## Math in Engineering## Mathematics and Architecture |

## How are the principles of ratio, proportion, and symmetry applied to architecture? |

The definition of a *ratio* is a comparison by division of two quantities expressed as the same unit measurement. For example, a building that is 200 feet wide and 100 feet tall has a ratio of 2:1 (200:100) between its width and height; it is also seen as the fraction ½. Such a relationship was understood as far back as ancient Greece and Rome, when people used mathematics to give structure and aesthetics to buildings. This is especially important in architecture, in which building design is based on complex mathematical ratios.

*Proportion* is an equation stating that two ratios are equal. Every proportion has four terms, with the first and fourth terms being the extremes; the second and third terms are called the means. In each proportion, the product of the means equals the product of the extremes. The Greeks and Romans often used proportions in their buildings and other structural designs. (The Roman architect Vitruvius was also instrumental in praising the virtues of proportion and symmetry in architecture; for more about Vitruvius, see above.) During the Renaissance, architects applied proportion (and other mathematical formulas) to produce aesthetically pleasing buildings— beauty that still holds true today.

Although there are other types of *symmetry,* the most common is line symmetry, in which a line divides an object, line, or other structure into two equal halves (an example in nature would be the wings of a butterfly). If a line of symmetry is drawn, each point on one side of the line has a corresponding point on the opposite side of the line. If you connect these two points, the line is perpendicular to the line of symmetry. There is a more mathematical way of defining symmetry: Two points are symmetric about a line if the line is the perpendicular bisector of the segment joining the two points. Symmetry was used by ancient (and modern) architects to maintain visual and sometimes structural balance of a building or structure.