Mathematics is an area of knowledge which is usually considered being very theoretical, one may wonder how a specific branch of mathematics called geometry relates to the visual arts. In general, the Merriam Webster dictionary defines geometry as a mathematical “branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids” In easier terms, geometry deals with shapes, both two-dimensional and three-dimensional shapes, as well as the general relation between points, lines and three-dimensional shapes. All mathematical processes can be graphed in a two-dimensional plane (Euclidean space) or a three-dimensional plane. This brings us closer to how visual mathematics and geometry …show more content…
This means that the placement of the various elements in a piece of art, the relationship between those shapes, colours and proportions is what is called composition. Whereas most artists would simply arrange the composition by aesthetic preference or conceptual thought, geometric ratios can be found in the composition of most artwork.
Fibonacci Sequence:
The Fibonacci sequence is a mathematical sequence in which “each number is the sum of the two preceding numbers” The sequence was found by an Italian mathematician called Fibonacci, thus the name Fibonacci sequence. This sequence can be visualised by showing each number as a rectangle and drawing a line through the two opposite sides of the rectangles.
Figure 1: The Fibonacci
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The golden ratio can be found anywhere in nature and can be used by artists to create more natural movement and composition. Composition is in fact the same as the relation between objects which is part of the definition of geometry.
Perspective
Geometry can be used to create an effect that is very hard to achieve for many artists. Many artists illustrate three-dimensional objects or sceneries on two-dimensional surfaces. Using geometric ratios, artists can successfully create depth as well as realistic perspective.
One-point perspective:
One-point perspective is a mathematical method to “represent[] three-dimensional objects and space on a two-dimensional surface by means of intersecting lines that are drawn vertically and horizontally and that radiate from one point”. The horizon line is often illustrated as a horizontal line of the page. On the horizon line is a point where all the lines converge onto one single point. This can be better illustrated by looking at a drawing created using one-point
For me geometry is the most basic concern for artists. Roland Shearer quotes poet Apollinaire where he explains, “geometry is to the plastic arts what is to the art of the writer”. This is not to say that artists are geometers, because most of us are far from it (Shearer 1992:143).
Mathematics is the central ingredient in many artworks. While notions of infinity and parallel lines brought “perspective” to the artistic realm in creating realistic representations of depth and dimension, mathematics has influenced art in a more definite way – by actually becoming art. The introduction of fractal geometry and tessellations as creative works spawned the creation of new and innovative genres of art, which can be exemplified through the works of M.C Escher. Escher’s pieces are among the most recognized works of art today. While visually stimulating and deeply meaningful, his art reflects many ideas of mathematics through geometry, symmetry, and patterns.
Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between Euclidean, and Non-Euclidean Geometry is the assumption of how many lines are parallel to another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that line.
An example of the difference in the abstract geometry and the measurement geometry is the sum of the measures of the angles of a trigon. The sum of the measures of the angles of a trigon is 180 degrees in Euclidian geometry, less than 180 in hyperbolic, and more than 180 in elliptic geometry. The area of a trigon in hyperbolic geometry is proportional to the excess of its angle sum over 180 degrees. In Euclidean geometry all trigons have an angle sum of 180 without respect to its area. Which means similar trigons with different areas can exist in Euclidean geometry. It is not possible in hyperbolic or elliptic geometry. In two-dimensional geometries, lines that are perpendicular to the same given line are parallel in abstract geometry, are neither parallel nor intersecting in hyperbolic geometry, and intersect at the pole of the given line in elliptic geometry. The appearance of the lines as straight or curved depends on the postulates for the space.
On first thought, mathematics and art seem to be totally opposite fields of study with absolutely no connections. However, after careful consideration, the great degree of relation between these two subjects is amazing. Mathematics is the central ingredient in many artworks. Through the exploration of many artists and their works, common mathematical themes can be discovered. For instance, the art of tessellations, or tilings, relies on geometry. M.C. Escher used his knowledge of geometry, and mathematics in general, to create his tessellations, some of his most well admired works.
A. Line – An irregular soft line is used to establish the eye level/horizon in background between the sky and the field. The diagonal lines define the field and are in an opposite direction of the diagonal lines that define the trunk off the tree in the middle of the picture. The diagonal lines in the field, the horizon line, and the edge of the painting create a linear perspective.
