(Math 110) Through the analysis of a single area of mathematical application, such as art, music, or politics, we will see how fundamental and traditionally studied mathematical elements are found in the underlying structure of problems, solutions, relationships, and works of expression and creativity. Once familiar with the mathematical building blocks of a certain area of application, the students will then synthesize these into an original contribution of their own, which may be a mathematical description of a phenomenon, a solution to a problem, a work of art, etc., depending upon the particular area of application. Students will find a number of ways to be successful in the class, including through presentation, homework and quizzes, projects, and examination. Specifically, we will discuss functions, graphs, trigonometry, probability, statistics, and logic.
In each of our class meetings, we will discuss a different topic that showcases the connections between mathematics and art. We will
- explain the impact of a mathematical understanding of linear perspective in art
- discuss the use of symmetry and the golden section in works of art as well as in nature
- explore how artists have attempted to illustrate complex mathematical ideas in their work, leading to entirely new genres of art such as Cubism
- review some of the different types of geometries, and how artists like MC Escher have visually described them
- explore how ancient societies viewed mathematics, and its impact on their works of art
- explain how geometry is used to create large scale works, such as architecture, and how these methods have changed over time
- explain how ancient and modern societies have used mathematics and celestial navigation to create maps and better understand their world
- view some examples of how computers can be used to create art based on mathematical algorithms
You will be required to complete one project throughout the semester. You can choose to create a research based or creative project, with slightly different requirements for each.
Research Option Requirements
Abstract: A typed abstract of your proposed research topic will be due by the end of the first week of class. The abstract should be a one to two paragraph summary of your chosen topic. A clear connection between mathematics and art should be present. If your abstract is accepted, you may begin work on your project. Otherwise, you will need to resubmit until an acceptable abstract is submitted. At least one reference must be included with your abstract.
Paper: A final paper will be due the last day of the semester. Your final paper should be typed, two to three pages in length, double spaced, with 12 point font and 1.5” margins. The paper should clearly reflect the topic submitted in your abstract and class presentation. You should include at least one paragraph that mentions why you had interest in the chosen topic. A clear explanation of the connection between mathematics and art should be evident and supported by at least three references. The included bibliography may be in any format. The paper should be entirely original, with any facts or quotations properly cited.
Presentation: A 10 minute presentation will be scheduled near the end of the semester. Your presentation should closely reflect the material in your paper, including an explanation of the sig- nificance of the topic and its relevance to the course. You are highly encouraged to use relevant visual aids in the form of handouts, slides, overheads, board work, posters, etc.
Creative Option Requirements
Project Proposal: A typed proposal of your plan will be due by the end of the first week of class. The proposal should be a one to two paragraph summary of your chosen topic. A clear connection between mathematics and art should be present. If your proposal is accepted, you may begin work on your project. Otherwise, you will need to resubmit until an acceptable proposal is submitted.
Paper and Project: Your project and a final paper will be due the last day of the semester. Your final paper should be typed, one to two pages in length, double spaced, with 12 point font and 1.5” margins. A thorough description of your project should be given, including a list of materials, computer programs used or written, etc. Be sure to explain how your project incorporates both mathematics and art. If you create a physical work (model, drawing, etc) you will be allowed to keep it. All work should be original. If you use any references in your paper, you should include a bibliography (a bibliography is otherwise not required for the creative project option).
Presentation: A 10 minute presentation will be scheduled near the end of the semester. Your presentation should demonstrate the work you created, along with an explanation of its relevance to the course.
Proposals for Creative Work
Arranged alphabetically by speaker
A Anderson, Naturally Occurring Spirals
For my project I chose to connect math and art through spirals. In class we discussed the Fibonacci spiral, I will be discussing this and other mathematical spirals that appear in nature. I will research the ways in which the spirals are present and draw the element myself and apply the spiral method to see if it holds true.
V Arciniega, Regular Tessellations
I found the lecture on tiling and tessellations very interesting. I am going to do my creative project using the tessellation technique. I will cut out a shape on a square index card and trace it on an 8’ x 12’ card stock. I will translate my shape onto the card stock and make sure to cover the entire sheet without having overlaps or gaps. After, I will color a pattern by using three different colored pencils. My creative project will have a regular tessellation pattern made by a repeating regular polygon with a line of symmetry.
D Butera, Tessellations
I would like to do a creative project demonstrating to the class how to do a tessellation. I will have examples from artists- such as MC Escher and Robert Fathauer on a slideshow, and then I would show the class step-by-step how they could create their own tessellation and have a step-by-step example of my own.
