Last Updated on December 22, 2022

Architecture is not an easy field to work in, which is why it is essential for the aspiring architect to master a variety of math concepts. At home concepts such as fractions and metric conversions are vital, while skill s like decimal division, square roots, and decimals must be mastered at school. Having a strong conception of all these elements will help any architect save time and do more work. Want to know more about Architecture Math Requirements, how much math is there in architecture, calculus in architecture problems, is calculus needed for architecture, math subjects in architecture philippines, geometry in architecture & does architecture require math

Have you been looking through the internet for information on architecture degree courses? Do you often get confused by the conflicting information you see on it online? You need not search further as you will find the answer to this question in the article below.

Read on to get the latest information on architecture degree entry requirements, what subjects are required to become an architect, march. for students with a non-architecture bachelor’s degree, how to become an architect, architecture degree online, architect requirements in high school and architecture degree salary. You will find more up to date information on architecture math requirements related articles on Collegelearners.

## architecture degree math requirements

Think of the home you live in, the school you attend, and stores you shop in. Architects are involved in every phase of the building process, from conception to construction. Requiring a wide range of abilities, architects need strong math skills, a keen eye for detail, an exceptional ability to communicate, superb problem-solving skills, and the capacity for critical thinking.

If helping design and build the places we inhabit sounds like an inviting career path, keep reading to learn more about what it takes to become an architect.

**What Does an Architect Do?**

Architects design both the interior and exteriors of the buildings where we live, learn, work, and shop. These individuals oversee every step of a structure’s development; they determine the needs of a project, estimate costs and viability, create structural specifications, and manage construction to ensure adherence to the architectural plans. It’s an architect’s responsibility to make sure a building is structurally safe, functional, in compliance with rules and regulations, and in line with the client or community aesthetic.

**How Much Do Architects Make?**

According to the Bureau of Labor Statistics (BLS), in 2018, the average salary of an architect was $79,380. The highest paid architects worked in the government sector and earned a median wage of $92,940. The BLS predicts that the demand for architects will increase by 8% between 2018 and 2028, higher than the 5% growth rate for all occupations.

**How to Become an Architect**

Architects need to balance competing needs, such as function vs. safety vs. cost, along with duties like design and supervision—all while communicating their needs to diverse teams. Consequently, on the path to becoming an architect, a person will explore a wide variety of academic fields.

### High School

Most high schools lack programs specific to architecture; however, that doesn’t mean a student can’t start preparing themselves for a career as an architect early in their academic careers. Math classes such as geometry, algebra, calculus, and trigonometry are all recommended for aspiring architects. Similarly, science classes such as physics are beneficial for understanding concepts such as force, compression, and tension. It’s also important to explore the arts; drawing, painting, sculpting, and photography all help build the ability to visualize and conceptualize.

There are a handful of AP courses that are particularly beneficial for those hoping to be accepted into a competitive college architecture program. AP Calculus, Physics, and 3D Art and Design will all help wow colleges and build skills necessary for the rigorous coursework ahead.

Keep in mind that some undergraduate architecture programs may have special course requirements (such as 4 years of math, with calculus recommended, and 1 year of physics). Be sure to check the requirements at the schools you’re interested in.

Similarly, some colleges may require portfolios, such as the Cornell College of Art, Architecture, and Planning. They ask for “15-20 slides with no more than two dedicated to the same project.” The portfolio should contain sketches and fully developed work, but doesn’t need to be entirely focused on architecture. The website states that “painting, printmaking, sculpture, photography, video, woodworking, and other crafts can convey artistic experience and aptitude.” Spend some time building a portfolio so you can put your best food forward come application time.

An architecture-focused summer program is also an excellent way to explore the field before committing to a degree path in college—Cornell, Syracuse, Frank Loyd Wright’s Taliesin, UNC Charlotte, and UCLA all have programs aimed at would-be architects early in their academic careers.

### College

There are two paths students can take toward a career in architecture in college. One path is to achieve a traditional undergraduate degree before going on to earn a master’s degree. In the architectural field, this is referred to as an M. Arch. The other path is to earn a B. Arch., which is a five-year program accredited by the National Architecture Accrediting Board (NAAB). Students graduating with a B. Arch. qualify to take the Architect Registration Examination (ARE)—a key step on the path to earning an architectural license.