When looking at an art piece such as a landscape oil painting by Albert Bierstadt—American artist who created Yosemite Valley. “In 1859, he traveled westward in the company of a land surveyor of the U.S. government, returning with sketches that would result in numerous finished paintings” (wikipedia). The artist shows incredible attention to detail throughout the landscape. The use of reflections is extremely effective and visually accurate; the reflections in the water—clarity of trees, shrubs, pebbles, and rocks—might be one of the most noticeable features in this piece. The relative brightness creates the warmth felt in this painting. The interposition as well as the allusion of three dimensions provides a high level of depth. Aerial perspective is correctly used, furnishing the effect of distortion—the...
Mathematics is everywhere we look, so many things we encounter in our everyday lives have some form of mathematics involved. Mathematics the language of understanding the natural world (Tony Chan, 2009) and is useful to understand the world around us. The Oxford Dictionary defines mathematics as ‘the science of space, number, quantity, and arrangement, whose methods, involve logical reasoning and use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis of mathematical operations or calculations (Soanes et al, Concise Oxford Dictionary,
Euclidean Geometry is the study of plane and solid figures based on the axioms and theorems outlined by the Greek mathematician Euclid (c. 300 B.C.E.). It is this type of geometry that is widely taught in secondary schools. For much of modern history the word geometry was in fact synonymous with Euclidean geometry, as it was not until the late 19th century when mathematicians were attracted to the idea of non-Euclidean geometries. Euclid’s geometry embodies the most typical expression of general mathematical thinking. Rather than simply memorizing basic algorithms to solve equations by rote, it demands true insight into the subject, cleaver ideas for applying theorems in special situations, an ability to generalize from known facts, and an
Lines are paths or marks left by moving points and they can be outlines or edges of shapes and forms. Lines have qualities which can help communicate ideas and feelings such as straight or curved, thick or thin, dark or light, and continuous or broken. Implied lines suggest motion or organize an artwork and they are not actually seen, but they are present in the way edges of shapes are lined up.
According to the Merriam-Webster Dictionary “Fibonacci Numbers are integers in the infinite sequence 1, 1, 2, 3, 5, 8, 13 … of which the first two terms are 1 and 1 and each succeeding term is the sum of the two immediately preceding.” This is also known as a recursive sequence, because each number is a function of the preceding two numbers. Fibonacci numbers are contained in a few formulas. They are also numbers in shapes, such as triangles and rectangles. Fibonacci number is found throughout nature, art, music, and science.
The foundations of mathematics are strongly rooted in the history and way of life of the Egyptian people, dating back to the fourth millennium B.C. in Egypt. Egyptian mathematics was elementary. It was generally arrived at by trial and error as a way to obtain desired results. As such, early Egyptian mathematics were primarily arithmetic, with an emphasis on measurement, surveying, and calculation in geometry. The development of arithmetic and geometry grew out of the need to develop land and agriculture and engage in business and trade. Over time, historians have discovered records of such transactions in the form of Egyptian carvings known as hieroglyphs.
The Fibonacci Series was discovered around 1200 A.D. Leonardo Fibonacci discovered the unusual properties of the numeric series, that’s how it was named. It is not proven that Fibonacci even noticed the connection between the Golden Ratio meaning and Phi.
This means that math work with numbers, symbols, geometric shapes, etc. One could say that nearly all human activities have some sort of relationship with mathematics. These links may be evident, as in the case of engineering, or be less noticeable, as in medicine or music. You can divide mathematics in different areas or fields of study. In this sense we can speak of arithmetic (the study of numbers), algebra (the study of structures), geometry (the study of the segments and figures) and statistics (data analysis collected), between
Throughout math, there are many patterns of numbers that have special and distinct properties. There are even numbers, primes, odd numbers, multiples of four, eight, seven, ten, etc. One important and strange pattern of numbers is the set of Fibonacci numbers. This is the sequence of numbers that follow in this pattern: 1, 1, 2, 3, 5, 8, 13, 21, etc. The idea is that each number is the sum of its previous two numbers (n=[n-1]+[n-2]) (Kreith). The Fibonacci numbers appear in various topics of math, such as Pascal?s Triangle and the Golden Ratio/Section. It falls under number theory, which is the study of whole or rational numbers. Number Theory develops theories, simple equations, and uses special tools to find specific numbers. Some topic examples from number theory are the Euclidean Algorithm, Fermat?s Little Theorem, and Prime Numbers.