Tessellations are a form of geometry and a gridding pattern to create a repeating pattern like a wallpaper pattern. They match up exactly so there are no gaps and all of the pictures are the same.
R Claros, Analytical Cubism Through Vector Illustration
For my creative project I'd like to explore analytical cubism through vector illustration. My goal is to create 3 different portraits. I'm basing this project of Pablo Picasso's Study of a Bull. I would follow the guidelines of analytical cubism (reducing natural shapes into geometrical planes and flattening the image. In order to add depth to the image I will be using different shades and tints to counteract 2 dimensional flatness. I will be using Adobe Illustrator for this assignment and I will also be looking at other artists, e.g. Tim Biskup.
J Desiderio, Aperiodic Tilings and Frieze Patterns
My project will be a drawing that includes both aperiodic tiling as well as frieze patterns. The aperiodic tiling will be centered at the middle of the page, which will include both reflection and rotational symmetry. Furthermore, the frieze patterns will act as a border for the aperiodic tiling, which will be found on the perimeter of the drawing. The tools used in order to create this drawing will include a pencil, colored pencils, sharpie markers, a ruler, as well as a compass.
S Eddins, Net Sculptures
Janet Echelman is known for her permanent installation pieces in cities all over the world. Her weightless pieces seem to defy gravity. Echelman collaborates with lace makers, engineers, even NASA scientist to make sure that her works of art are not only beautiful but last. Using design software, Echelman can see how to form the shapes she desires out of netting. The program uses equations to determined how the netting needs to be hung and supported. A combination of math and science allows for Echelman to create beautiful and unique art. I will research her process and create a smaller model of my own.
R James, Mathematics and Ballet
For this project I will be choreographing a Ballet trio that uses mathematic principles to guide it. Ballet does naturally use many ideas from math such as angles, lines and geometric shapes. While I am constructing this piece I will be consciously making choices with the three dancers to show one of these areas in their movement. Exploring the way that three dancers can move on the stage. What shapes and patterns they can create in a visually pleasing and interesting way will have its own challenges. I will be looking at shapes and options that math provides. Another aspect of this project are the angles and lines that the human body can create. This is an important element to Ballet and in that makes all dancers very concerned about this mathematic idea. I am also preparing this piece for a prestigious international Ballet competition call Youth American Grand Prix. I look forward to seeing how adding this concentration on mathematic ideas will refine the work and make it more precise and pleasing. The work will be two minutes and forty five seconds long and I plan on videoing it and showing the full work during my presentation. I will also be discussing the important role math has in Ballet and show some specific lines and angles dancers create with video and photos.
D Mattson, Mirrored Anamorphic Art
My project will be based on the idea of anamorphic art. I will be drawing a distorted message and at a slight angle it will become clear to whoever is holding the project. Point of view changes everything in every aspect of life as it will in my project.
H Metzger, False Curvature
For my creative project, I plan on utilizing a strong square as a base with hooks to be able to utilize string from the different points. My objective is to show how out of straight lines unexpected curves appear. Then to look how my design from the string relates to the Bezier curves.
A Petrosoniak, Irregular Tessellations
Tessellations are the tiling of a plane by using geometric shapes without allowing for white space to show. These creations are similar to puzzles and therefore pose a challenge to the artist. In high school I learned about this form of art and was given a project to create a simple or complex tessellation. Through the accompanied study of M.C. Escher, I quickly became inspired with the work of patterning and tiling. I found this art form to be very difficult throughout the project. Given this opportunity again, I would like to challenge myself further to a new creation.
In this project I am planning on challenging myself by using an animal as the subject and repeating pattern. As a Canadian, I wanted to choose and animal that represented my country and something that is important to me. Experimenting with a few different ideas, I have decided to use a moose as my subject matter. This will be challenging because of the antlers but I’m excited to tackle this confrontation. In terms of the intended colour scheme, I will be using browns to represent the moose. I will use the inverse technique to help distinguish separation among the moose. This project will either be done by using coloured pencil or by using acrylic paints. I am very excited about this project and challenge.
A Pyawasit, String Sculpture
I have been fascinated by a recent artist Henry Moore along with his creations which imply mathematical artistry. His use of three-dimensional sculptors comprised of stone (or wooden frames) with string which are manipulated into symmetric or mobious designs. Henry Moore's obsession with geometry intertwining with art has inspired him along with countless others to create intricate sculptures and drawings even today. I personally find his abstract string art which borders the golden spiral or infinite creations to be memorizing as they are elaborate.
