You, my friend, are living in a world created by triangles.

Since the dawn of time, this humble three-sided polygon has quietly shaped the evolution of human civilisation. As you gaze around, you can likely spot a triangle or two tucked within your surroundings. This may be of no surprise to you. Externally, the triangle governs the material construction of our world, underpinning the foundations of countless engineering and architectural designs.

Yet these more obvious physical constructions are just one contribution of this pointy, three-sided shape to modern society. Indeed, it is where the role of the triangle remains invisible that it harnesses the most power.

Triangles have played an integral role in sailing and modern navigation systems, having enabled us to explore all corners of the Earth. Beyond this, let us not forget the massive contributions this shape has made to the development of 3D modelling, used everywhere from graphic design and animation to CGI. All thanks to the simple, unassuming triangle.

The physical, the navigational and the digital. Three key sides of the triangle’s influence in shaping the modern world.

The Physical

The triangle's importance in the physical world stems from its inner strength. Unbeknownst to many, it is the strongest two-dimensional shape that exists, with its power amplified in three-dimensional polyhedrons derived from triangles. How can this unique strength be explained? 

Consider applying force to one corner, or apex, of a triangle. This force is distributed down either side of the triangle and as these sides are compressed, the base is stretched outwards. Weight can therefore be evenly dispersed across the shape, preventing it from bending and breaking (Saint Louis Science Center, 2020). It is for good reason that the triangular shape underpins many fundamental principles of architecture and design.

Perhaps the most iconic of the structures that utilise this shape are the Pyramids of Giza, one of the Seven Wonders of the Ancient World. Constructed in the early 25th Century BCE, they housed the tombs of ancient Egyptian pharaohs and are the last remaining Wonder that exists today. The tallest of the Pyramids, known as the Great Pyramid, originally soared as high as 147 metres above the ground, though today erosion has reduced it to 138 metres (Encylopedia Britannica, 2024a). This architectural feat was monumental for its time, and to this day, how exactly the Pyramids were constructed remains a hotly contested debate amongst archeologists and engineers. One proposition is that large ramps were used in conjunction with a complex system of ropes, sledges and levers to haul stone blocks up (Handwerk, 2023). Whatever the method of construction may have been, these ancient wonders have stood the test of time for over 4500 years - a remnant of one of humanity's first advanced civilisations that exemplifies the scale, strength and resilience of construction made possible by triangles.

Triangles also play a crucial role in the construction of seemingly dissimilar shapes. This is highlighted in the case of geodesic structures - spheres constructed from a network of triangles approximating a rounded shape, like a soccer ball. First developed in the 20th Century by architect Richard Buckminster Fuller, these domes are lightweight and able to distribute stress across large, arching structures (Encylopedia Britannica, 2024b). Since Fuller’s earliest constructions, these domes have been widely utilised in the construction of stadiums, planetariums and even "glamping" accommodations. One notable example is the Eden Project - the world's largest biodome botanical garden in the United Kingdom, housing thousands of plant species over 5.5 acres of land (Eden Project, 2024). The interconnectedness of the triangles allows for maximum sunlight exposure across wide spaces, creating an ideal environment for plant photosynthesis and cultivation.

Intriguingly, Fuller's use of triangles in this innovative manner led to a breakthrough in the far-away field of synthetic chemistry. Scientists Robert Curl, Harold Kroto and Richard Smalley discovered the nanomaterial Buckminsterfullerene, or “the Buckyball”, after the scientists realised the structure's similarity to Fuller's geodesic spheres (The Stanford Libraries, 2024). This led to the discovery of a new class of materials known as fullerenes. The scientists were subsequently awarded the 1996 Nobel Prize in Chemistry for elucidating this molecule’s structure (The Stanford Libraries, 2024).

Balancing power with versatility, triangles form the crux of our built environments at both an atomic and architectural level.

The Navigational

Remember those sine and cosine formulas your maths teacher insisted had important real world applications? Turns out they weren’t kidding.

Triangulation is the process of finding an unknown location of an object by forming a triangle between this object and two other reference points. Sine, cosine and tangent, the main trigonometric ratios, are used to relate the sides and angles formed within a right triangle and hence, determine the position of an unknown point.

For centuries, humans have turned to triangles as a means to find their ways. Sailors, in particular, have long used landmarks and celestial objects like the stars to orient themselves at sea. By observing the angle between known locations (or stars) and using basic trigonometry, navigators could calculate distances and determine their precise location.

Moving to a more global scale of navigation becomes a bit more complicated, as the Earth is a sphere and not a flat surface (although some may beg to differ…).

