M.C. Escher was a prominent artist who used tessellations in his works. There are several books available showing the use of geometry in his sketches, lithographs, and woodcuts. We will use some of the same techniques used by Escher. The basis of these tessellations will be the regular polygons. The sides of these polygons will be changed, and the changes will be translated, rotated, or reflected to another side. The resulting shape will tessellate.
A translation is a movement is a specific direction, without turning or reflecting. Use a tessellation tracer to draw a square, make a change to one of its sides and use tracing paper to copy the modified square. Slide the original paper so that the change is translated to the opposite side and copy it, as shown below right. Use a new sheet of tracing paper to copy this shape until at least 12 shapes are traced. Shade or color the tessellation as shown below.
This same process can be applied to a square when two adjacent sides are changed. Start with a square and change two adjacent sides, trace the changes and translate these changes to the opposite sides as shown below. Form the tessellation by tracing this shape at least 12 times and shade or color alternating shapes.
This process may be applied to a parallelogram when two adjacent sides are changed. Start with a parallelogram and change two adjacent sides, trace the changes and translate these changes to the opposite sides as shown below. Form the tessellation by tracing this shape at least 16 times and shade or color alternating shapes. Adding internal features to the tessellating shape creates a more interesting tessellation.
This same process can be applied to a hexagon when three adjacent sides are changed. Start with a hexagon and change three adjacent sides, trace the changes and translate these changes to the opposite sides as shown below. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes. In this case, several internal features were added.
One side of a square can be changed and this change can be rotated 90° to an adjacent side. This same change can be rotated 180° and 270° so that it appears on all sides. Start with a square and change one of the sides, trace the change and rotate the change to the remaining sides as shown below. Each rotation requires that the center of rotation be the vertex between the source side and the destination side. Form the tessellation by tracing this shape at least 16 times and shade or color alternating shapes.
One side of a 60° rhombus can be changed and this change can be rotated 120° and 60° to the remaining sides. Start with a 60° rhombus and change one of the sides, trace the change and rotate the change to the remaining sides as shown below. Each rotation requires that the center of rotation be the vertex between the source side and the destination side. Form the tessellation by tracing this shape at least 15 times and shade or color alternating shapes.
One side of a hexagon can be changed and this change can be rotated to the remaining sides. Start with a hexagon and change one of the sides, trace the change and rotate the change to the remaining sides as shown below. Each rotation requires that the center of rotation be the vertex between the source side and the destination side. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes.
All sides of a hexagon can be changed and these changes can be rotated to adjacent sides. Start with a hexagon and change three alternating sides, trace the changes and rotate each change to an adjacent side as shown below. Each rotation requires that the center of rotation be the vertex between the source side and the destination side. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes.
The half-sides of a triangle can be rotated to the remaining half-side. Start with a triangle and change half of one side, rotate that change to the other half of that side, and rotate the entire side to all other sides. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes.
All of the half-sides of a triangle can be rotated to each remaining half-side. Start with a triangle and change half of each side, and rotate that change to the other half of that side. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes.
The half-sides of a quadrilateral can be rotated to the remaining half-side. Start with a quadrilateral and change half of each side, and rotate each change to the other half of that side. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes.
It is possible to form a tessellation by changing a half side of a triangle, and then rotating and reflecting the change to the other half side, followed by rotating and reflecting a second side to the third side. Start with an equilateral triangle and change half of one side. Rotate and reflect the changed half side to the remaining half of that side. Change an entire second side and rotate and reflect that change to the third side. Form the tessellation by tracing this shape at least 18 times and shade or color alternating shapes.
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