Perfect buildings: the maths of modern architecture | south-park-episodes.info
There is a relationship between mathematics and architecture. That relationship is, at times, a partnership where one draws equally upon the other. Sometimes it . Math has various roles in south-park-episodes.info maths, geometry has a crucial part for for south-park-episodes.infoects uses geometry to define the spatial form of the buildings. Do you have to be good at math to be an architect? with the array of dimensions, quantities, area, volume and other geometric relationships.
Designing such enormities is a delicate balancing act. A building not only needs to be structurally sound and aesthetically pleasing, it also has to comply with planning regulations, bow to budget constraints, optimally fit its purpose and maximise energy efficiency.
Mathematics and architecture
The design process boils down to a complex optimisation problem. It's in the way this problem is solved that modern architecture differs most from that of the ancient Egyptians: Maths describes the shapes of the structures to be built, the physical features that have to be understood and, as the language of computers, forms the basis for every step of the modelling process. The SMG's job is to help architects create virtual models of their project.
We then help them to model it using CAD computer aided design tools, or we develop tools for them. Image courtesy Brady Peters. With the help of computers you can model pretty much every aspect of a building, from its physics to its appearance. Computer models can simulate the way the wind blows around the building or sound waves bounce around inside it. Graphic programs can explore different mathematical surfaces and populate them with panels of different textures.
And all the information you get from these models can be pulled together in what is probably the most important innovation in architectural CAD tools in recent years: An architect's model of 30 St Mary Axe.
Parametric modelling has been around since the s, but only now are architects fully exploiting its power. The models allow you to play around with certain features of a building without having to re-calculate all the other features that are affected by the changes you make. This makes them extremely powerful design tools.
Take the Gherkin shown on the left as an example. If you decided to make the building slightly slimmer, this would have a knock-on effect on some other features. You'd have to re-calculate its out-lining curves and the angles of its diamond shapes, for example. This is quite a lot of work and even when it's done, you'd still have to draw a new sketch, either by hand or by re-programming your computer. Parametric models do all this for you. They allow you to change a variety of geometrical features while keeping fixed those features you have decided should not change.
The models function a bit like spreadsheets: In response to a change the software regenerates the model so that pre-determined relationships are maintained, just like a spreadsheet re-calculates all of its entries.
Equipped with the digital tools provided by the SMG, a design team can explore a huge range of design options in a very short period of time.
The team can change geometric features of a building and see how the change affects, say, aerodynamic or acoustic properties. They can explore how complex shapes that are hard to build can be broken down into simpler ones, and they can quickly calculate how much material is needed to estimate the cost.
The results are buildings that would have been impossible only a few decades ago, both because their complex shapes were next to impossible to construct and because of the degree to which they exploit science to interact optimally with their environment.
The Gherkin The Gherkin is one of the projects the SMG was involved with and is a prime example of how geometry was chosen to satisfy constraints. Going by the official name of 30 St Mary Axe, the building is metres tall, three times the height of the Niagara Falls.
There are three main features that make it stand out from most other sky-scrapers: All these could easily be taken as purely aesthetic features, yet they all cater to specific constraints. A major problem with buildings of the Gherkin's size is that air currents sweeping around them create whirlwinds at their base, making their immediate vicinity an uncomfortable place to be.
To address this problem, the SMG advised the architects to use computer models which, based on the mathematics of turbulence, simulate a building's aerodynamic properties.
The model showed that a cylindrical shape responds better to air currents than a square one and reduces whirlwinds. The fact that the tower bulges out in the middle, reaching its maximal diameter at the 16th floor, also helps to minimise winds at its slimmer base. A model of air currents flowing around the Gherkin. But even if you're not being ruffled by strong winds, standing next to a sky-scraper can be eery.
It dwarfs you, it eclipses shorter buildings and it takes away the sunlight. Each half-rectangle is then a convenient 3: The inner area naos similarly has 4: The stylobate is the platform on which the columns stand.
