Many of us recall those old, red plastic 3D viewers from younger days. With each clunky click to rotate the wheel of images, we were transported to a far-off location, immersed in a photo of the place. But why was it so darn cool to use? Why are they still around today? And how exactly did it all work? Well to start, a little bit of mathematics, of course!

We see the world in three dimensions because most of us have two eyes spaced a particu lar distance apart. This gives each eye a slightly different view of the world in front of us. The brain reconciles the difference between these slightly different views from each eye, and perceives that difference as depth. While this is just one of the ways we see and perceive depth (there are many more!), it’s also is the basic principle behind a popular type of 3D photograph. This idea and many others are explored in MOPA’s current exhibition which looks at the history, science and creative uses of 3D photography over its history.

Nowadays, it seems each month there is a new device, headset or app offering the ability to explore our world in 3D, augmenting our view with more information or immersing us in an exciting experience. And while a lot of the new technology in these devices (read: smartphone!) were unthinkable just thirty years ago, the fundamentals of 3D photography at their most basic (two images from slightly different angles presented side by side to recreate depth) are relatively unchanged from a 150 years ago, and it’s all thanks to a little bit of mathematics. Don’t believe us? Give it a try!

Click here for PDF and Discussion Worksheet: Making a StereoCard Lesson

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This project includes the outline for two math-focused museum tours designed by elementary school teachers from the San Diego Unified School District. The tours were created as part of a five part workshop on making California Common Core math connections in an art museum space. Each tour was tested with the authors students onsite at the Mingei International Museum and was part of a larger lesson implemented in the classroom.

The following museum tour outlines were designed by elementary school teachers who participated in a five part professional development workshop on furthering California Common Core Mathematics standards through the use of art museum visits and experiences.

The workshop focused on understanding the new math standards and sharing current implementation strategies, followed by experimenting with mathematics learning in the museum space. Each teacher created and prototyped a museum tour to implement with their students at Mingei International Museum, supporting a math lesson/project that would take place back in their classroom. The tour was lead by a museum docent based on the outline created by the teacher.

The tours were designed using work that was currently on view at the Museum as part of the exhibition, **SELF-TAUGHT GENIUS – Treasures from the American Folk Art Museum**, an exhibition that included works on loan from the American Folk Art Museum in New York.

The first tour was created for a 3rd/4th/5th combination class at The Museum School, created by the teacher Emily Watson. The students were studying geometry, so the tour focuses on looking at architectural models within the exhibition using Common Core geometry vocabulary. Some of the term they explored in the Museum space included: quadrilateral, area, parallel/perpendicular lines, symmetry and vertex.

The second tour was created for two 3rd grade classes at Valencia Park Elementary, Elise Samaniego and Jennifer Teruya. This tour was connected to a larger lesson on area and perimeter using the quilts in the exhibition as primary resource material. Back in the classroom students looked at more pictures of quilts, used them as a subject to measure area and perimeter and made their own quilt squares that would be joined together to create a classroom quilt.

First Tour: Math Art Combo Tour Outline, Museum School

Second Tour: Math Art Combo Tour Outline, Valencia Park

]]>Author: Julia Marshall

Year: 2010

Journal: Art Education

Volume: 63

Issue: 3

Pages: 13-19

Audience(s): Educators, Artists

Content: Art, Education, Academic Curriculum, Contemporary Art

Creating and evaluating art goes beyond investigating the artistic methods used. Art is intimately connected with other disciplines and through some of the strategies presented by Marshall this connection is brought to the forefront. Art is a vehicle that allows concepts to cut across standard disciplinary boundaries and invites conversation about science, the environment, mathematics, history, and so forth. The five strategies Marshall describes to achieve this goal of integration are depiction, extension/projection, reformatting, mimicry, and metaphor.

