# Elements of Architecture

## Geometric Systems in Architecture

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## Mathematics in Architecture

This unit attempts to address the students inability to see the world specially by l)providing students with activities that will help them to see objects specially 2)make students aware of how space dictates the forms around us and 3)point out some distinct patterns that appear in nature and architecture.

### Unit Work

If you hand a student 18 cubic units and instruct him to build a rectangular prism using the 18 units, the student will create one of four prisms. (see figure 1) Each prism has the same number of cubic units therefore each prism has the same volume even though the shapes of the prisms are different. Students can easily recognize that the volume*of*each prism is determined by the product of the linear measure of length, width and height. Students can discover all possible combinations of side measurements given a volume by writing the factors of the volume number and determining which three factors can be used to construct a prism with this volume.

(figure available in print form)

Through classroom experience we discover many middle school students have difficulty applying a formula to a given set of information and very few students will be able to explain why this formula is true. By constructing prisms with a given number of cubic units, students can derive the formula and therefore understand why these formulas work. Using the tactile senses of the students will help us reach these students who do not learn well visually or audibly. This simple exercise will help students begin to see volume as a three dimensional measurement.

We may also help students take another step into understanding the space around them through use of this simple exercise. Give each student 3 toothpicks and instruct each student to construct four triangles. Most students will fumble with those toothpicks and build only one triangle in a single plane. How many students will step out of the plane of the desk and build a tetrahedron with the toothpicks? I would venture to guess that those students who are able to describe skew lines or have mastered the Rubick’s Cube may quickly complete this exercise. The key to this activity is to step out of the 2-dimensional plane in which we compute our arithmetic and draw our geometrical figures and look at our world specially. (see figure 2)

Hold the 3 sticks at an apex to form the tetrahedron. Consider the table top to contain the fourth triangle.

(figure available in print form)

We can use this simple exercise to make students aware of the many avenues they may explore to solve problems. Children love to manipulate their world, to build and create. We can apply this natural ability and tendency to manipulate to the geometry class by providing exercises that will make sense of the 3-dimensional figures we use in finding volume.

If we secure the apex of the construction in figure 2 and provide more toothpicks to build a full tetrahedron, (a small bit of modeling clay at each apex will do the trick) students can feel the strength of this collection of triangles. Man has determined that the triangle is the most stable system. Triangular braces are used to strengthen bridges, buildings, shelves, and a host of other constructions. Many students may argue that a square base or cubic system is more secure. An interesting exercise is to give half the students toothpicks and clay with instructions to build a 3-dimensional system using squares (a cubic system) while the remaining students construct a 3-dimensional system with triangles (a tetrahedral system). The system constructed with triangles will be stronger and more stable than the cubic system. (see figure 3)

Cubic System | Tetrahedral System |

Figure 3

### Pattern Generation

We have begun to show students how to manipulate space to construct specific figures. It is important for the students to recognize that many of the geometric figures exist in nature and in the world around them. Let us first instruct students to manipulate squares. A very simple pattern develops. (see figure 4). Three dimensional cubes may also be created when squares are joined together. (figure available in print form)Figure 4

More interesting are the patterns and forms that are generated when equilateral triangles are arranged. Two patterns emerge, a pattern in which the triangles are adjacent to one another to form, more or less, a line, and a pattern emerges when six triangles are arranged around a center point, a hexagon appears. (figure 5)

(figure available in print form)

repeating triangles | hexagon |

These are clearly 2-dimensional patterns that are identifiable in many tile patterns seen in kitchen floors, tile floors, stain glass windows and in ornamentation on buildings.

If we remove one triangle and arrange those 5 triangles around a central point, the only possible construct is a 3-dimensional cup shape. This in the only pattern that can be constructed. The figure will not lie in an 2-dimensional plane. (see figure 6) Seven triangles centered around the center point would result in a curved or saddled shape. (Lesson 4) No other shapes are available when the triangles are adjoined. The three shapes, the 2-dimensional linear pattern of hexagons, the cupped shape of pentagons, and the undulating saddle with 7 triangles, are the only possible constructions.

(figure available in print form)

5 triangles (pentagon) to form a cup shape

Figure 6

Students can manipulate these triangles to discover whether any other patterns can emerge. All other patterns can be reduced to these cubic, hexagonal, or pentagonal shapes. It is also interesting to note that the same geometric shapes or systems can be derived by looking at other collections of shapes. Take, for example, a set of circles placed in a repeating pattern as in figure 7. If we connect the centers, our square pattern will appear.

(figure available in print form)

square system | hexagonal system |

Figure 7 | Figure 8 |

(figure available in print form)

tetrahedral 4 point system

(figure available in print form)

Figure 10

It is amazing to see how the same systems and patterns repeat themselves in nature. Students should play and build models to help them understand how these systems appear by themselves, not by some unnatural application of mathematics or science.

### Natural Patterns

These same patterns can be found in nature. For the young scientist, it is known that all living things have a carbon base and the carbon atom itself allows 4 opportunities to bond with other elements. The methane molecule described by a carbon atom at the center and 4 hydrogen atoms can take on the appearance of the tetrahedron with the carbon atom at the center. (see figure 11)(figure available in print form)

methane molecule | > as a tetrahedron |

Since all living things are carbon based, all other carbon compounds can be described with tetrahedrons.

The crystal structure of minerals found outdoors or on display in museums also exhibit the same kinds of patterns and systems. These crystals reflects the growth of the mineral based on the atomic structure of the mineral. We have reviewed the basic cubic, tetrahedral, and hexagonal patterns. The structure of crystal are described by these same terms. This is not to say that a
*
perfect
*
crystal structure can always be found, but traces of the systems can be identified if we have a good specimen. We can study a specimen of pyrite, or fool’s gold, and see that the crystals are cubic. Halite, or rock salt, also exhibits cubic crystals.

Quartz, a mineral common to our area, can be seen in the form of hexagonal crystals while fluorite, a mineral that possesses the property of glowing in ultra-violet exhibits distinct tetrahedral crystals. These crystal formations are indications of the growth patterns of the minerals. In general tetragonal crystals are often long and slender or needle-like. They are characteristically four sided prisms or pyramids. Hexagonal crystals are generally column or prism-like with hexagonal cross sections, while the cubic system exhibits crystals that are very blocky or ball-like in appearance with many similar, symmetrical faces. Again, these same geometrical patterns appear in all of nature.

Students can walk about the city streets and find many examples of repeating patterns. Ornamentation on windows can take the form of square patterns. Tiling on floors generally take the form of hexagonal patterns or some other combination of squares and hexagons. The same patterns persist. The same patterns are dictated by space to appear in nature, in art, in mathematics and in architecture. There is a sense of cohesiveness, of some master plan, an order to our universe. It is evident in the world around us and is manifested in architecture. It will not take a great deal of effort to open the eyes of our students to the knowledge available to them. Their natural curiosity will take over once they’ve become aware of the 3-dimensional world around them.