# Mathematics And Architecture 1700

′S Essay, Research Paper

A look at scientific/mathematical developments of the 1600’s and 1700’s helps place the Central European Baroque church in context: Desargues, Bernoulli, Leibniz and Scieferenna, all mathematician or architect, all dealt with theories of synthesis and convergence. The effect of the new mathematical ideas were on architecture was a gradual transformation of space from pure, static and isolated to composite, dynamic and interpenetrating.

Architects used geometrical methods as plan generators. Transformational operations were of utmost importance, including area, rotation, reflection, translation, and coordinate transformation. Although we have probably one of the greatest architectural minds in Michelangelo contributing work during these times, we’re going to look deeper into the roots of the architectural and mathematical advances during this era.

Let’s begin by looking at the work of the Jacob and Daniel Bernoulli, mathematicians during the times of mid 1600’s to late 1700’s.

Jacob (Jacques) Bernoulli

Born: 27 Dec 1654 in Basel, Switzerland

Died: 16 Aug 1705 in Basel, Switzerland

Jacob Bernoulli was one of the sons of Johann (II) Bernoulli. Following the family tradition he took a degree in law but his interests were in mathematics and mathematical physics. Jacob Bernoulli was the first to use the term integral. He studied the catenary, the curve of a suspended string. He was an early user of polar coordinates and discovered the isochrone.

Daniel Bernoulli

Born: 8 Feb 1700 in Groningen, Netherlands

Died: 17 March 1782 in Basel, Switzerland

Daniel Bernoulli was a Swiss scientist who was born in the year 1700. He discovered that fast moving air exerts less pressure than slow moving air. Daniel Bernoulli’s most important work considered the basic properties of fluid flow, pressure, density and velocity, and gave their fundamental relationship now known as Bernoulli’s principle. He also established the basis for the kinetic theory of gases. His most important work was Hydrodynamica, which considered the basic properties of fluid flow, pressure, density and velocity, and gave their fundamental relationship now known as Bernoulli’s principle. In this book he also gave a theoretical explanation of the pressure of a gas on the walls of a container. Bernoulli’s principle can be seen most easily through the use of a venturi tube (see Figure below). The venturi will be discussed again in the unit on propulsion systems, since a venturi is an extremely important part of a carburetor. A venturi tube is simply a tube which is narrower in the middle than it is at the ends. When the fluid passing through the tube reaches the narrow part, it speeds up. According to Bernoulli’s principle, it then should exert less pressure. Let’s see how this works.

These two mathematicians greatly influenced the era of building during their times. If not only in the elements of geometrical proportions and scale, but also with the human endeavor aspect of moving forth as a “technological” advancement. Jacob had discovered the isochronous curve, which when illustrated, shows the derivation of a complex mathematical curve from a simple hyperbola diagram. The experimentation that Jacob underwent during these periods pushed further the ideas and geometrical associations found in areas like the Piazza of St. Peter’s, in Italy (1656 1667) and el Piazza del Campidoglio, Rome, Italy (1650 to 1657)- done by Michelangelo and Bernini. The direct relationship between these mathematicians and the actual buildings are rare, yet the influence at the time is a rich co-dependency of each subject. A closer look at certain examples will lead us to different examples of how these geometries and influential advancements associate themselves with their buildings.

The Louvre

Architect: Pierre Lescot, Claude Perrault

Location: Paris, France

Built 1546 to 1878

Building Type: palace, art museum

Context: urban, river front

The Louvre is one of many buildings during this era

that demonstrated the geometrical proportions that

were being displayed through mathematical endeavor

The Karlskirche, begun in 1715, in Vienna, is dedicated

to Carlo Borromeo, the Italian cardinal and saint of the

Counter Reformation. What is most extraordinary about

this structure is the successful coherence of its design

despite a seemingly irreconcilable eclecticism. In front

of a longitudinally placed oval nave stands an unusually

wide facade composed of a bizarre combination of

elements. A Corinthian hexastyle temple portico on top

of a stepped podium, archaeological in its fidelity to

Roman temple fronts, represents the entrance to a vast

temple with the grandeur and power as that of Olympia

itself. In construction, the building possesses the many

distributing structural elements from a diagram that

seemed to have been derived from a geometrical diagram.

A look at the fa ade will show the pedimented main

accent and serene confidence of order, but will fail

to show the untraditional flat roof that the architects

chose to implement. The mathematical implementations

that the architect involve in this otherwise simple building

can be seen in the proportional scale- perfectly measured

so that the observer is emphasized this “overwhelming”

sense of massiveness.

