The Mathematics of Design – Fibonacci, Fractals & Polyhedra
It is common for people to group themselves into two categories; those who are good at art or design, and those who are good at math or science.
The reason for this is that many people believe that the skills needed to be successful in creative services do not relate to the skills that are required to be successful in the analytical.
Even though mathematics is on the list of the most challenging college classes, in reality, nothing could be further from the truth.
Many design concepts, such as symmetry, have direct ties to mathematical concepts and discoveries.
So, if you are an artist or a designer, there is a good chance that you are already incorporating math into your work, you just may not be doing it consciously.
Here are some very specific ways that the mathematics of design has not only influenced brands but has acted as a true game-changer.
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The Fibonacci Sequence
Consider this sequence numbers; 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.
This special sequence of numbers is what is known as the Fibonacci sequence.
If you take a closer look, you will notice that every number, after the seed numbers 0 and the first 1, is the sum of the two numbers prior.
If you were to continue this pattern, you would add 55 and 34 to make 89.
At first glance, it may be difficult to see how any of this relates to art, but in fact, this sequence plays a significant role in art and design.
Imagine if each number is a shape. Let’s say the second number 1 represents a square that is a 1-inch square. Then, let’s say that the 55 is a 55-inch square.
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The combination of these squares can be used to form what is known as the golden rectangle.
You can find this rectangle in paintings such as the Mona Lisa and both classical and modern architecture.
Many web designers incorporate the golden rectangle in their designs by using the PHI calculator.
This calculator will help you to determine, for example, if you were to put the logo here (use your imagination) how the other elements of your page should be placed and sized.
Of course, this is not only limited to web design work. You could apply this to any design project.
Now, imagine the shapes representing circles. When arranged in certain ways, these can form the basis of starbursts, flowering patterns, branching, and more.
A perfectly formed spiral is based on the Fibonacci sequence. Not only do these things occur in art, but they also occur in nature.
In fact, the golden ratio, which is the reduction of numbers in the Fibonacci sequence that comes as close to zero as possible can be used to create the perfect spiral.
Many of Leonardo Da Vinci’s works incorporated the golden ratio in the mathematics of design. Even the human face closely follows the Fibonacci Sequence.
Other works that contain the golden ratio include the Great Pyramids and the Parthenon. Some people even believe that the Apple computer logo uses the Fibonacci sequence, but experts dispute this claim.
Fractals
Fractals are repeating patterns that can be created via the mathematics of design, but also appear in nature.
The most notable characteristic about fractals is that the repeating pattern can be noticed regardless of scale.
To better understand this, imagine using a microscope and turning up the magnification on an object.
With fractals, no matter how closely you zoom in, you continue to see the same repeating patterns. This is known as self-similarity.
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However, not all self-similarity is the same. For example, exact self-similarity means that the patterns within the fractal are perfectly identical.
On the other hand, quasi-self-similarity means that the patterns very closely resemble one another, but not perfectly.
Fractals can be created using computer software and mathematical formulas, but some of the most compelling examples of fractals come in nature. Here are just a few examples:
● Crystals ● Snowflakes ● Blood Cells ● Citrus Fruit ● Waves ● Mountain Ranges ● Fault Lines ● DNA ● Pineapple ● Animal Color Patterns
You can see an example of fractal patterns when you look at a window after a light frost. The similar patterns that repeat in the ice crystals are fractals.
In art, some of the best-known examples of the use of fractals can be found in Jackson Pollock’s paintings.
In fact, computer analysis has been done on his works that have determined the presence of fractals. This is in spite of the fact that at first glance, they often appear to be quite random.
It is believed that his particular style of creating his paintings is the cause of this. In spite of this, for the most part, fractals in the art are created digitally. In fact, digital art and animation rely on fractals as their foundation.
Web designers and graphic artists frequently use fractal images. If you have seen repeating patterns in background images on websites, for example, these are based on fractals in many cases.
Many believe that the repeating patterns in fractals are both soothing and aesthetically pleasing.
If you would like to learn how to create fractals and how to use them in your designs, there are plenty of tools and examples available to you on the internet.
Polykleitos Canon
Polykleitos was a famous Greek sculptor. In fact, he is widely regarded as one of the fathers of the classic Greek style of sculpture seen in many museums around the world.
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His contributions do not stop there. He also wrote a Canon on symmetry regarding the sculpting of the male form. This canon represented his thoughts on aesthetics and artistic perfection.
He believed that all parts of the body appearing in a sculpture should be clearly distinct from one another, and proportioned using mathematics.
For example, the size and proportion of the first knuckle of the little finger would be the basis for which the proportion of the rest of the finger, the finger would be the basis of the proportion of the palm of the hand, and so forth.
This was done by treating the tip of the finger as a square unit of measurement.
His work at determining the best ways to reach a perfect balance in sculpture is still influential today. In fact, Polykleitos son, and namesake grew up to be a famous architect.
It is hard to imagine that his father’s work on proportion did not influence him greatly.
