base shape is a quadrilateral, specifically a
rectangle. The top has been modified and translated to the bottom. Both the left
and right sides are unique modifications. An adjacent white tile is created by
rotating a shaded tile 180° about a vertex of the base rectangle and translating
vertically. This couplet repeats horizontally within a row. Adjacent horizontal
rows in the tessellation are vertical translations with alternating colors.
11.b. The base shape is a quadrilateral, specifically a rectangle. A side has been
modified
and then translated to form the other side. The top modification is
reflected about
a vertical line, translated vertically, and translated horizontally to
create the
bottom. Within a horizontal row, this base tile repeats via horizontal
translation
with alternating colors. Adjacent horizontal rows in the tessellation are
created by
reflecting a tile about a vertical line and translating vertically.
11.c.
90°
to
Consider a white tile first. The base shape is a quadrilateral, specifically a square.
A given side has been modified. This modification has been rotated clockwise
to form an adjacent side. This newest side is rotated clockwise 90° and reflected
to create a third side parallel to the first. This third side is rotated clockwise 90°
create a fourth side parallel to the second. The four adjacent shaded tiles are
created by rotating a white tile clockwise and counterclockwise 90° about two
nonadjacent vertices of the base square. These same vertices are used to create
horizontal adjacent white tiles by rotating the original white tile 180°. Adjacent
horizontal rows in the tessellation are created by reflecting an entire row about a
horizontal line.
11.d. The base shape is a triangle, specifically an equilateral triangle. A side has been
modified to have a rectangular tab. This side is rotated 60° to form the adjacent
side. The remaining side is a unique modification. Each vertex is surrounded by
six tiles, three white and three shaded. The plane can be tessellated by rotating
tiles of the same color 120° about a given vertex.