One strategy that struck me in particular was the practice of reformatting. Through reformatting, a concept is viewed from another vantage point and then conveyed in a visual format. An example presented by Marshall is charting one’s emotional world as a geographical map. The geographical map within itself presents several questions such as, where is north and west? Are there bodies of water? What is the terrain like on the map? What is the scale and size of the map? These questions could allow students to think as a cartographer, geographer, or historian. Examining one’s emotional world allows students to think like a psychologist or a philosopher. Reformatting challenges the way in which we normally view a particular idea or concept and encourages students to be inquisitive about the world around them and to not be hesitant about viewing it through different lenses and perspectives. The process of reformatting may elicit new and imaginative ideas that are more powerful when intimately connected with art than if the ideas were to stand-alone. The process of art making affords students with the opportunity to fuse their own experiences and learning together.

]]>Authors: Robert Bosch, Urchin Colley

Year: 2013

Journal: Journal of Mathematics and the Arts

Volume: 7

Issue: 3-4

Pages: 122-135

Audience(s): Researchers, Educators, Mathematicians, Artists

Content: Mathematics, Art, Mosaics

Description: This paper discusses how Truchet tiles can be modified for use in halftoning to create figurative mosaics from greyscale images. In the 18^{th} century, Father Truchet published his mathematical and artistic investigations, ‘Memoire sur les combinaisons,” on the use of square tiles to create aesthetically pleasing designs. A Truchet tile is a square that is divided in half down a diagonal and then one side is colored black. There are four distinct Truchet tiles (alternating halves between black and white, using both diagonals) that can be arranged to create infinitely many different patterns. The authors create flexible Truchet tiles whose relative brightness/darkness can be adjusted by flexing at the midpoint of the diagonal that separates black from white. Figurative mosaics are then created by assigning brightness values to each of the pixels in the matrix of the desired greyscale target image. The authors extend this idea to create flexible hexagonal tiles, and also change the white/black distribution to create star tiles.

Date Created: 7/29/15

Document DOI or URL: doi:10.1080/17513472.2013.838830

]]>Authors: Veronika Irvine, Frank Ruskey

Year: 2014

Journal: Journal of Mathematics and the Arts

Volume: 8

Issue: 3-4

Pages: 95-110

Audience(s): Researchers, Educators, Mathematicians, Artists

Content: Mathematics, Topology, Braid Theory, Lattice Theory, Graph Theory, Art, Lace, Craft

Description: This paper discusses the development of a mathematical model for making bobbin lace, a traditional fiber art for creating lace by hand by braiding together many strings using specific patterns. The authors give an overview of the craft and its history, then review some relevant areas in mathematics for modeling other kinds of fiber arts, including the topology of textiles and braid theory. They then describe a mathematical model for creating bobbin lace, using graph and lattice theory. This model, in turn, yields new lace patterns, not found in existing catalogues of traditional bobbin lace patterns. Finally, the authors present these novel lace patterns to lacemakers to elicit feedback on the aesthetics of the design. Overall, the arch of the narrative presented in this paper represents an interesting interplay between mathematics and the arts, with a longstanding craft tradition inspiring a mathematical model that, in turn, points to new directions for the craft.

Date Created: 7/29/15

Document DOI or URL: doi:10.1080/17513472.2014.982938

]]>Authors: Carola-Bibiane Schönlieb, Franz Schubert

Year: 2013

Journal: Journal of Mathematics and the Arts

Volume: 7

Issue: 1

Pages: 29-39

Audience(s): Researchers, Educators, Mathematicians, Artists

Content: Mathematics, Simulation, Art

Description: The paper begins by reviewing the use of randomness in the creation of pictorial art. In particular, it describes the contribution of randomness in the painting of Pollock, Warhol, and Rothko, as they incorporated accidental and unplanned strokes in their works of art. Then it describes diverse computer-based RNG (Random Number Generators) and how their output can generate images by the random specification of the position, color, and texture of their elements. The authors illustrate how randomness can be incorporated into “computer supported, algorithm based art production.”