Polyhedra
A polyhedron is a three-dimensional structure that consists of a collection of polygons that are joined along their edges.
Polyhedra have been incorporated into art and design for centuries. For example, Salvador Dali’s painting of The Last Supper depicts Jesus and his disciples within a dodecahedron, which is a type of polyhedron.
A polyhedral net is a net that has been unfolded for printing. Another one of Salvador Dali’s works, Corpus Hypercubus uses a polyhedral net as the cross.
Polyhedra was specifically addressed in the book Education on Measurement written by Albrecht Durer.
He was a German printmaker during the Renaissance, who wrote the book to educate others about perspective, While some of his thoughts on that subject were a bit off base, his discussion of polyhedra and polyhedral nets were quite insightful.
These nets contain fascinating patterns that can often be seen in mosaics and other art forms where geometric patterns are common.
Mathematics of Design And Other Art Forms
Even if artists and designers do not use mathematical formulas in their work, the results often reflect using the mathematics of design.
If you notice symmetry, geometric patterns, balance, or proportion on a website, there is a mathematical principle behind those things.
Even the use of positive and negative space has mathematical origins. What’s even more interesting is that these principles do not only impact images.
Even the way that text is displayed on a screen can involve the use of mathematics.
For example, the principle that text is more readable when there is plenty of white space is primarily the use of positive and negative space to create something that is more pleasing to the eye and easier to process.
Conclusion
Mathematics goes hand in hand with art and design because the human brain appreciates and understands consistency.
It is for this reason, that symmetry, balance, exactness, and proportion are such important parts of a design.
It is also why patterns frequently appear in art and design.
If you’ve ever visited a website or looked at a work of art that just seemed off to you, and was troubling to look at, there is a good chance that the artist or designer failed in one of these areas.
It is one thing when this is done by the mathematics of design, for example, painting something that lacks symmetry for effect.
Unfortunately, when it is simply the result of poor execution, it becomes problematic.
Hopefully, some of the information here will help designers to incorporate the mathematics of design principles into their work in more ways.
The results will likely be more aesthetically pleasing as well as being easier to use.
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.
Conclusion. Mathematics goes hand in hand with art and design because the human brain appreciates and understands consistency. It is for this reason, that symmetry, balance, exactness, and proportion are such important parts of a design. It is also why patterns frequently appear in art and design.
In fact, many of the core skills in art and math are closely related. Both disciplines require spatial reasoning skills and the ability to recognize patterns. Artists and mathematicians use geometry in their work — including shapes, symmetry, proportion, and measurement.
fractal, in mathematics, any of a class of complex geometric shapes that commonly have “fractional dimension,” a concept first introduced by the mathematician Felix Hausdorff in 1918. Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth.
Math is used by designers to coordinate and structure their designs. It is frequently used to determine the best use of space, where to place various pieces of furniture, or how much storage space is required. Area calculations are critical for this aspect.
Often, many artists benefit from mathematical observations, such as the golden ratio, to make their artwork realistic and breathtaking. Mathematics can also explain angles and perspective. Thus, there is a large amount of math involved in art.
The control systems can be represented with a set of mathematical equations known as mathematical model. These models are useful for analysis and design of control systems. Analysis of control system means finding the output when we know the input and mathematical model.
Fractal art is achieved through the mathematical calculations of fractal objects being visually displayed, with the use of self-similar transforms that are generated and manipulated with different assigned geometric properties to produce multiple variations of the shape in continually reducing patterns.
In combinatorics: Design theory. A design is a set of T = {1, 2, . . .,υ} objects called treatments and a family of subsets B1, B2, . . .,Bb of T, called blocks, such that.
Mathematics can help us to draw real-life objects. The regularity of natural patterns can lead artists to use mathematical concepts in works of art. Many plants have very interesting and beautiful leaves and some mathematical patterns can be found in their structures.
They are tricky to define precisely, though most are linked by a set of four common fractal features: infinite intricacy, zoom symmetry, complexity from simplicity and fractional dimensions – all of which will be explained below.
D = log N/log S. This is the formula to use for computing the fractal dimension of any strictly self-similar fractals. The dimension is a measure of how completely these fractals embed themselves into normal Euclidean space.
The Elements of Design are the things that artists and designers work with to create a design, or composition. The Elements are: line, shape, space, value, color and texture.
Everything you can see has a design. When you describe something you see, you use words that tell about the lines, shapes, colors, textures, and spaces. Line, shape, color, texture, and space are the basic elements of design.
Effective design centres on four basic principles: contrast, repetition, alignment and proximity. These appear in every design. This article provides a brief overview of the basic principles discussed in this series. Although the companion articles explore each principle separately, they are all interconnected.
Some graphic designers use fractal methods to reproduce very realistic-looking landscapes and other natural phenomena. These, too, seem to trigger something in our perception. We somehow recognize them as being “natural” and connect with them emotionally.
Maths is a necessary part in many undergraduate courses in graphic design. This covers units such as arithmetic, calculus, and perhaps perspective geometry and statistics.
