How to Use Various Drawing Tools to Test or Draw Figures With Translational Symmetry

Tile uses translational symmetry to create a cohesive surface pattern.

Translational symmetry is the repetition of an object across a plane. The translational symmetry can be composed of moves or translations and rotations. Mathematicians use translational symmetry to define two-dimensional surfaces, and architects use translational symmetry to define forms for wallpaper, masonry, tile work, domes and repetitive structures. Although modern mathematicians and architects use computers to draw and analyze figures with translational symmetry, the disciplines have traditionally used a ruled straightedge and a compass for this purpose. In either case, the method is the same, requiring a description of direction and distance to determine axes of rotation and surface tiling.

Things You'll Need

• Ruled straightedge
• Compass
• Optional computer aided design (CAD) application

Instructions

1. Draw an element you would like to analyze or draw with translational symmetry. Draw points at the critical areas or corners that define the shape. Choose one point that is unique on the shape and determine the location of a similar point on a congruent shape somewhere on the surface plane.

2. Draw a line to connect the unique point with the similar point. If the points and the shapes are in the same orientation, the line represents the direction and magnitude of the translation, or move. All points of the two shapes can be mapped using the angle and distance of the line. If the two shapes are not in the same orientation, go to Step 3.

3. Draw a second line from another unique point on the first shape to the similar point on the congruent shape.

   If the two lines intersect within or between the two shapes, then they have translational symmetry by two-dimensional rotation. If this is the case, go to Step 4.

   If the two lines are parallel, but the shape is flipped, the two shapes have translational symmetry by three-dimensional rotation. If this is the case go to Step 5.

4. Draw a circle with the first unique point as the center and the distance to the congruent shape's similar point as the radius; repeat the circle with the center at similar point and the radius at the first unique point. Draw a line between the two points of intersection of the two circles.

   Draw another circle with the second unique point as the center and the congruent shape's second similar point as the radius. Draw yet another circle with the congruent shape's second similar point as the center and the original shape's second unique point as the radius. Draw a line between the two points of intersection of the two circles.

   The intersection of the two lines drawn from the circle intersections is the center of two-dimensional rotation for the translational symmetry.

5. Bisect the two parallel lines connecting the flipped shapes. Draw a circle with the center at the first shape's unique point and the radius to the congruent shape's similar point. Then, draw a circle with the center at the congruent shape's similar point and the radius to the first shape's unique point. Draw a line connecting the two intersections of the circle. The resulting bisecting line is the axis of three-dimensional rotation for the translational symmetry. Three-dimensional rotation results in flipping the object, creating bilateral symmetry.

References

• "Plane and Solid Geometry"; J.M. Aarts; 2008.