S. Carlo Alle Quattro Fontane

Architect: Frencesco Borromini

Location: Rome, Italy

Date: 1638 to 1641

Building Type: church

Context: urban

Style: Baroque

Let’s look at a building more detailed and driven towards

a more complex subject. If we look at Bernoulli’s

Isochronous curve, we see motion and fluidity yet a highly

structured environment. Here we have Borromini’s

S. Carlo alle Quattro, completed in 1682, after Borromini’s

death. At this time, while at the age of his early

30’s, Bernoulli was investigating the mathematical figures

of his experimental curve. In Borromini’s S. Carlo, the interior conformed to the growing penchant in the seventeenth

century’s distinct design methods. Although he was well within the circle of the more noted architects during this time, he was very much an outcast as his work outshined others for its flair and originality. Borromini, however had a large reverence towards

the geometrical orders and he followed them well- yet consistently contributing in the architectural sense as well as the detailed

artistic sense. Despite the difficulty with his general approval, Borromini chose to let the interior bow to the trend of the 17th

The early 1600’s were filled with forms and dialectic of classicism as constituted by the legacy of antiquity and its modern enrichment in the two centuries since Brunelleschi. These forms followed a potent language derived from the use of a broadcast of engineered advancements. The social and political nature of these “messages” were also broadcast to the public, showing the architecture as articulate objects- each owned and created by a distinct architect. This led to the Roman Baroque period. Before our case study mathematicians were even born, their paths were destined to impact the architectural world in a way so as to say that through intellectual observation, buildings could harmoniously envelop a concept while still obeying the “classical” orders. As seen above, dynamism is a word well associated with the Roman Baroque .energetic formation and the fall towards an oval or elliptical shape. Columns and pilasters paid reverence to the scale and the event of sequences that placed them in the overall plan.

The next period after the European Architecture of 1600-1750 was a bold attempt to refine the new world of architecture- a new order. During the times of the 1800’s historians associated themselves with the ways that the economical and technological forces affected building production- especially that of the mathematical advances.

These advances led to the ability to use new materials and new techniques in construction. Even after our case studies have long passed away, the same path that was paved before their creation is the same path being led towards newer and younger mathematicians to continue the progress. This is clearly reflected in the works of many buildings during this era. During this time metals were used more as well as cast -iron for elements such as columns.

Monticello

Architect: Thomas Jefferson

Location: Virginia, United States

Date: Varied- look below

in our very own United States during these times considering

a new order of architecture that was greatly influenced by the

progress in mathematics.

Thomas Jefferson spent most of his adult life designing

and redesigning Monticello, which was constructed over

a period of forty years. The self-taught architect designed

Monticello after ancient and Renaissance models, and in

particular after the work of Italian architect Andrea Palladio.

Intentionally it was a far cry from the other American homes

of its day. Jefferson felt that the architecture of {this land}

had been “leapt over and under-designed”.

Monticello was largely completed in 1782; the first floor of

the house featured a bedroom, parlor, drawing room, and

dining room. In 1796, walls of the original home

were knocked down to make room for an expansion that would

essentially double the floor plan of the house. The second

Monticello was largely completed in 1809, the year Jefferson

retired from the Presidency.

Perhaps Monticello is best known, however, for the way

in which Jefferson designed his house “with a greater eye to

convenience.” Its geometrical figures are outstanding measures

of an architects fascinations within order. Just as Jefferson’s

closet once featured an innovative “turning machine” which

held his clothes, other rooms in the house contain similar devices.

The dining room, located directly over the wine cellar, contains

two dumbwaiters which carry wine up from the basement.

The Parlor has a set of “magic” doors, so that as one door opens

or closes, the other follows automatically. And the Cabinet, in

which Jefferson wrote letters, performed scientific experiments,

and designed buildings, was filled with state-of-the-art devices,

including a copying machine, to help him in his work. The technological

advances found in Jefferson’s Monticello are high signs of the movement

of engineering and its influence of mathematics. The final product is a unique amalgam of beauty and function that combines the best elements of the technological world and old worlds with a fresh American perspective.

Piazza di Spagna, or the Spanish Steps

Architect: Alessandro Specchi

Location: Rome, Italy

Date: 1721 to 1725

Building Type: plaza, stairway

Context: urban

Style: Italian Baroque

The Spanish Stairs were built to unite Via del Babuino with Via Felice,

the first great street planned by Sixtus V in 1585. Their junction is crossed

at an approximately right angle by Via Condotti, which defines the direction

toward St. Peter’s and the Vatican. The very rich and varied solution

ultimately employed by De Sanctis in 1723-26 as a solution to it’s final

design is based on a simple doubling in depth of the central theme from

the Ripetta: a protruding volume flanked by convex stairs and a straight

flight in front. The upper unit presents the theme in its basic form; the

lower constitutes an articulate and lively variation. During this period in Italy

there was an unending reverence towards secular architecture and these stairs

helped tie in the notion of a geometrical undulation within a design.

Piazza del Campidoglio

Architect: Michelangelo

Location: Rome, Italy

Date: 1650-1654

Building Type: plaza, piazza,

urban open space, stairway

Context: urban plaza

Style: Italian Renaissance

Michelangelo designed the project and his Piazza del Campidoglio

is one of the most significant contributions ever made in the history of urban planning. It’s elliptical courtyard with central figure sculpture stands as a geometrical work of art, with the possibility of leading future architects to further the use of such complex curvaceous forms. A short walk to the south (starting out south-west) from the Piazza Venezia are the Cordonata steps, also by Michelangelo.