**B. Arch.: **For students entering college with clear intentions of becoming an architect, a B. Arch. program is the most direct route through college, as graduates of these five-year programs qualify for licensure. The first two years of a B. Arch. program focus on the history of the discipline and basic building skills, such as fundamental design concepts and understanding of material properties. The final three years of a B. Arch. program cover topics like architectural theory, building technology, and computer-aided design and drafting.

**M. Arch.:** For students who don’t discover their passion for architecture until college—or who aren’t prepared to settle on an architectural career path as an undergraduate—there is the M. Arch., or master’s degree in architecture. Students come to M. Arch. programs from all backgrounds; some schools will even offer separate tracks for students entering an M. Arch. program with bachelor’s degrees in architecture, and those coming from unrelated subjects. Regardless, students in M. Arch. programs will study courses covering topics the theoretical, technological, historical, and cultural aspects of architecture. The length of time a student spends in an M. Arch. program is determined by a student’s previous architectural education.

### Internship

All states require a student to complete an internship before becoming licensed, the majority of which employ the Intern Development Program (IDP) administered by the National Council of Architectural Registration Boards (NCARB) and the American Institute of Architects. The IDP program requires 5,600 hours of state- and NCARB-approved work experience in four core areas: pre-design, design, project management, and practice management.

**Certification and Licensure **

All states require architects to pass the Architect Registration Examination (ARE), a test administered by the National Council of Architectural Registration Boards, before becoming licensed to practice. The ARE is a test of a person’s knowledge, skill, and ability in the field of architecture. The NCARB also offers a national certification, which makes it easier on those looking to operate in multiple states

## What Math Classes Do Architects Take in College?

To become an architect, you must complete a degree program in architecture, participate in an internship and pass the Architect Registration Exam. Architects must have a strong knowledge of mathematical principles, so they can effectively plan and design buildings and other structures. Students must take several math classes in college to obtain a degree in architecture.

### Algebra, Geometry and Trigonometry

Algebra, geometry and trigonometry are prerequisites for taking Calculus, and Calculus is required to complete a degree program in architecture. Some students complete the algebra, geometry and trigonometry requirements in high school and can immediately start with calculus courses in college. Architecture students who didn’t take courses such as Geometry and Algebra II with Trigonometry in high school must take those classes or a similar ones in college.

### Calculus

Students pursuing a degree in architecture must take calculus courses. At the University of Illinois and Brigham Young University, students in an architecture degree program must take both Calculus I and II. The University of Illinois allows students to take beginning or intermediate physics courses in place of Calculus II, if they choose. Calculus and physics courses help students calculate structural issues, so they can design buildings that will hold up under the weight of materials and withstand interior and exterior forces.urn:uuid:85009e1c-254e-d909-939f-d909254e8500

### Probability and Statistics

Some architectural degree programs require students to take a math class in probability and statistics, which helps architects analyze data such as geological and geographical information, structural specifications and construction optimization. When an architect estimates costs for labor, materials and machinery, he often uses statistical analysis to determine the best value for the money. Some computer software and modeling programs are designed to help architects analyze statistical data quickly and effectively so they can make informed real-world decisions

### Linear Programming

Linear programming math classes help students learn to evaluate variable factors that affect design and construction. For example, linear programming enables architects to determine whether the cost of a particular design element will pay off in the long run or if it’s too expensive for the function and purpose it provides. Architects often design structures according to budgetary constraints, so linear programming makes cost and outcome analysis possible. Depending on the university, linear programming courses may be listed as part of the math department or the technology department.

### Considerations

According to the U.S. Bureau of Statistics, architecture students must complete a training period, usually in the form of an internship that lasts three or more years. Many students and graduates work for architectural or engineering firms to satisfy the requirement. Students who complete an internship while they’re still in school can apply that experience toward the 3-year training requirement. Graduates are not permitted to take the Architect Registration Exam until the internship is complete. Once a graduate passes the Architect Registration Exam, he may apply for state licensure as a practicing architect.

### 2016 Salary Information for Architects

Architects earned a median annual salary of $76,930 in 2016, according to the U.S. Bureau of Labor Statistics. On the low end, architects earned a 25th percentile salary of $59,000, meaning 75 percent earned more than this amount. The 75th percentile salary is $99,790, meaning 25 percent earn more. In 2016, 128,800 people were employed in the U.S. as architects.