As for the terms of the creation I will make which will be a piece inspired by Henry Moore's work I would also hope that I could put a twist on to the piece inspired also by M.C. Escher's works on three-dimensional illusions.
Arranged alphabetically by speaker
H Adams, The Art of Nathaniel Friedman, Helaman Ferguson, and Koos Verheof
I have found that I am interested in how artists use mathematics to either create art, or find mathematics in art naturally. I have decided that I want to study various artists and how they have found and used a relation between the two subjects. Three artists that interested me in the research process were; Nathaniel Friedman, Helaman Ferguson, and Koos Verheof. For each artist, I will provide a brief history of their studies and backgrounds; however, I will mostly describe the work that they do relating the two subjects.
All three of these artists made discoveries that linked mathematics and art, and used these to produce work that was exceptionally detailed and had many aspects of mathematics involved in their creation. I will provide examples of some of each artist’s work and I will clearly define each mathematical principle involved in their research and production of their work. Lastly, I will discuss how their information went on to influence other artists, mathematicians or people.
W Adams, A History of Mathematics and Art
I plan on doing an overview of the history of math and art. I will involve things that occurred in ancient times such as the pyramids and the Parthenon. I will also involve artists from the Renaissance such as Paolo Uccello and Leonardo De Vinci. I will even cover more recent topics such as the Penrose tiles and the Eden project and other more recent artists in industrial and modern times such as Salvador Dali and M.C. Escher.
A Brinkman, A Study of Leonardo da Vinci
Leonardo da Vinci was an Italian philosopher. He used a large amount of mathematics in his artwork. Some pieces, that show the connection between art and mathematics is the Mona Lisa, Vitruvian Man, The Last Supper and many of his drawing for inventions, he dreamt up. These connections are very important due to how innovative da Vinci’s ideas were at the time.
M Contestabile, Structure Advancements Throughout Time
Advances in engineering have come as a result of mathematical progress. As new methods of math were developed, larger and more complicated structures have been able to be built. In the past with little knowledge of math pyramids were the most stable structure used. Pyramids are based on simple solid geometric forms, with square base and less material higher up, the pyramid will be pushing down distributing its weight more creating more stability. As more mathematics were explored, a greater understanding of architecture was shown. During the Renaissance both symmetry and proportions were emphasized while creating structures. Also, buttresses were utilized to create a form of support and symmetry throughout the buildings. Past structures have been so limited by a heavy base, recently mathematical developments have begun to create a way for taller more modern buildings, like the skyscraper. Now that there is a way to distribute more of the weight and make bases lighter the structures are able to move upward.
D Dorris, Constructing Egyptian Pyramids
The project that I would like to pursue would be a research paper that looks at how the Egyptians constructed the pyramids. In order to build a pyramid, one must consider many factors. For starters, it could simply fall apart if all of its measurements are not in line. This makes it rather difficult to construct the perfect pyramid, especially without the use of today’s technology (Root). It is clear that many of the pyramids built did not last. What was the difference between these pyramids, and the ones which still stand today?
The answer to this question seems to lie in the techniques, instruments, and architects of each individual pyramids (Grimault). All of these elements had to work together in order to create the perfect pyramid structure. Alone, the math required to make a pyramid that stands is very impressive, but the sheer size of the pyramids leaves us to question how the Egyptians obtained the tools and mathematics to create these structures (Dief).
E Kocieneiwski, The Golden Ratio in Art
I find the golden ratio an interesting subject to research. This mathematical calculation can be applied in art to create harmonious proportions. Famous paintings by Leonardo da Vinci, Salvador Dali and many others incorporate the golden ratio either intentionally or unintentionally due to its aesthetic nature. The golden ratio is a universal law and is naturally occurring. It can be found in the arrangement of branches along stems of plants, in leaves, the veins and nerves of animals, even in the proportions of the human body. I plan to write about the history of the calculation and the different applications such as the golden spiral and rectangle, its occurance in nature and how it relates to the proportions of a human body and of course how it can and has been utilized in the creation of artwork and architecture. I plan to focus the research of my paper to the use of the golden ratio in famous artworks.
