Geometry & Art
Link to full Project Presentation
The purpose of this project is to investigate how geometric transformations, including translation, rotation, reflection, scaling, and their combinations, are employed to produce unique patterns and how these patterns have been used in contemporary art both historically and currently. The aim of this project is to bridge the gap between geometry and art, demonstrating how geometric principles not only support the production of aesthetical patterns but also reflect deeper aesthetic, cultural, and philosophical values.
This project explains the use of geometric patterns in art, looking at both classical and contemporary applications. It also includes artworks and visuals from the past to the present day, which explore the use of geometric patterns in art.
Figure 1: Mathematics in art: A. Dürer's copper plate engraving Melencolia I, 1514
Introduction to Geometry in Art
Artists have used geometric shapes in different cultures from past to present. Geometric patterns are used in a wide variety of art fields, including architecture, interior design, fine arts, textiles, ceramics, and metalwork. Complex designs are usually created by combining, repeating, and using symmetrical shapes such as squares, circles, and straight lines. These patterns establish a connection between geometry and art [2].
Ancient Egyptian architecture used geometric shapes on temple walls, ceilings, and columns. However, the fact that geometric patterns are also seen in ordinary objects shows the Egyptians’ interest in complex geometry. The Great Pyramid of Khufu and the Saqqara Pyramid are among the most important geometric structures of Egypt. This shows us that geometry is used for art and also shows that geometry is important for society and structural success.
Figure 2: Great Pyramid of Khufu
Figure 3: Sakkara Pyramid
Especially the geometric patterns used in the sculptures, vases, kilims and mosques of the Greek and Ottoman periods in the past are remarkable [3][4][5]
Figure 4: The Dipylon Amphora
Figure 5: Islamic Kilim
The collision of geometry and art has been used to shape compositions in everything from architecture to abstract paintings. Shapes such as circles, squares, and triangles are used, especially for balance and symmetry. Geometry, which has been used in many different cultures, shows that patterns can also have spiritual meaning in different religious works of art, such as mosques.
Although geometric shapes were used to add depth to two-dimensional art in spaces before the 20th century, the use of geometric shapes in art has decreased in the 20th century. However, today, geometry has started to be used again in digital and algorithmic art [6].
Figure 6: Algorithmic Art by sprit11 on Pixabay
Geometric Transformations in Art
Changes to geometric figures that alter their size, shape, or position are known as geometric transformations [8]. There are four main types of geometric transformations: translation, rotation, reflection, and dilation, which are also used in art.
Translation is the type of geometric transformation that is shifting an object in a specific direction without rotating or changing its size. Rotation is the type in which the object turns around a fixed point called the “center of rotation”. Rotation is also known as a circular pattern in art since it turns around a fixed point. An object is flipped across a line known as “the line of reflection” known as reflection in geometric transformation. It brings symmetry to the design by mirroring the shapes and lines. Dilation is the type in which we enlarge or reduce the size of an object without shanging its shape [8][9].
Figure 7: Translation, rotation, reflection, and dilation of an object.
Maurits Cornelis Escher, a mathematician who lived in the 19th century, made use of geometric transformations in his art works. One of his most important efforts where transformations can be seen clearly is the Day & Night, which can be seen in Figure 8.
Figure 8: Day&Night, Wood Cut Print
Historical Use of Geometry in Art
The cuneiform of the Summers is considered the starting point for the use of geometry in art, architecture, design, mosaics, and textiles. The Summers civilization, which lived in ancient Mesopotamia between 5000-3000 BC, is considered the first ancient geometry examples with cuneiform, which is considered the first written system on clay tablets [10]
Figure 9: Cuneiform Geometry Problems
After the Egyptians began using mathematics for land surveying and construction, they used geometric shapes in their structures, such as the pyramids. Although the Greeks had geometric shapes later, Euclid's Theorem, written by the Greek Euclid in 300 BC, which carries the characteristics of geometry, continues to be used today [11]. The Greeks were one of the first civilizations to apply geometric principles to sculpture, textiles, and decorative vases, as well as to more traditional arts such as painting. Geometric designs were also widely used, especially in their vases.
Figure 10: Egypt Pottery
Figure 11: Greek Decorative Art
Geometric shapes and transformations were used especially in mosaics during the Roman period. Today, there are many works belonging to different civilizations from past periods in the Eastern Anatolia Region, which is within the borders of Mesopotamia. Some of them can be visited in the Gaziantep Mosaic Museum.
Figure 12: Roman Mosaics (Photos taken from Mosaic Museum, Gaziantep, 2024)
In Islam, human figures in art began to be replaced by animals and geometric shapes. These figures are often seen in buildings, places of worship, ceramics, girih tilings, woodwork, and rugs [5].
Transformation techniques used in the past to use geometric shapes in different architectures and artworks are realized by following a certain algorithm when viewed from the perspective of informatics. Today, the relevant techniques may be realized with programming languages with the help of user-ready libraries and software. It is seen that the arkworks produced in this way appear as three-dimensional as well as two-dimensional works [12].
It is evident that each of the works depicted in the figures above uses a variety of geometric shapes and approaches.
Geometric Transformations in Contemporary Art
At the beginning of the 20th century, cubism was an art movement that began in Paris and revolutionized painting and visual arts [14]. In cubism, by using geometric transformation methods in the objects used, by changing their size and position, three-dimensional artworks can be created instead of two-dimensional. One of the pioneers of this movement is Pablo Picasso. He drew attention with his three-dimensional artworks, especially with the geometric shapes he used [15].
Figure 14: Three Musician, Pablo Picasso
Figure 15: Stars, M.C. Eschers
Maurits Cornelis Escher is one of the artists who used mathematics in his fascinating arts by using basic patterns in his tessellations, applying translations, reflections, and rotations to obtain a greater variety of patterns. He elaborated these patterns into animals, birds, and other figures [17].
In mathematics, a fractal is defined as “a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension[18]”. Benoit Mandelbort was one of the first mathematicians who used fractals in computer graphics at the end of the 20th century. He transformed the fractals that mathematicians had been working on since the 17th century into computer graphics, thus paving the way for today’s algorithmic art.
Figure 16: Mandelbrot set with 12 encirclements
Optical art is created by repeating geometric shapes in black and white. This method brings the dynamics of sight, creates movement and color, and causes the eye to get confused.Bridget Riley is one of the famous artists who uses this technique in her art works [20].
Figure 17: Square movement
Today, with the development of computer systems, three-dimensional historical artifacts can be created by using geometric shapes in an algorithmic way, which is called visual art [21].
Conclusion
We can see that geometry has been used by many artists in many different ways from the past to the present, simply by using only translation, rotation, reflection, and dilation methods. It is quite impressive to see that in every period of history, geometric shapes and mathematics could be used to move from two-dimensional to three-dimensional works, and that the shapes used actually added depth to the works.