## Do You Need Math To Become An Architect

### What do I need to do to become an architect?

To get a better understanding of the architecture math requirements, it is essential to comprehend the necessity of architecture math. Generally, students enroll themselves in architecture programs because they are fascinated by designing and construction. The math requirement for architecture programs is considerable because you may need to deal with complicated figures and shapes in a calculation for architectural design.

To become an architect, you will need to complete a five year architecture degree which is recognized by the Architects Registration Board (ARB).

To take a recognised degree you will need: five GCSEs A*- C including English, maths and science and three A levels. Some universities prefer a maths or a science subject. Many also require a portfolio of work, so an art & design based A level can be helpful. Entry requirements vary so check carefully with the the Architects Registration Board.

Other level 3 courses (e.g. science or engineering) may be acceptable for entry to a recognised degree at some universities – check with them direct.

You can get into this job through:

a university course

an apprenticeship

working towards this role

University

You’ll need to complete:

a degree recognised by the Architects Registration Board (ARB)

a year of practical work experience

another 2 years’ full-time university course like BArch, Diploma, MArch

a further year of practical training

a final qualifying exam

Many course providers will also want to see a portfolio of your drawings and sketches.

Entry requirements

You’ll usually need:

5 GCSEs at grades 9 to 4 (A* to C), or equivalent, including English, maths and science

2 to 3 A levels, or equivalent, for a degree

More information

equivalent entry requirements

student finance for fees and living costs

university courses and entry requirements

Apprenticeship

You can get into this role through an architect degree apprenticeship.

Entry requirements

You’ll usually need:

4 or 5 GCSEs at grades 9 to 4 (A* to C) and A levels, or equivalent, for a higher or degree apprenticeship

To become an architect, you must complete a degree program in architecture, participate in an internship and pass the Architect Registration Exam. Architects must have a strong knowledge of mathematical principles, so they can effectively plan and design buildings and other structures. Students must take several math classes in college to obtain a degree in architecture.

### Algebra, Geometry and Trigonometry

Algebra, geometry and trigonometry are prerequisites for taking Calculus, and Calculus is required to complete a degree program in architecture. Some students complete the algebra, geometry and trigonometry requirements in high school and can immediately start with calculus courses in college. Architecture students who didn’t take courses such as Geometry and Algebra II with Trigonometry in high school must take those classes or a similar ones in college.

### Calculus

Students pursuing a degree in architecture must take calculus courses. At the University of Illinois and Brigham Young University, students in an architecture degree program must take both Calculus I and II. The University of Illinois allows students to take beginning or intermediate physics courses in place of Calculus II, if they choose. Calculus and physics courses help students calculate structural issues, so they can design buildings that will hold up under the weight of materials and withstand interior and exterior forces.

### Probability and Statistics

Some architectural degree programs require students to take a math class in probability and statistics, which helps architects analyze data such as geological and geographical information, structural specifications and construction optimization. When an architect estimates costs for labor, materials and machinery, he often uses statistical analysis to determine the best value for the money. Some computer software and modeling programs are designed to help architects analyze statistical data quickly and effectively so they can make informed real-world decisions

### Linear Programming

Linear programming math classes help students learn to evaluate variable factors that affect design and construction. For example, linear programming enables architects to determine whether the cost of a particular design element will pay off in the long run or if it’s too expensive for the function and purpose it provides. Architects often design structures according to budgetary constraints, so linear programming makes cost and outcome analysis possible. Depending on the university, linear programming courses may be listed as part of the math department or the technology department.

## Architecture Math Requirements

### What kind of math do you need for architecture?

**Geometry, algebra, and trigonometry** all play a crucial role in architectural design. Architects apply these math forms to plan their blueprints or initial sketch designs. They also calculate the probability of issues the construction team could run into as they bring the design vision to life in three dimensions

### Can I be an architect if I’m bad at math?

**Not really**. If you understand general geometry and physics you are good; having addition, subtraction, multiplication and sometimes division skills are encouraged. Aspiring architects should challenge themselves with as much math as they can handle (plus the class one further than they can handle)

### How is math used in architecture?

Math **helps us to determine the volume of gravel or soil that is needed to fill a hole**. We rely on math when designing safe building structures and bridges by calculating loads and spans. Math also helps us to determine the best material to use for a structure, such as wood, concrete, or steel

## formulas in architecture

Excel is more than just digital graph paper. It’s a serious tool for analyzing and computing data. In order to access this power, however, you need to understand formulas.