Some Mathematical Principles of Architecture
As in other classical Greek temples,  the platform has a slight parabolic upward curvature to shed rainwater and reinforce the building against earthquakes.
The columns might therefore be supposed to lean outwards, but they actually lean slightly inwards so that if they carried on, they would meet about a mile above the centre of the building; since they are all the same height, the curvature of the outer stylobate edge is transmitted to the architrave and roof above: Islamic architecture The historian of Islamic art Antonio Fernandez-Puertas suggests that the Alhambralike the Great Mosque of Cordoba was designed using the Hispano-Muslim foot or codo of about 0.
In the palace's Court of the Lionsthe proportions follow a series of surds.
There is no evidence to support earlier claims that the golden ratio was used in the Alhambra. The very large central space is accordingly arranged as an octagon, formed by 8 enormous pillars, and capped by a circular dome of The building's plan is thus a circle inside an octagon inside a square.
Mughal architectureFatehpur Sikriand Origins and architecture of the Taj Mahal Mughal architectureas seen in the abandoned imperial city of Fatehpur Sikri and the Taj Mahal complex, has a distinctive mathematical order and a strong aesthetic based on symmetry and harmony.
The white marble mausoleumdecorated with pietra durathe great gate Darwaza-i rauzaother buildings, the gardens and paths together form a unified hierarchical design.
The buildings include a mosque in red sandstone on the west, and an almost identical building, the Jawab or 'answer' on the east to maintain the bilateral symmetry of the complex. Islamic architects created a wealth of two-dimensional tiling patterns centuries before western mathematicians gave a complete classification. The first mentioned type of architecture Salingaros mentions in this quote is the pyramid and here we have marked disagreement between experts on the how much geometry and number theory the architects used.
Much has been written on the measurements of this pyramid and many coincidences have been found withthe golden number and its square root. There are at least nine theories which claim to explain the shape of the Pyramid and at least half of these theories agree with the observed measurements to one decimal place.
This is a difficult area, for there is no doubt about certain astronomical alignments in the construction of the pyramid.
Also regular geometric shapes were sacred to the Egyptians and they reserved their use in architecture for ritual and official buildings. That they had a goddess of surveying, called Seschat, shows the religious importance placed on building. However, no proof exists that sophisticated geometry lies behind the construction of the pyramids. One has to make decisions as to whether the numerical coincidences are really coincidences, or whether the builders of the pyramids designed them with certain numerical ratios in mind.
Let us look at just one such coincidence involving the golden number. Is this a coincidence? The authors of [ 23 ], however, suggest reasons for the occurrence of many of the nice numbers, in particular numbers close to powers of the golden number, as arising from the building techniques used rather than being deliberate decisions of the architects.
Arguments of this type have appeared more frequently in recent years. Even if deep mathematical ideas went into the construction of the pyramids, I think that Ifrah makes a useful contribution to this debate in [ 4 ] when he writes: But the first gardener in history to lay out a perfect ellipse with three stakes and a length of string certainly held no degree in the theory of cones!
Nor did Egyptian architects have anything more than simple devices -- "tricks", "knacks" and methods of an entirely empirical kind, no doubt discovered by trial and error -- for laying out their ground plans. The first definite mathematical influence on architecture we mention is that of Pythagoras. Now for Pythagoras and the Pythagoreans, number took on a religious significance.
The Pythagorean belief that "all things are numbers" clearly had great significance for architecture so let us consider for a moment what this means. Taken at face value it might seem quite a silly idea but in fact it was based on some fundamental truths.
Pythagoras saw the connection between music and numbers and clearly understood how the note produced by a string related to its length. He established the ratios of the sequence of notes in a scale still used in Western music.
By conducting experiments with a stretched string he discovered the significance of dividing it into ratios determined by small integers. The discovery that beautiful harmonious sounds depended on ratios of small integers led to architects designing buildings using ratios of small integers.