R Lewis, Methods of Creating Anamorphic Art
This project will explore the subject of anamorphic art. Anamorphic art will be defined and the origins of anamorphic art will be discussed; specifically Leonardo da Vinci's influence through his work known as Leonardo's Eye. Julian Beever and Kurt Wenner are both prominent modern anamorphic artists known for their sidewalk drawings. This report will explore how each artist began producing anamorphic art and also identify some of their works. Anamorphic art must be viewed from a very specific angle so it will be explained how, when viewed from a different angle, the images appear distorted. When anamorphic art is projected onto a mirrored surface there are mathematical formulas to determine how this projection should be drawn. These formulas will be identified and explained for anamorphic art projected onto a cylindrical surface.
C Massari, Natural Tessellations
I would like to further research Tessellations naturally occurring in nature. We see non-manmade tessellations almost every day, and we don’t even notice it. Tessellations relate to mathematics, as well as art, and I will explain the relationship between the two subjects as I give natural examples.
Tessellations can also be simply categorized between regular and irregular. We do see regular tessellations in nature, but I believe we see more of irregular tessellations, and that makes us notice them day by day less. I will show examples of each type of tessellation in nature throughout my presentation.
M Monaco, Origami and Mathematics
For my research project, I will investigate the relationship between mathematics and the art of origami. In my final paper, I plan to explain how the attempts to understand the basic principals of paper folding have enriched mathematics. The particular topics I will discuss are the axiomatic treatment of paper folding and some interesting theorems that follow.
Additionally, I will explain how mathematical techniques have advanced the art of origami itself by helping designers create very intricate models. In particular, I will explore how the problem of circle packing is related to designing the initial version of the model called the base.
A Novea, The Mathematics and Art of George William Hart
I am going to write a research paper about George William Hart who is an American geometer who uses mathematics to create art. He graduated with a B.S. in Mathematics and a Ph.D. in Electrical Engineering and Computer Science, both from Massachusetts Institute of Technology. His artistic work consists of sculptures, computer images, toys, and puzzles. He has also published many different works including the textbook, Multidimensional Analysis, the online publication, “Encyclopedia of Polyhedra”, as well as over sixty academic articles. He is also the co-founder of North America’s only Museum of Mathematics in New York City. My research will focus on his “Encyclopedia of Polyhedra” which is an online publication on geometrical figures he created, which are solids in three dimensions with flat faces and straight edges.
C Padovano, Methods of Creating Anamorphic Art
The idea behind anamorphic art is very mathematical. The image is distorted to the point where it can only be visible from one vantage point or from its reflection on a mirrored surface. The first efforts at this technique were during the period of the Renaissance, but became extremely popular during the Victorian era. Many artists have now used this technique to make three dimensional chalk art.
Artists are able to make these illusions by either using as distorted grid, or by looking at the mirrored image and drawing on a flat surface. If you have a projector you can do a six-step process to produce your image, which is a mathematical process. They now have program that an artist can use to print the grid with the image on it for mirrored reflections. This program automates the anamorphic images for you so you don’t have to do the math behind it by yourself like they did back in the day.
I Pertz, Leonardo da Vinci and the Golden Ratio
This research paper takes an inside look at an example of the implementation of mathematics and art by a famous artist. Leonardo Da Vinci is perfect for this topic because of his use of the golden ratio in some of his most famous works, including the Mona Lisa. One of the most famous artists of all time, he was an innovator as well. Da Vinci was one of the first to implement mathematics into art. He set a precedent with the use of the golden ratio that would change art as we see it. This mathematical instrument would help to make the perfect geometrical rectangle and other shapes to use in art pieces. The golden ratio is still used to this day. Several different sources such as journals, interviews, and art reviews were looked at to examine the details of Da Vinci’s work. This paper will take an inside look behind the surface layer and purvey the distinct instruments of artistic creation.
M Steele, The Eden Project
The Eden Project, located in England, is a main attraction and encompasses a number of different elements that makes it visually appealing and mathematically built. Its buildings are made using hexagon and pentagon tiling. The Fibonacci Sequence also was considered in the construction. Furthermore, the Project takes environmental steps to better the Earth and offers education to its visitors. The architecture and mathematics that were put into the attraction shows the relation of the two different fields of study.
W Thompson, The Evolution of Origami
Just like in the world of math, laws govern and dictate the possibilities. I would like to write a report on the evolution of origami from "Traditional" to "Modern" forms through the use of Math from Yashizowa, to Robert Lang, and into the various artists of today. By discussing the history of origami from its original form to the complex pieces of art we see being developed today with an emphasis on how math made this possible.