Mathematics and Architecture Designs

Winnifred Morgan

It is well known that virtually all disciplines e.g.. Music, and Language, Mathematics or Science are introduced to children at a very early age as distinct, isolated subjects that have no connection (inter-relationship) among them. Consequently, although children may have a ready grasp of architectural ideas before fourth grade, they cannot see the relationship between the elements of design and symbols and the Mathematics of the design. From as early as first grade, steeples, columns, special window shapes, colors, and figurative ornaments have being singled out and illustrated by students. Such drawings litter the walls of classrooms; and people, stars, trees, animals, houses, and buildings are symbolically sized, simplified, and composed together to tell a story. Here the "golden" opportunity for integrating architectural concepts with Science and Mathematics is ignored, leaving students at a disadvantage.

The introduction phase of Architectural understanding is excluded from most, if not all modern educational programs at both the elementary and high school levels, despite the overwhelming presence of buildings in the everyday life, of cities and suburbs and despite the unique role of architecture in orienting people to personal and public space and history. Indeed, Architectural orientation is also excluded from college and university curriculum except for those students who choose to pursue a career in the discipline. And yet we spend most of our lives in and around buildings, knowing very little about them or how we are influenced by them. We make daily decisions related to architecture such as selecting color for a room, re-arranging the furniture, plan gardens or even buy a house. How much easier these decision making processes would have been if we were exposed to some basic fundamental architectural knowledge.

Symbols remain the visual principle in architectural design, serving as the key elements in the design of memorials. Their presence is indispensable to our understanding of the history of communities. But the Mathematics of such symbols must not be ignored for without mathematics we would have many lopsided buildings, and poorly articulated designs and patterns.

Because Mathematics and Architecture have traditionally been treated by the curriculum as an either/or polarity, students are denied the opportunity to see the unique inter-relationship between the two disciplines. If the gap is to be bridged between ideas from Architecture and Mathematics, the teacher must pose tasks that will engage the students' intellect and foster a better understanding of the concepts and procedures of both disciplines that will stimulate them to make the necessary connection between the disciplines.

This unit seeks to bridge the existing deficiency of the curriculum and aim at highlighting the importance of pure Geometry that is utilized in Architectural plans, designs and facades. Briefly stated the general objectives of this unit are:

1. To provide teachers with hands-on-activities to experiment with their Math class,
2. To establish the link between topics in Mathematics and Architecture,
3. To develop students awareness of the interdependence of the mathematical concept of ratio, proportion and geometry in Architectural designs and constructions.
4. To develop students ability to apply mathematical knowledge to problem solve architectural procedures, plans and pattern forming processes.
5. To facilitate and encourage a greater appreciation for Architecture by students, teachers and parents.
This unit is being organized to be presented to tenth and eleventh grade students who have been exposed to Algebra and Geometry. They meet for one forty-five minutes period per day, five days per week. The unit hopes to utilize between ten to fifteen classes depending on the ability grouping of the students.

Architecture clearly articulates the principles of harmony, order and similarity. Therefore, the activities will incorporate mathematical principles such as symmetry, similarity, Pythagorean Theorem, congruence, ratio, proportion and measurement. These basic elements of geometry reinforces the concepts of order and continuity. To design or build a structure we must use simple geometry. To create order or patterns we much use symmetry. Symmetry is defined in Math as a rotation for translation of a plane figure which leaves the figure unchanged but alters its position. Architecture defines the concept as the interchange of two elements A and B. The four basic forms of symmetry used in architecture to reinforce the concepts of grouping, order and patterning are:

1. Translation: defined as the parallel movement of a plane-figure from one position to another.
(figure available in print form)
2. Rotation: defined as the movement of a plane figure or object around an axis.
(figure available in print form)
3. Reflection: defined as the bending or folding back of an object upon itself.
(figure available in print form)
4. Glide reflection which is a combination of translation and reflection.
(figure available in print form)
These principles are demonstrated in various ways in architectural designs and buildings. Below are a few examples of the ideas of symmetry that are found in buildings and pattern designs:
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
The use of similarity and repetition (congruent elements) are also utilized to create a sense of coherence in buildings. Similarity of details evident in the dimensions of the walls, corners, cornices, windows and doors counterbalances individual differences of size and proportions of houses.
(figure available in print form)
(figure available in print form)
The concept ratio is used in Mathematics to describe a comparative relationship between two quantities. Proportion is used to equate ratios. In architecture these concepts are used in designs and buildings to create balance, patterns and beauty. There are in buildings an amazing ratios of equivalent dimensions such as window ratio to door or the ratio of columns to the wall space between the columns, etc. Architects utilized proportion to express rationally and beauty. Pythagoras used numbers to demonstrate basic order in space. He believed that all things could be denominated as a whole integer and was therefore beautiful by definition. He used the magic numbers 1, 2, 3, 4 to explain order in space: point, line, plane, solid
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
He utilized the system of proportion which is commensurate with sizes to develop his Tetra chord or "Perfect Golden Triangle." The principle combines ratios into larger units which in turn forms clear ratios. This allows architects to choose any preferential relationships they desire. Usually the ratios used are 1:1, 2:3, or 4:5 since they are more apparent to the eye than a ratio of say 7:8.
(figure available in print form)
(figure available in print form)
(figure available in print form)
If proportions are not observed perfectly, the visual image of architecture will be defective. Proportion by itself does not create beauty in architecture. The constitutive principles of architecture derive from regularity, symmetry, and a combination of all the other elements.