Design Thinking is: A solutions-based approach to solving problems. An iterative, non-linear process. A way of thinking and working. Supported by a collection of strategies and methods.
Math is often found in art, but many mathematicians feel the math itself is art, that the formulas are elegant and the graphs are beautiful. “You find these problems to work on and they become fascinating,” Johnson said. “You want to solve the problems for their own sake and so you can share it with your colleagues.”
Some artists excel in maths. Leonardo da Vinci would be one famous example! But not all creators are Leonardo Da Vinci. An individual's lack of progress in math could be genetic, or a general lack of interest due to their environment or a bit of both.
Visual artists and painters, especially, draw upon mathematical concepts in their works, and some of the most famous works of art have mathematical elements. Architecture, sculpture, monuments and other forms of art also work with math in creative ways.
Mathematics makes our life orderly and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and even effective communication skills.
Even if they can, it is often simpler and faster to use a computational method to find a numerical solution. The real power of equations is that they provide a very precise way to describe various features of the world. (That is why a solution to an equation can be useful, when one can be found. )
Clouds, mountains, coastlines, cauliflowers and ferns are all natural fractals. These shapes have something in common - something intuitive, accessible and aesthetic.
Fractals provide a systematic method to capture the “roughness” of some objects. This method to capture roughness has uses in a wide variety of fields ranging from programming to medicine. Here in the Faculty of Mathematics, students and researchers are actively working on fractal geometry.
A fractal is a pattern that the laws of nature repeat at different scales. Examples are everywhere in the forest. Trees are natural fractals, patterns that repeat smaller and smaller copies of themselves to create the biodiversity of a forest.
A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process.
The three different types of shapes- geometric, natural and abstract. Geometric shapes are triangles, squares, rectangles and circles. Geometric shapes are regular and structured, and make excellent building blocks for design. Natural shapes are plant, animal or human, and are irregular and fluid.
In fact, it's what design thinking is all about: finding fresh, creative solutions to problems, but in a way that puts people and their needs first. ...
Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold in 1965. Spirograph.
There are many uses for drawing. Drawing is a form of communication that preceded writing and that continues to serve as another form of communication. "Drawings can do amazing things. They can tell stories, educate, inspire, reveal, entertain, and inform.
Etymology. The term "fractal" was coined by the mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.
A fractal is a geometric shape that has a fractional dimension. Many famous fractals are self-similar, which means that they consist of smaller copies of themselves.
Put simply, fractals are never-ending patterns made by the same shape repeated. Its size, position or angle may change but the shape is the same. Examples of Fibonacci numbers in nature include the numbers of petals in flowers, the spiral in a snail's shell and the spiral pattern on the outside of a pineapple.
Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.
A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.
The golden spiral is based on the golden ratio. Symbolised by the character φ (Phi), it's found when a line is split in such a way that the larger part divided by the smaller part is equal to the whole part divided by the larger part -- a ratio of (rounded) 1.618.
Starting with the simplest concepts, math will surround you from day one. Visualization is incredibly important to succeed as a designer, and you will need basic math skills to create credible visual explanations using computer programs (and don't forget that computer programs themselves are based on math).
Interior designers are frequently required to be mathematically proficient. They must understand basic geometry and how to calculate measurements in order to create accurate floor plans.
Algebra is used in architecture to calculate important values. For example, algebra can be used to calculate how large to make certain parts, on the basis of the size of the building design. This can enable architects to make more efficient designs that cost less.
In Mathematics, a pattern is a repeated arrangement of numbers, shapes, colours and so on. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern. Sometimes, patterns are also known as a sequence.
Apart from the courses in Art & Design, Commerce, Business Management, Art & Humanities, there are many other courses without Maths after Class 12th that you can do.
You can still become a graphic designer without going down the usual degree route. With a good dose of diligence and patience, and a world of information at your fingertips (online tutorials, short courses, classic books, training videos, boot camps), you can guide yourself into the profession.
Design school, like many other programs, see's itself as the most difficult of all degrees. It is challenging and rewarding- knocking you down with each all-nighter and building you up with each strong design critique. In many ways, it forms you as a person.
Jobs in writing, publishing, public relations, advertising, and communications also require virtually no math at all. If you love history, literature, civics, foreign languages, or art, you may find your sweet spot teaching a favorite subject in private or public schools.
Algebra is divided into different sub-branches such as elementary algebra, advanced algebra, abstract algebra, linear algebra, and commutative algebra.
Geometry is an integral part of design from start to finish. Architects use geometry to divide space when generating schematic designs. Artists use repetitive sequences like fractals or cubes to create rich patterns or abstract images.
The algebraic method is a collection of several methods used to solve a pair of linear equations with two variables. The most-commonly used algebraic methods include the substitution method, the elimination method, and the graphing method.
Introduction: My name is Edwin Metz, I am a fair, energetic, helpful, brave, outstanding, nice, helpful person who loves writing and wants to share my knowledge and understanding with you.
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