If you’re like me, you started using Excel as a way to create nice looking tables of data – things like building programs or drawing lists. Lots of text and some numbers. Nothing too crazy. If I was feeling a little bold, I’d add a simple formula to add or subtract some cells. That’s about it.

I knew I was using only about 10% of the software but I wasn’t sure what else it could do or how I could access the other functions. I’d heard about formulas but they seemed really confusing. Plus, I was an architect, not a bean counter.

It wasn’t until I ran into a number problem that I realized the true power of Excel. I needed to analyze the leasable area for a large mixed-use project I was working on. We were getting different area numbers from the developers. Since no one likes losing area, I had to dig through the data to figure out what was going on.

So I rolled up my sleeves, took a deep breath and plunged into the world of Excel formulas. A few hours later, I had a lean and mean spreadsheet that accurately calculated the leasable area. Using the formulas I had built, we could quickly play out scenarios for our client. This spreadsheet ended up being a really useful tool during the design phase.

**Get started with Excel formulas**

Inserting a formula into a cell is real easy. Just type an equals sign (=) followed by the formula. You can click the formula icon to open the “Insert Function” dialog.

You can also access all of the Excel functions through the “Formulas” ribbon. All of the formulas are grouped into categories. Click the category you want then select the formula from the list. This will open a dialog where you can input the formula parameters.

What’s the best way to learn Excel formulas? I’m a big believer in learning by doing. Take a spreadsheet you’ve created and see how you can make it better with formulas. Not sure which ones to try? Here’s my list of 12 Excel formulas every architect should know:

**1. SUM**

Adds together all the values in the specified range. The range can be a single column or multiple columns. You can even specify individual cells by using a comma to separate the values.

*=SUM(A5:A25)*

**2. IF**

Returns one value if a condition is true and another if the condition is false. Useful for getting a quick overview of your data. You can also use AND or OR within the IF statement to build complex logic.

*=IF(A2>B2, “NEED AREA”, “AREA OK”)*

**3. SUMIF**

Performs the SUM function only on instances that meet certain criteria. Use SUMIFS to specify multiple criteria.

*=SUMIF(A1:A7, “>0″)*

*=SUMIFS(A1:A7, A1:A7, “>100″, A1:A7, “<200″)*

**4. COUNT, COUNTA, COUNTBLANK**

Counts the number of items in the specified range. COUNT only counts numbers, not text or blank cells. COUNTA counts cells that are not empty. This includes number, text and other types of data. COUNTBLANK counts only cells that are blank.

*=COUNT(A5:A25)*

*=COUNTA(A5:A25)*

*=COUNTBLANK(A5:A25)*

**5. COUNTIF**

Similar to COUNT but will count only instances that meet the specified criteria. Use COUNTIFS to specify multiple criteria. For instance, rooms that are greater than 200 SF but less than 500 SF.

*=COUNTIF(A1:A8, “>100″)*

*=COUNTIFS(A1:A8, “>100″, A1:A8, “<200″)*

**6. AVERAGE**

Returns the average or arithmetic mean of the specified range of cells.

*=AVERAGE(A5:A25)*

**7. MIN**

Returns the smallest number in the specified range of cells. This might be useful for finding the smallest area in a programming spreadsheet.

*=MIN(A5:A25)*

**8. MAX**

Similar to MIN but this formula returns the largest number in a range of cells.

*=MAX(A5:A25)*

**9. VLOOKUP**

VLOOKUP helps Excel function more like a database than just a spreadsheet application. With it, you can search for values based on other values, which can be located in another part of the worksheet or in a completely different worksheet. In the formula, you need to specify the key value, the range of values you want to search, and the column number of the value you want to return. VLOOKUP is a little tricky to use so I highly recommend checking out this step-by-step guide.

*=VLOOKUP(B3,$A$17:$B$20,2)*

**10. ROUND**

Rounds a number to a specified number of digits. Can also use ROUNDUP and ROUNDDOWN to specify the direction of rounding.