One of the basic concepts in mathematics that students utilize everyday is that of measurement. The concept is used in architecture to denote scale. Size and relationships of size are important qualities that are most likely to be distorted in designing a structure or a pattern or making a plan. The chances of such errors are greatly minimized when there is a scale to relate the size of the sketch or plan to the actual. For example, the house floor plan below makes no sense unless it is accompanied with a scale. Measurement therefore become a vital link to good architectural designing.

(figure available in print form)
Constructing a model of a public building would be great, but that would require a longer time span and mechanical skill than is available for the duration of this unit, so we will concentrate our efforts on analyzing the proportions of selected buildings, including the Center Church on the New Haven Green and other public buildings and if time permits build motifs for a facade for a public building.

"Architecture is at once a Science and an Art. As a Science it requires knowledge; as an ART is demands talent. To learn a Science requires listening, understanding and assimilating." 1 Accordingly, the purpose of Architecture is not just for pleasure or decoration for public and private utility but the happiness and preservation of individuals and society. The study of Architecture will not only stimulate historical investigations but will furnish valuable data on climate, structure of places, nature of materials, growth of technology and industrialization, and civilization of different peoples and epochs.

"Building a shelter against sun, wind, or rain is a fundamental human need, but it goes beyond functional necessity." 2 It is a creative partnership with the natural world through which man develops and utilizes his intellect and skills. Architecture, like history, is a process in which many circumstances combine to produce certain results. The availability of material, the existing technology and the particular cultural, economic and political circumstances, influence the Architectural "forms" evident in Western Architecture. Material used include glass, marble, timber, ceramics, a variety of stone including limestone, sandstone, granite, flint and brick, concrete, tiles, and mosaic.

The Architectural achievement of the Greeks has brought a distinct and exquisite refinement of detail and proportion to the existing structural system. Briefly stated, Classical Architecture is a combination of the temple architecture of the Greek and the religious, military and civil architecture of the Romans. The decorative elements of a classical building derive directly, and indirectly from the Greeks and the aim is to achieve a demonstrable harmony of parts by the use of proportions. Classical Architecture is built on a traditional system of the five orders, the Doric, Tuscan, Ionic, Corinthian and Composite orders.

(figure available in print form)
The development of Architecture reached the point where great buildings required the services a Specialist designer. The sheer power of Gothic Architecture indicates the product of an increasingly secular society, the mathematical and building knowledge and the talents and skills of a master-mason and a team of highly skilled specialist craftsmen. Gothic buildings represent a transition point in history between the church-dominated early Middle Ages and the period referred to as the Renaissance.

The basic elements of Architecture are the elements of construction and the elements of space. The elements of construction include brick or stone, columns (post), beams (entablature), walls, roof, platform (foundation), rafters, joist and pedestal. Elements of space include windows, doorways, pergola (covered walkway), aedicule, vents, intercolumnation, courtyard, fire court, rooms (dinning room, bedroom etc.), cortile (enclosed courtyard inside building) atrium (room which functions as a fire protector between two buildings) hallways, and pediment.

(figure available in print form)
(figure available in print form)
The design vocabulary include the following: plan, elevation, section, solid void, figure, ground, proportion (the ratio of parts to the whole), scale (the ratio to an external unit measure), and grids. Since we will be analyzing the facade of the New Haven Center Church, which is purely Gothic on the interior. I will just comment briefly on the style. Gothic architecture is considered a phase of classical architecture. It had tentative beginnings in the 18th century and was firmly established in 1834 in England. It became accepted as an appropriate style for the country houses of the wealthy, romantic and eccentric. The style was not particularly popular in the Unites States, where there was a strong Classical tradition. However, there are many such buildings in New Haven and Center Church on the Green is unique in the use of that style for its ornaments.

The discipline inherent in the proportions and patterns of a building highlights the relatedness of parts. The Pythagorean triangle is frequently found in the building pattern and is used to approximate the relations of the heights of the steeples to the other parts of the building. By completing the following activities, students will not only be taught Math but will be forced to consider architectural designs and pattern making.


Content retrieved from:
http://teachersinstitute.yale.edu/curriculum/units/1993/1/93.01.07.x.html


Comments (0)

Be the first person to comment

When you are finished viewing curriculum units on this Web site, please take a few minutes to provide feedback and help us understand how these units, which were created by public school teachers, are useful to others.
THANK YOU — your feedback is very important to us! Give Feedback