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Geometry of Architecture: Exploring the Relationship Between Math and Design

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Geometry is the backbone of architecture, providing the framework for buildings and structures. Discover more about Geometry of Architecture in our article.
Building Exterior

Have you ever stood in front of a building in awe and wondered how the architect made it look so great?

The answer lies in the geometry of architecture. The connection between geometry and architecture has been around for thousands of years and will continue in the future.

In this blog post, we will explore:

  • The connection between geometry and architecture
  • The origin of geometry in architecture
  • How architects have used geometry of architecture in the past and now.
  • The most famous geometric concepts and shapes used in architecture.

Let’s start by asking: What is geometry in architecture?

What is Geometry in Architecture?

To understand geometry of architecture, we must first understand what geometry is.

Geometry is the branch of mathematics that studies shapes, sizes, and positions of objects in 2D and 3D. So what is geometry of architecture?

Geometry of architecture is the art and science of designing buildings based on geometric rules. Geometry of architecture isn’t just about how a building looks from the outside (the building exterior); it’s also about how we experience and plan its interior spaces.

I think appreciating great geometric compositions is in our genes as humans. I believe this is why, when designing, we strive to achieve perfect geometric arrangements. — It’s not on purpose, but we strive toward geometric shapes.

To understand this inherited tendency to appreciate great geometric compositions of architecture, we have to go back in time and see:

What is the origin of geometry in architecture? and How did our ancestors appreciate geometry of architecture?

The origin of Geometry in architecture

To understand the geometry of architecture, we first need to understand the origin of geometry itself.

Vitruvian Man
Vitruvian Man

The origin of Geometry

In his book “De Architectura,” Vitruvius, the ancient Roman architect and writer, famously claimed that buildings that are good have three conditions: commodity, firmness, and delight.

  • Commodity is a building’s ability to fit its purpose
  • Firmness is its ability to stand up.

But the third condition,”delight,” is probably the most subjective and hard to measure.

Vitruvius believed that the path to delight was to create structures connected with nature. And geometry was our primary tool for understanding nature’s constructions. We were able to learn about things like symmetry and the sizes of things’ parts in relation to each other from nature.

This understanding of geometry opened the gate to apply it to human creations, including our finest works of art.

One of the most famous examples using geometry in the artistic realm is Leonardo da Vinci’s Vitruvian Man.

This drawing attempts to figure out how the human form can be simultaneously inscribed within a square and within a circle.

One question arises in this process is whether geometry can be used to describe the human form or if the human’s form is actually born from geometric principles?

But geometry isn’t just essential for understanding the human form, but also architecture.

The origin of Geometry in Architecture

The landmark of trevi fountain in Italy
Trevi Fountain

Geometry has been a part of architecture since ancient times.

But during the Renaissance, geometry became a more formal subject and was more fully integrated into architecture.

Architects such as Leon Battista Alberti and Filippo Brunelleschi, who were active during the 15th century, were particularly interested in the mathematical principles underlying architecture and used geometry to design buildings that were more structurally sound and aesthetically pleasing.

Renaissance-era architect Alberti believed that geometric principles govern nature and all of art.

Alberti is the first one to think of architect as a separate person from a builder or carpenter.

He wanted to raise the status of architecture to be on par with the other fine arts, so it was essential to distinguish the person who knew the buildings through study, intellect, and geometry. Not just by building them.

In the 17th century, the German mathematician and astronomer Johannes Kepler made one of the most important contributions to geometry of architecture.

Kepler’s work on the laws of planetary motion showed how important geometry is for understanding the physical world. These laws made architects think differently about how to design buildings.

Historical use of Geometry of Architecture

The Great Pyramid of Giza
The Great Pyramid of Giza

The ancient Egyptians, Greeks, and Romans appreciated geometry of architecture like no other. You can see it evident in their buildings and structures still standing today.