*=ROUND(7.86, 1) results in 7.9*

*=ROUNDUP(7.23, 0) results in 8*

*=ROUNDDOWN(8.85, 1) results in 8.8*

**11. FLOOR and CEILING**

These two functions round a number up (CEILING) or down (FLOOR) to the nearest specified multiple. Useful when rounding currency.

*=FLOOR(A1, 10)*

*=CEILING(A2, 0.25)*

**12. CONCATENATE**

Use the CONCATENATE function to join two cells together. This function is great for piecing together text that resides in separate columns. You can also use an ampersand (&) instead of typing out CONCATENATE.

*=CONCATENATE(B1, “, “, A1)*

*=A3& ” ” & B3*

**A few more things about formulas**

Named ranges are great to use with formulas. Rather than typing the cell range (like A3:B4), you can enter the name (like “Level1Area”). Plus, if the range changes, just update it once in the “Name Manager”. You don’t need to update each formula.

You can review all the available formulas by going to the “Formulas” ribbon and clicking one of the icons in the “Function Library” section. All of the formulas are organized by category. Likewise, you can click the “Insert Function” button to see all the available functions.

You can include one formula in another formula. This is known as “nesting function”. In Excel 2013, you can nest up to 64 functions.

Copying and pasting formulas can sometimes be tricky. By default, Excel will increment the cell ranges when you paste a formula. Sometimes this is useful, particularly if you’re using SUM to add up a row of values. However, sometimes you want to calculate specific cells. In order to do this, use a “$” before the cell to designate it as an absolute reference. For example, if I want to multiply cell B4 with cell D3, I would type my formula as “=B4*D3″. Now, if I want to copy this formula down the column but I still want to multiply by cell D3, I would type the formula at “=B4*$D$3″. This designates cell D3 as an absolute reference so Excel doesn’t increment it.

**How about you?**

How do you use formulas in your spreadsheets? If so, what’s your favorite formula? Leave a comment below!

## Mathematics and architecture

**Mathematics and architecture** are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.

In ancient Egypt, ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces and mausoleums were laid out with specific proportions for religious reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about the infinite in Hindu cosmology. In Chinese architecture, the tulou of Fujian province are circular, communal defensive structures. In the twenty-first century, mathematical ornamentation is again being used to cover public buildings.

In Renaissance architecture, symmetry and proportion were deliberately emphasized by architects such as Leon Battista Alberti, Sebastiano Serlio and Andrea Palladio, influenced by Vitruvius‘s *De architectura* from ancient Rome and the arithmetic of the Pythagoreans from ancient Greece. At the end of the nineteenth century, Vladimir Shukhov in Russia and Antoni Gaudí in Barcelona pioneered the use of hyperboloid structures; in the Sagrada Família, Gaudí also incorporated hyperbolic paraboloids, tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. In the twentieth century, styles such as modern architecture and Deconstructivism explored different geometries to achieve desired effects. Minimal surfaces have been exploited in tent-like roof coverings as at Denver International Airport, while Richard Buckminster Fuller pioneered the use of the strong thin-shell structures known as geodesic domes

## Religious principles[edit]

### Ancient Egypt[edit]

See also: Golden ratio § Egyptian pyramidsBase:hypotenuse (b:a) ratios for pyramids like the Great Pyramid of Giza could be: 1:φ (Kepler triangle), 3:5 (3:4:5 triangle), or 1:4/π

The pyramids of ancient Egypt are tombs constructed with mathematical proportions, but which these were, and whether the Pythagorean theorem was used, are debated. The ratio of the slant height to half the base length of the Great Pyramid of Giza is less than 1% from the golden ratio.^{[51]} If this was the design method, it would imply the use of Kepler’s triangle (face angle 51°49′),^{[51]}^{[52]} but according to many historians of science, the golden ratio was not known until the time of the Pythagoreans.^{[53]} The Great Pyramid may also have been based on a triangle with base to hypotenuse ratio 1:4/π (face angle 51°50′).^{[54]}

The proportions of some pyramids may have also been based on the 3:4:5 triangle (face angle 53°8′), known from the Rhind Mathematical Papyrus (c. 1650–1550 BC); this was first conjectured by historian Moritz Cantor in 1882.^{[55]} It is known that right angles were laid out accurately in ancient Egypt using knotted cords for measurement,^{[55]} that Plutarch recorded in *Isis and Osiris* (c. 100 AD) that the Egyptians admired the 3:4:5 triangle,^{[55]} and that a scroll from before 1700 BC demonstrated basic square formulas.^{[56]}^{[f]} Historian Roger L. Cooke observes that “It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem,” but also notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle’s sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor’s conjecture remains uncertain; he guesses that the ancient Egyptians probably knew the Pythagorean theorem, but “there is no evidence that they used it to construct right angles.”^{[55]}

### Ancient India[edit]

Further information: Architecture of India and Vaastu ShastraGopuram of the HinduVirupaksha Temple has a fractal-like structure where the parts resemble the whole.