Date Created: 7/29/15

Document DOI or URL: doi:10.1080/17513472.2013.769833

]]>To kick off the work of the Design Lab we left the friendly confines of the Tinkering Studio at the Fleet Science Center and made our way out to the fountain with our big protractors in hand. This activity requires 3 people: one person to hold the protractor, and two more people who each hold an end of the string that forms the angle. The two string holders must remain fixed; we used small cones to mark their locations in an effort to minimize any unintentional migration (see the red cones in the photo below).

Their primary job is to dole out and retract the string as the protractor moves. The person with the protractor tries to move in such a way that the selected angle formed by the strings remains constant. Each time a location is found that preserves the angle, a cone is set down as a marker (see the yellow cones above).

There are two main parameters that affect the system: 1) the fixed distance between the string holders (ie the distance between the two red cones above), and 2) the angle selected on the protractor. Each team of three adjusted the parameters as they saw fit, or as curiosity dictated, and then tried to find as many points, on both sides of the string holders, as they could. For each trial (ie every time we changed the angle or the distance between the string holders) we scribbled quick notes and observations, and took photos to record the collection of locations. As teams succeeded in finding points, attention began to shift towards the resulting pattern or shape formed by the locations. Was it a peanut, an hour glass, an ellipse, etc?

What makes this mathematical experience so much different than the experience of sitting at a desk with a pencil? Lets consider what is involved in finding a point. By design, this activity requires the cooperation of at least three people, so communication and coordination among the team is critical.

How should the protractor move to maintain the angle?

Who needs to change the tension in their string?

How should the tension be changed: increased or decreased?

To engage in these questions and respond to them requires learning to integrate the perceptual and motor. The string holder feels the tension of the string in their hand, the braid of the string tugging your skin, perhaps an urge to move away from their fixed spot as the tension increases, pulling them. Release string.

Is the angle being maintained? Peer down the string to the protractor, is the string still straight? If it’s not, what needs to be done? Should you change the tension in the string? Should the protractor move? If so, where?

If you’re holding the protractor, how should you move to find a new point? Should you look in the direction you intend to move, or try to maintain a visual of the protractor and strings? Don’t trip over your own feet! Uh oh, you feel the protractor pulling you. Is it coming from the left side or the right? Should you request more string, or move differently?

To further explore what shape is produced by the patterns, we created a small scale, desktop device, PEGI. To learn more about PEGI, click here.

*Written by* Bohdan Rhodehamel

*Contact Bohdan at bohdanr561@yahoo.com*

Pegi was created to further explore the shape created by the points, but in a way that dramatically reduces the scale and allows one to work individually, if desired. Pegi contains three components: a slotted bar, two pegs to insert in the slots, and angles of varying measures along with their supplements. All parts are made from acrylic cut on a laser cutter.

One begins by selecting the distance between the pegs and inserting them in the slots of the bar (this distance is the secant), and then choosing an angle to work with. The angle slides between the two pegs, maintaining contact with both pegs . Sliding the vertex through the pegs creates a collection of points that form the same contours (ie circular arcs) found using the protractor and string.

*Written by* Bohdan Rhodehamel

*Contact Bohdan at bohdanr561@yahoo.com*

the circumcenter is located at the point whose distances from A, B, and C are equal to each other. As we moved A, B, or C, the circumcenter will move as well in order to keep equal distances to the three points. To make things easier let us call the circumcenter “circum.”

Trying out motions of the points A, B, and C and noticing where circum goes is a way of exploring how circum responds to these motions. We created stories, such as one in which A, B, C were named after participants in the lab and circum worked as their angel guardian. You can read this story here, as well as a subsequent response written by Circum.

Since the distance between Circum-to-A and Circum-to-B must be equal to each other it turns out that Circum is forced to be on the line perpendicular to the one connecting A and B, passing through the midpoint. All the points on this line are equidistant to A and B:

In addition, the distance Circum-to-C must also be equal to the ones Circum-to-A and Circum-to-B, therefore C must be on a circle centered on Circum:

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