The great pyramid of Giza is just one marvelous example. We have no idea how they built it; its construction remains a mystery to scholars and historians, but one thing we are sure of is that it is a true geometric wonder:

  • The great pyramid base is a perfect square, each side measuring about 230 meters (755 feet), with an error of only a few centimeters.
  • The pyramid’s sides are inclined at an angle of 51 degrees and 51 minutes, forming an almost perfect pyramid shape.
  • The pyramid’s interior is also geometrically fascinating. Its descending passage is aligned with the pyramid’s north face and is precisely 26.6 degrees downward, reflecting the angle of the pyramid’s slope.

The appreciation of geometry is not only limited to the ancient Egyptians but also extends to almost every nation and culture.

Let’s look at some of these examples:

Examples of geometry in Architecture

1️⃣ The Parthenon:

The Parthenon
The Parthenon

The Parthenon in Athens, Greece, is another famous example of geometry of architecture. The use of geometry is evident in the harmony and balance of the temple design. For example:

The building’s columns, friezes, and pediments are all based on precise geometric calculations, with the ratio of the building’s height to its width being a perfect 9:4. 

2️⃣ The Taj Mahal:

The Taj Mahal
Taj Mahal

The Taj Mahal in India is one of the best examples of geometry of architecture. Many people even take it a step further and think of Taj Mahal as the most beautiful buildings in the world due to its geometry. 

Geometry is evident In Taj Mahal, from the tiniest details, such as patterns, to the largest, such as the building’s symmetry.

Let’s move on to the now: How do architects use geometry today?

How do architects use geometry?

The use of geometry in modern architecture is not as strict as in the past.

Architects today see geometry as a tool rather than a master. 

They still follow the rules of geometry to create balanced and harmonious designs, but they also try to break and bend the rules whenever possible. Why has this happened?

In my opinion, I think this transformation happened for two reasons:

1️⃣ Technology advancement 

Today’s architects have access to advanced technology, and building techniques allow them to make more complex and creative designs.

2️⃣ The evolution of architecture

Architecture today is not limited to traditional, rigid forms. Architects now frequently use curves, irregular shapes, and organic forms to create more dynamic buildings. These designs often require more flexibility and creativity than strict geometric forms.

Geometry is still crucial in architecture — But it’s no longer the master. Instead, it’s just one of the tools that architects use today.

What are some other tools that architects use to get ideas besides geometry?

Architects today draw inspiration from various sources and tools, not just geometry.

1️⃣ Nature

Geometry in aloe swirl plant
Aloe Swirl Plant

Nature is a significant source of inspiration for architects. Architects often draw inspiration from their shapes, patterns, and colors.

For example, an architect might look at how a tree branches out and create a building based on it rather than using geometry.

2️⃣ 3D Softwares

Thanks to technological advancements, modern architects can use various 3D modeling programs to generate limitless ideas, create parametric solutions, and simulate how a building will look under different conditions.

This freedom allowed architect to raise above traditional geometry

3️⃣ The Rise of Ai

Artificial Intelligence is the futureThere is no doubt in that

AI is already drastically changing every single field; architecture is no exception.

You already see interesting architectural concepts generated by AI. Some started to call themselves “AI architects” in their bios.

Architects now use AI technology to come up with new ideas and concepts that didn’t exist in nature before.

So, why do architects still use geometry if they have so many tools now?

Importance of geometry in Architecture

1️⃣ Structural integrity

First and foremost, geometry is essential to ensuring the structural integrity of a building. Architects use mathematical principles and geometric calculations to ensure that structures are safe, stable, and resistant to external forces such as wind and earthquakes.

2️⃣ Sense of proportion and scale

Geometry also plays a critical role in creating proportion and scale in architecture. By using geometric principles such as the golden ratio, architects can create harmonious and balanced designs, creating a visually appealing experience for the user.

3️⃣ The user experience

Geometry isn’t just about the visual appeal but also affects the user experience of the building.
Geometry can help to create functional floor plans that allow for optimal flow and use of space.