Vaastu Shastra, the ancient Indian canons of architecture and town planning, employs symmetrical drawings called mandalas. Complex calculations are used to arrive at the dimensions of a building and its components. The designs are intended to integrate architecture with nature, the relative functions of various parts of the structure, and ancient beliefs utilizing geometric patterns (yantra), symmetry and directional alignments.^{[57][58]} However, early builders may have come upon mathematical proportions by accident. The mathematician Georges Ifrah notes that simple “tricks” with string and stakes can be used to lay out geometric shapes, such as ellipses and right angles.^{[12][59]}Plan of Meenakshi Amman Temple, Madurai, from 7th century onwards. The four gateways (numbered I-IV) are tall gopurams.

The mathematics of fractals has been used to show that the reason why existing buildings have universal appeal and are visually satisfying is because they provide the viewer with a sense of scale at different viewing distances. For example, in the tall gopuram gatehouses of Hindu temples such as the Virupaksha Temple at Hampi built in the seventh century, and others such as the Kandariya Mahadev Temple at Khajuraho, the parts and the whole have the same character, with fractal dimension in the range 1.7 to 1.8. The cluster of smaller towers (*shikhara*, lit. ‘mountain’) about the tallest, central, tower which represents the holy Mount Kailash, abode of Lord Shiva, depicts the endless repetition of universes in Hindu cosmology.^{[2]}^{[60]} The religious studies scholar William J. Jackson observed of the pattern of towers grouped among smaller towers, themselves grouped among still smaller towers, that:

The ideal form gracefully artificed suggests the infinite rising levels of existence and consciousness, expanding sizes rising toward transcendence above, and at the same time housing the sacred deep within.

^{[60]}^{[61]}

The Meenakshi Amman Temple is a large complex with multiple shrines, with the streets of Madurai laid out concentrically around it according to the shastras. The four gateways are tall towers (gopurams) with fractal-like repetitive structure as at Hampi. The enclosures around each shrine are rectangular and surrounded by high stone walls.^{[62]}

### Ancient Greece[edit]

Further information: Greek architecture, golden ratio, Pythagoreanism, and Euclidean geometryThe Parthenon was designed using Pythagorean ratios.

Pythagoras (c. 569 – c. 475 B.C.) and his followers, the Pythagoreans, held that “all things are numbers”. They observed the harmonies produced by notes with specific small-integer ratios of frequency, and argued that buildings too should be designed with such ratios. The Greek word *symmetria* originally denoted the harmony of architectural shapes in precise ratios from a building’s smallest details right up to its entire design.^{[12]}

The Parthenon is 69.5 metres (228 ft) long, 30.9 metres (101 ft) wide and 13.7 metres (45 ft) high to the cornice. This gives a ratio of width to length of 4:9, and the same for height to width. Putting these together gives height:width:length of 16:36:81, or to the delight^{[63]} of the Pythagoreans 4^{2}:6^{2}:9^{2}. This sets the module as 0.858 m. A 4:9 rectangle can be constructed as three contiguous rectangles with sides in the ratio 3:4. Each half-rectangle is then a convenient 3:4:5 right triangle, enabling the angles and sides to be checked with a suitably knotted rope. The inner area (naos) similarly has 4:9 proportions (21.44 metres (70.3 ft) wide by 48.3 m long); the ratio between the diameter of the outer columns, 1.905 metres (6.25 ft), and the spacing of their centres, 4.293 metres (14.08 ft), is also 4:9.^{[12]}Floor plan of the Parthenon

The Parthenon is considered by authors such as John Julius Norwich “the most perfect Doric temple ever built”.^{[64]} Its elaborate architectural refinements include “a subtle correspondence between the curvature of the stylobate, the taper of the naos walls and the *entasis* of the columns”.^{[64]} *Entasis* refers to the subtle diminution in diameter of the columns as they rise. The stylobate is the platform on which the columns stand. As in other classical Greek temples,^{[65]} 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 kilometre and a half 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: “all follow the rule of being built to delicate curves”.^{[66]}

