Scale Drawings and Models

Scale drawings are based on the geometric principle of similarity. Two figures are similar if they have the same shape even though they may have different sizes. Any figure is similar to itself, so, in this specific case, the similar figures do actually have the same size as well as the same shape.

Scale Drawing and Models in Geometry

In traditional Euclidean geometry, two polygons are similar if their corresponding angles are equal and their corresponding sides are in proportion. For example, the two right triangles shown below are similar, because the length of each side in the large triangle is twice the length of the corresponding side in the small triangle.

In transformational geometry, two figures are said to be similar if one of them can be "mapped" onto the other one by a transformation that expands or contracts all dimensions by the same factor. Such a transformation is usually called a dilation, a size change, or a size transformation. For example, in the preceding illustration, the small triangle can be mapped onto the large triangle by a dilation with scale factor 2. The ratio of the length of a side in the large triangle to the corresponding side in the small triangle is 2-to-1, often written as 2:1.

Another example is that a square with sides of length 2 centimeters (cm) is similar to a square with sides of length 10 cm, since the 2 cm square can be mapped onto the 10 cm square by a dilation with magnitude (scale factor) 5. This simply means that each dimension of the 2 cm square is multiplied by 5 to produce the 10 cm square. We call 5 the scale factor or magnitude of the dilation and we say that the ratio of the dimensions of the larger square to those of the smaller square is 5:1 (or "5-to-1"). The smaller square has been "scaled up" by a factor of 5.

A figure can also be "scaled down".* Starting with a square with sides of length 10 cm, we can produce a square whose sides have length 2 cm by scaling the larger square by a factor of ⅕. Note that all squares are similar because any given square can be mapped onto any other square by a dilation of magnitude b /a where b /a is the ratio of the sides of the second square to that of the first. This is not true for polygons in general. For example, any two triangles whose corresponding angles are unequal cannot be similar.

*This process of scaling up or down to produce similar figures with different sizes has numerous applications in many fields, such as in copying machines where one enlarges or reduces the original.

Maps

One familiar type of scale drawing is a map. When someone plans a trip, it is convenient to have a map of the general region in which that trip will take place. Obviously, the map will not be the actual size, but, if it is to be helpful, it should look similar to the region, but on a much smaller scale. Any map that is designed to help people plan out a journey will have a "legend" printed on it showing, for example, how many miles or kilometers are represented by each inch or centimeter. It would be terribly confusing to the traveler to have the scale vary on different parts of the same map, so the scale is uniform for the entire map, which returns to the mathematical concept of similarity. The ratio of each distance in the real world to its corresponding distance on the map will be the same for all corresponding distances between the two.

It is no coincidence that mathematicians have adopted the word "mapping" to refer to the correspondence between the points in one set and the points of another set. The dilation referred to in the earlier paragraph is often called a "similarity mapping."

Models

Almost everyone is familiar with toys that are scale models of cars, airplanes, boats, and space shuttles, as well as many other familiar objects. Here again, similarity is applied in the manufacturing of these scale models. A popular series of scale model cars has the notation "1:18" on the box. This is a ratio that tells how the model compares in size to the real car. In this case, all the dimensions of the model car are the size of the corresponding dimensions of the real car. So the model is as long, as wide, and as high as the real car. If the real car is 180 inches long, then the model car will be as long, as long or 10 inches in length. Because all the dimensions of the model car are scaled from the real car by the same factor, the model looks realistic.

This same similarity principle is used in reverse by the people who design automobiles. They usually start with a sketch, drawn more or less to scale, of the car they want to design; translate this sketch into a true scale drawing, often with the help of computer-assisted design software; and build a scale model to the specifications in the drawing. Then, if everything is approved by the executives of the automobile company, the scale drawings and models are given to automotive engineers who design and build a full-size prototype of the vehicle. If the prototype also receives approval from senior executives, then the vehicle is put into full-scale production.

Other Uses

Scale drawings and models are also very important for architects, builders, carpenters, plumbers, and electricians. Architects work in a similar manner as automotive designers. They start with sketches of ideas showing what they want their buildings to look like, then translate the sketches into accurate scale drawings, and finally build scale models of the buildings for clients to evaluate before any construction on the actual structure is begun. If the client approves the design, then detailed scale drawings, called blueprints, are prepared for the whole building so that the builders will know where to put walls, doors, restrooms, stairwells, elevators, electrical wiring, and all other features of the building.

On each blueprint is a legend similar to that found on a map that tells how many feet are represented by each inch on the blueprint. In addition, a blueprint tells the scale of the drawing to the actual building. All of the people involved in the construction of the building have access to these blueprints so that every feature of the building ends up being the right size and in the right place.

Artists such as painters and sculptors often use scale drawings or scale models as preliminary guides for their work. An artist commissioned to paint a large mural often makes preliminary small sketches of the drawings she plans to put on the wall. These are essentially scale drawings of the final work.

Similarly, a sculptor usually makes a small scale model, perhaps of clay or plaster, before beginning the actual sculpture, which might be in marble, granite, or brass. Mount Rushmore, one of the great national monuments in the United States, is a gigantic sculpture featuring the heads of Presidents George Washington, Thomas Jefferson, Abraham Lincoln, and Theodore Roosevelt carved into the natural granite of the Black Hills of

South Dakota. The sculptor of this imposing work, Gutzon Borglum, first created plaster scale models of the heads in his studio before ever going to the mountain. His models had a simple 1:12 scaling ratio; one inch on the model would represent one foot on the mountain. He made very careful measurements on the models, multiplied these measurements by 12, and used these latter values to guide his work in sculpting the mountain. When Borglum's work was finished, each of the faces of the four presidents was more than 60 feet high from the chin to the top of the head.

From arts and crafts to large-scale manufacturing processes, the use of similarity in scale drawings and models is fundamental to creating products that are both useful and pleasing to the eye.

see also Computer-Aided Design; Mount Rushmore, Measurement of; Photocopier; Ratio, Rate and Proportion; Transformations.

Stephen Robinson

Bibliography

Chakerian, G. D., Calvin D. Crabill, and Sherman K. Stein. Geometry: A Guided Inquiry. Pleasantville, NY: Sunburst Communications, Inc. 1987.

Yaglom, I. M. Geometric Transformations. Washington, D.C.: Mathematical Association of America, 1975.

Internet Resources

"Creating Giants in the Black Hills." Travelsd.com. <http://www.travelsd.com/parks/rushmore/creation.htm>.

Scale Drawings. New York State Regents Exam Prep Center. <http://regentsprep.org/Regents/math/scale/Lscale.htm>.