So now that we know the importance of geometry in architecture:

what are the most famous geometric concepts architects use in their buildings?

Geometrical Concepts in Architecture

Various geometric concepts make up the architecture we admire today. Let’s define and talk about some of the most common ones used in architecture:

1️⃣ Symmetry 

La galleria Vittorio Emanuele II
La galleria Vittorio Emanuele II

Symmetry is a geometrical concept that architects use to create balance and harmony in architecture. 

Symmetrical designs feature a balanced composition of shapes and patterns on either side of a central axis. 

You can see it in buildings like the Taj Mahal.

The Taj Mahal is famous for its perfectly symmetrical design, with identical structures and gardens on either side of the central mausoleum.

You can learn more about symmetry in design in these two article:

Balance in Design & Asymmetrical balance

2️⃣ Tessellation

Geometry of architecture represented in Alhambra Palace
Alhambra Palace

Tessellation is when you repeat the same shape to fill a plane without gaps or overlaps. 

Architects use tessellation to create intricate and detailed patterns in a building’s design. 

A famous example of tessellation in architecture is the Alhambra in Granada, Spain, which features stunning tessellated patterns in its intricate tile work.

3️⃣ Golden Ratio 

The Golden Ratio is a mathematical concept that architects often use to create proportion and balance. 

The ratio is approximately 1.618, which we believe to be the most visually pleasing proportion for the human eye. 

The golden ratio has geometric shapes that architects use in their buildings, like the Parthenon in Athens, Greece, which has a facade with Golden Ratio proportions.

Geometric Shapes in Architecture

Geometric shapes in architecture are more than just an aesthetic choice. Each geometric shape has its own functional and structural implications that can impact the overall design of a building.

Let’s start with a famous shape used in architecture: the rectangle.

1️⃣ Rectangles 

Rectangles are the most common geometric shape used in architecture. 

They are simple and easy to work with, which makes them a popular choice for building design. 

Architects use rectangles to make clean lines and a sense of order in a design. 

Rectangles are also the most common shape in internal spaces such as rooms, hallways, and corridors. Examples of buildings that use rectangles in their design include the Empire State Building in New York City and the National Gallery of Art in Washington, D.C.

2️⃣ Triangles

Triangles is the most powerful geometric shape because it can’t be distorted in any direction. Architects use triangles more in landmarks and public installations than in buildings such as the Eiffel Tower in Paris or the Pyramids.

3️⃣ Circles and Spirals

The Guggenheim Museum
The Guggenheim’s design is as famous as the collection of art it contains

There are no perfect circles in the world, made by people or by nature. Even the sun or planets are quite circular – they are oblong spheroids.

Circles and Spirals are not as common in architecture as rectangles, but they still have their place.

Architects use circles to create a sense of continuity and flow in a building’s design.

Examples of buildings that use circles in their design include the Guggenheim Museum in New York City.

4️⃣ Squares 

Squares are similar to rectangles in that they are easy to work with and create a sense of order in a building’s design.

But because squares have interesting geometric features, they can make an interesting contrast when used with other shapes like triangles or circles.

5️⃣ Hexagons

Architects don’t use hexagons as much as they do other shapes, but they’ve become more popular in recent years.

Hexagons add uniqueness and interest to a design when used correctly.

For example, see the Honeycomb Apartments in Slovenia and the Denver Art Museum in Colorado, which include hexagons in their design.


1️⃣ Video

Want to learn more about the fascinating connection between geometry and architecture? Check out this YouTube video


In conclusion, there has been a link between geometry and architecture for thousands of years, and it will remain important in the future. Geometry of architecture is the art and science of designing buildings based on geometric rules, and it’s not only about how a building looks from the outside but also about how we experience and plan its interior spaces.

Picture of Bahaa Aydi

Bahaa Aydi

I'm Bahaa, a Licensed architect specialized in Interior design & Archviz • Sharing my design days with you to help you design better space