A more advanced form of triangulation known as trilateration underpins the Global Positioning System (GPS) in order to determine three-dimensional coordinates of a receiver. Instead of angles, GPS utilises the time taken for radio signals sent from satellites to reach a receiving device on Earth. A connected system of navigation satellites circles the Earth, each sending out signals with the location and time it was sent by that satellite. By measuring the delay between the time of signal reception and the broadcast time, the distance from the receiver to each satellite can be computed (Federal Aviation Administration, 2024). Once distances to at least three satellites are known, the receiving device can determine its own three-dimensional position, employing similar techniques to triangulation.

GPS data is not only used to guide your Google Map directions. Analysing the positions of satellite stations and their movements is a crucial tool for monitoring volcanic and seismic activity (Murray & Svarc, 2017). Recent breakthroughs have even suggested that there may be a future for utilising the GPS to detect earthquakes before they happen (Rao, 2023).

From the seas to the skies, triangles allow us to push the boundaries of exploration while always guiding us home to safety.

The Digital

What does connect-the-dots have to do with triangles or 3D modelling? 

A connect-the-dots drawing begins with nothing but some labelled dots. Yet as each dot is joined by a straight line, a complex and curved picture emerges. The more dots you use, the smoother the picture looks. 

Consider now trying to design a three-dimensional surface. Just as you might use dots to approximate a curve, triangles serve as building blocks for constructing complex surfaces. By taking enough triangles and joining them at their edges, we too can approximate intricate and multidimensional structures. 

In 3D modelling, objects are represented as meshes - models consisting of vertices (points in 3D space) connected by edges to form polygons and thus, the surface of an object (Stanton, 2023).

To define a flat surface oriented in a plane, a minimum of three distinct points are needed. Triangles are the simplest shape for constructing these planes as they are coplanar, meaning any three points in space will always form a flat surface (Licata & Licata, 2015). This makes them perfect for modelling complex 3D shapes out of interconnected triangles. Animation, gaming, graphic design and computer generated imagery (CGI) in movies are just some of the many varied applications that utilise these mesh modelling techniques to create intricate 3D models, with curved and highly detailed surfaces.

Additionally, there exist efficient computer algorithms that are optimised to dissect objects into hundreds of thousands of flat triangles. A complex, digital representation of any object can therefore be easily portrayed as a simple collection of points and triangles. Combined with their simple geometric properties, triangles can then be processed quickly by modern Graphics Processing Units (GPUs), optimising their performance in real-time applications. Add in lighting, shading and smooth deformation, and you will find yourself with an intricate, three-dimensional model.

Pointing the Way Forward

For too long, the triangle has been overshadowed by its more popular cousin, the square. Yet, what is a square? Two triangles put together. The simplicity of this three-sided shape allows it to integrate within our society, with its contributions often invisible to the naked eye. From the physical, to the navigational and the digital, modern human society is built on the triangle. Maybe that trigonometry class wasn’t so pointless after all.

References

Eden Project (2024). Eden Project's Mission. https://www.edenproject.com/mission/origins

Encylopedia Britannica (2024a). Great Pyramid of Giza. https://www.britannica.com/place/Great-Pyramid-of-Giza

Encylopedia Britannica (2024b). Geodesic Dome. https://www.britannica.com/technology/geodesic-dome

Federal Aviation Administration (2024). Satellite Navigation - GPS - How It Works. United States Department of Transportation. https://www.faa.gov/about/office_org/headquarters_offices/ato/service_units/techops/navservices/gnss/gps/howitworks

Handwerk, B. (2023). The Pyramids at Giza were built to endure an eternity—but how? National Geographic. https://www.nationalgeographic.com/history/article/giza-pyramids

Licata, J., & Licata, A. (2015). From triangles to computer graphics. ABC Science. https://www.abc.net.au/science/articles/2015/06/10/4251713.htm

Murray, J. R., & Svarc, J. (2017). Global Positioning System Data Collection, Processing, and Analysis Conducted by the U.S. Geological Survey Earthquake Hazards Program. Seismological Research Letters, 88(3), 916-925. https://doi.org/10.1785/0220160204

Rao, R. (2023). GPS satellites may be able to detect earthquakes before they happen. Space. https://www.space.com/earthquake-prediction-gps-satellite-data

Saint Louis Science Center (2020). The Secret Strength of Triangles. https://www.slsc.org/the-secret-strength-of-triangles/

Stanton, A. (2023). Exploring the World of 3D Modeling: Solid vs. Mesh Modeling. Cadmore. https://cadmore.com/blog/solid-vs-mesh-modeling-differences

 The Stanford Libraries (2024). What is a geodesic dome? Stanford University. https://exhibits.stanford.edu/bucky/feature/what-is-a-geodesic-dome