# The Symbolic Language of Architecture and Public Monuments

## Mathematics of Ornaments and Architecture

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## SECTION II: DESIGNS IN GEOMETRY

### Line Designs

Many African designs were made by using simple lines with the combination of symmetry and translation. These can be reproduced using simple geometric tools; the ruler and straight edge. The following are simple African line designs:*(figure available in print form)*

*(figure available in print form)*

### Daisy Design

Some African designs show evidence of flowers in the execution of the design. These designs were found on the Ashanti bronze urn in Ghana.*(figure available in print form)*

- Step l. Construct a circle. Select a point on the circle. Place the compass point at any point on the circle.
- Step 2. Without changing the compass setting, make an arc with center at the selected point.
- Step 3. Make arcs from each of the two new points of intersections located.
- Step 4. Make arcs from each of the two points of intersection.
- Step 5. Make arcs connecting the last two points of intersection.
- Step 6. Cut out the spaces between. Color to decorate. These daisy designs can be found in some Egyptian ornaments. This design was used in combination with other designs.

### Mandelas or Circular Designs.

Mandelas are circular designs arranged in layers radiating from the center. The word comes from Hindu Sanskrit, the language of India. Mandelas were used by the Hindus in meditation. Evidence of these circular designs can also be found in African design. The circles seem to be drawn without the use of any instruments. The difference between these and the Hindu mandela is that in the Hindus’ mandela the center of the circles are decorated, whereas the Africans’ are void of designs. The following show examples of a Hindu circle design and an African circular design.*(figure available in print form)*

### Evidence of Tessellation in African Designs

Mathematicians call tiling patterns tessellation. A tessellation is an arrangement of figures that fill the plane but do not overlap or leave gaps or spaces. If the same figure is used throughout the tiling it is called pure tessellation. The following show tessellation with spherical shapes.Some shapes are better for tessellation than are others. These are triangles and the polygon. Many tessellation in African designs are shown using different objects.

The theory of tessellation is extended to the creation of wall paper and pattern designs. Coordinate systems called lattices can be developed using polygonal shapes. These are examples.

*(figure available in print form)*

With these lattices and a combination of operations manipulating the designs there 17 patterns that can be generated.

The standard repeating movements and operations to generate different patterns are (a) a translation alone, (with no rotation or reflection). (b) the rotation alone through an angle. (c) the reflection alone and (d) the combination of reflection and translation together. This is known as glide reflection.

These four combination types may be combined with one another on any of the lattices to produce complicated repetitive patterns. When the combination is done systematically, the result will be the total number of ways of reflecting a given motif in the plane. The lines over which the motif is translated or repeated are called reflection lines and glide lines.

Patterns can be blocked out in unit cells. A unit cell is that portion of a pattern which when repeated by the translations alone , develops the entire pattern. If a copy of the motif is developed the repeated movement of the motif at appropriate intervals in two directions will construct the pattern. The type of pattern will depend on the type of lattice used.

Other forms of tiling.

The first spiral tiling was discovered by Vonderberg (1936)

*(figure available in print form)*

These tilings and with the operation of translation develop many decorative spirals. If the spiral is cut in half and the other half translated affixed number of units it yields a tiling that is different from the original.