The golden ratio was known in 300 B.C., when Euclid described the method of geometric construction.^{[67]} It has been argued that the golden ratio was used in the design of the Parthenon and other ancient Greek buildings, as well as sculptures, paintings, and vases.^{[68]} More recent authors such as Nikos Salingaros, however, doubt all these claims.^{[69]} Experiments by the computer scientist George Markowsky failed to find any preference for the golden rectangle.^{[70]}

### Islamic architecture[edit]

Further information: Islamic architecture and Golden ratio § ArchitectureSelimiye Mosque, Edirne, 1569–1575

The historian of Islamic art Antonio Fernandez-Puertas suggests that the Alhambra, like the Great Mosque of Cordoba,^{[71]} was designed using the Hispano-Muslim foot or *codo* of about 0.62 metres (2.0 ft). In the palace’s Court of the Lions, the proportions follow a series of surds. A rectangle with sides 1 and √2 has (by Pythagoras’s theorem) a diagonal of √3, which describes the right triangle made by the sides of the court; the series continues with √4 (giving a 1:2 ratio), √5 and so on. The decorative patterns are similarly proportioned, √2 generating squares inside circles and eight-pointed stars, √3 generating six-pointed stars. There is no evidence to support earlier claims that the golden ratio was used in the Alhambra.^{[10]}^{[72]} The Court of the Lions is bracketed by the Hall of Two Sisters and the Hall of the Abencerrajes; a regular hexagon can be drawn from the centres of these two halls and the four inside corners of the Court of the Lions.^{[73]}

The Selimiye Mosque in Edirne, Turkey, was built by Mimar Sinan to provide a space where the mihrab could be see from anywhere inside the building. The very large central space is accordingly arranged as an octagon, formed by eight enormous pillars, and capped by a circular dome of 31.25 metres (102.5 ft) diameter and 43 metres (141 ft) high. The octagon is formed into a square with four semidomes, and externally by four exceptionally tall minarets, 83 metres (272 ft) tall. The building’s plan is thus a circle, inside an octagon, inside a square.^{[74]}

### Mughal architecture[edit]

Main articles: Mughal architecture, Fatehpur Sikri, and Origins and architecture of the Taj MahalThe Taj Mahal mausoleum with part of the complex’s gardens at Agra

Mughal architecture, as 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.^{[11]}^{[75]}

The Taj Mahal exemplifies Mughal architecture, both representing paradise^{[76]} and displaying the Mughal Emperor Shah Jahan‘s power through its scale, symmetry and costly decoration. The white marble mausoleum, decorated with pietra dura, the great gate (*Darwaza-i rauza*), other 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. The formal charbagh (‘fourfold garden’) is in four parts, symbolising the four rivers of paradise, and offering views and reflections of the mausoleum. These are divided in turn into 16 parterres.^{[77]}Site plan of the Taj Mahal complex. The great gate is at the right, the mausoleum in the centre, bracketed by the mosque (below) and the jawab. The plan includes squares and octagons.

The Taj Mahal complex was laid out on a grid, subdivided into smaller grids. The historians of architecture Koch and Barraud agree with the traditional accounts that give the width of the complex as 374 Mughal yards or gaz,^{[g]} the main area being three 374-gaz squares. These were divided in areas like the bazaar and caravanserai into 17-gaz modules; the garden and terraces are in modules of 23 gaz, and are 368 gaz wide (16 x 23). The mausoleum, mosque and guest house are laid out on a grid of 7 gaz. Koch and Barraud observe that if an octagon, used repeatedly in the complex, is given sides of 7 units, then it has a width of 17 units,^{[h]} which may help to explain the choice of ratios in the complex.^{[78]}