Piazza of St. Peter’s

Architect: Bernini

Location: Vatican City, surrounded by Rome, Italy

Date: 1656 to 1667

Building Type: piazza, outdoor plaza, urban open space

Context: urban

Style: Baroque Neoclassical

Bernini built the Piazza of St. Peter’s for Pope

Alexander VII in 1656-67. It was conceived as a grand

approach to the church but had to be planned so that the

greatest number of people could see the Pope give his

prayer, either from the middle of the facade of the

chapel or from a window in the Palace of the Vatican.

The Plaza has four rows of simple and majestic Doric

columns-300 all together- to form an oval 650 feet across

the long axis marked by three monuments: laterally by

fountains propelling tall jets of water and in the center by an Egyptian obelisk. As they enter the piazza, the faithful are

embraced by “the motherly arms of the church,” Bernini’s own description of his Colonnade. The Colonnade becomes simultaneously a dramatic frame for the church, a nurturing enclosure for the crowds of faithful, and a stage for the

processions and other sacred spectacles on which, at this particular period, the Catholic Church so strongly depended for its appeal. Also, the area is constructed in such a way that even upon nearing its entrance, one feels the intended impression of the geometry.

Christ Church

Architect: Nicholas Hawksmoor

Location: Spitalfields, London, England

Date: 1715 to 1729

Building Type: church

Context: urban

Style: Georgian English Baroque

Christ Church was built on a longitudinal west-east axis and the west front

terminates the facing street with a monumental steeple, which punctuates

the simple rectangular massing of the body of the church. The steeple rises

directly behind a grand portico, raised by steps from the street. The portico

has a Palladian or Venetian form, an arched center flanked by two rectangular

openings on each side, which is repeated by the tripartite window behind the

chancel to the east.

The interior is an axially organized plan, with column screens all around

articulating the entrance. Four piers with half columns attached articulate a central

rectangle, marking a cross aisle between two side doorways. Together these

piers and the column screen give the linear plan an additional centralized reading.

Tall, flat-coffered ceilings allow space for clerestory windows to light the central

space and lower outer windows to light the side aisles.

The giant order of columns, extreme height of the steeple, and the tall interior volumes and window or door openings are proportioned as simple rectangular, semi-circular and circular forms. They tend to follow a pattern evoked by a numerical order found only in greater buildings of the Renaissance. The geometric simplicity of the giant scaled forms gives the church a somber monumental grandeur. It evokes basic and archetypal experiences of form and faith, due to it’s strict obedience to the order of its predecessors and involves the occupant into its grace by allowing one to actually feel it in stone.

The previous buildings must all be analyzed for the influence of mathematics on architecture from antiquity up to the present day; the difference between the criteria that is the basis for the design process must be recognized as some of the aforementioned display. Such criteria are the geometry, the structure’s function, the load bearing behavior of the structural elements and of the structure, the manufacturing technique and the choice of materials, and interior and exterior lighting and decoration. Lately new ecological and economical criteria have been added, but during the times of our case study mathematicians these different types of criteria were under scrutiny by the public for a number of reasons due to social ineptitude. This didn’t make things easier for the bearers of the engineering and technical issues- which were the architects. Although all these criteria are well-known, each era assigned different meanings to them. While modern man understands the term “function” as purely meeting the primary task; a bridge as a means to cross a canyon, or a roof and walls as protection from external influences of nature and mankind, the definitions of past eras far exceeded this understanding. A structure not only had to be of material usefulness psychologically, but also beneficial and intellectually a fruitful influence. Mathematics combined with technology and mechanics is a part of every design criterion. Each era applied this very differently to the structures, based on cultural understanding.”

When Joseph Plateau published his treatise on soap bubbles

and film in 1873, soap bubbles already had their own place in

literature and art. Plateau’s problem consists in taking a generic

curve in three-space and finding a surface with the least

possible area bounded by that curve. The empirical solution

may be obtained by dipping a tri-dimensional model of the

curve into soapy water, resulting in a form called a minimal

surface. When a soap bubble is blown, the soapy surface

stretches; when blowing ceases, the film tends toward

equilibrium. The sphere presents the least exterior surface

area of all surfaces containing the same volume of air.

Creations stemming from the technology necessary to

create a tri-dimensional soap bubble form can be seen in the

structure of this proposed project for a cenotaph for Sir Isaac

Newton, 1784, by ‘Etienne-Louis Boulle’e. The lower image is

a similar project by Boulle’e of the Bibliotheque du Roi. His

projects tested the waters for such vast structures and met a flat

opinion that these type building’s should be suited for hospitals,

theatres, and jail cells for their definition as public institutions.

Throughout the history of architecture there has been a quest for a system of proportions that would

facilitate the technical and aesthetic requirements of a design. Such a system would have to ensure a

repetition of the mathematical process in which the same manner we complete a simple arithmetic problem.

Now, it has been shown throughout history that the sciences drift apart and then return to each other a later time to re- orient the way a building should be designed. Be it as a symmetrical form or as an abstract piece of art, the bottom line is that this geometrical or mathematical essence must exist- especially in today’s buildings not only visually, to support the eye’s need for balance, but also structurally, to serve a better use in housing our needs.

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