### Christian architecture[edit]

Further information: Church architecture

The Christian patriarchal basilica of Haghia Sophia in Byzantium (now Istanbul), first constructed in 537 (and twice rebuilt), was for a thousand years^{[i]} the largest cathedral ever built. It inspired many later buildings including Sultan Ahmed and other mosques in the city. The Byzantine architecture includes a nave crowned by a circular dome and two half-domes, all of the same diameter (31 metres (102 ft)), with a further five smaller half-domes forming an apse and four rounded corners of a vast rectangular interior.^{[79]} This was interpreted by mediaeval architects as representing the mundane below (the square base) and the divine heavens above (the soaring spherical dome).^{[80]} The emperor Justinian used two geometers, Isidore of Miletus and Anthemius of Tralles as architects; Isidore compiled the works of Archimedes on solid geometry, and was influenced by him.^{[12]}^{[81]}Haghia Sophia, Istanbul

a) Plan of gallery (upper half)

b) Plan of the ground floor (lower half)

The importance of water baptism in Christianity was reflected in the scale of baptistry architecture. The oldest, the Lateran Baptistry in Rome, built in 440,^{[82]} set a trend for octagonal baptistries; the baptismal font inside these buildings was often octagonal, though Italy’s largest baptistry, at Pisa, built between 1152 and 1363, is circular, with an octagonal font. It is 54.86 metres (180.0 ft) high, with a diameter of 34.13 metres (112.0 ft) (a ratio of 8:5).^{[83]} Saint Ambrose wrote that fonts and baptistries were octagonal “because on the eighth day,^{[j]} by rising, Christ loosens the bondage of death and receives the dead from their graves.”^{[84]}^{[85]} Saint Augustine similarly described the eighth day as “everlasting … hallowed by the resurrection of Christ”.^{[85]}^{[86]} The octagonal Baptistry of Saint John, Florence, built between 1059 and 1128, is one of the oldest buildings in that city, and one of the last in the direct tradition of classical antiquity; it was extremely influential in the subsequent Florentine Renaissance, as major architects including Francesco Talenti, Alberti and Brunelleschi used it as the model of classical architecture.^{[87]}

The number five is used “exuberantly”^{[88]} in the 1721 Pilgrimage Church of St John of Nepomuk at Zelená hora, near Žďár nad Sázavou in the Czech republic, designed by Jan Blažej Santini Aichel. The nave is circular, surrounded by five pairs of columns and five oval domes alternating with ogival apses. The church further has five gates, five chapels, five altars and five stars; a legend claims that when Saint John of Nepomuk was martyred, five stars appeared over his head.^{[88]}^{[89]} The fivefold architecture may also symbolise the five wounds of Christ and the five letters of “Tacui” (Latin: “I kept silence” [about secrets of the confessional]).^{[90]}

Antoni Gaudí used a wide variety of geometric structures, some being minimal surfaces, in the Sagrada Família, Barcelona, started in 1882 (and not completed as of 2015). These include hyperbolic paraboloids and hyperboloids of revolution,^{[91]} tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. This varied mix of geometries is creatively combined in different ways around the church. For example, in the Passion Façade of Sagrada Família, Gaudí assembled stone “branches” in the form of hyperbolic paraboloids, which overlap at their tops (directrices) without, therefore, meeting at a point. In contrast, in the colonnade there are hyperbolic paraboloidal surfaces that smoothly join other structures to form unbounded surfaces. Further, Gaudí exploits natural patterns, themselves mathematical, with columns derived from the shapes of trees, and lintels made from unmodified basalt naturally cracked (by cooling from molten rock) into hexagonal columns.^{[92]}^{[93]}^{[94]}

The 1971 Cathedral of Saint Mary of the Assumption, San Francisco has a saddle roof composed of eight segments of hyperbolic paraboloids, arranged so that the bottom horizontal cross section of the roof is a square and the top cross section is a Christian cross. The building is a square 77.7 metres (255 ft) on a side, and 57.9 metres (190 ft) high.^{[95]} The 1970 Cathedral of Brasília by Oscar Niemeyer makes a different use of a hyperboloid structure; it is constructed from 16 identical concrete beams, each weighing 90 tonnes,^{[k]} arranged in a circle to form a hyperboloid of revolution, the white beams creating a shape like hands praying to heaven. Only the dome is visible from outside: most of the building is below ground.^{[96]}^{[97]}^{[98]}^{[99]}

Several medieval churches in Scandinavia are circular, including four on the Danish island of Bornholm. One of the oldest of these, Østerlars Church from c. 1160, has a circular nave around a massive circular stone column, pierced with arches and decorated with a fresco. The circular structure has three storeys and was apparently fortified, the top storey having served for defence.