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Digital Drawings with TurtleArt _ Exploratorium

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Tops are one of the oldest toys known to humankind. They are easy to make with common household materials like yogurt cups, bottle caps, pencils, cardboard cutouts and more! Making a top sparks a learner's curiosity and opens up many tinkering opportunities.

Gather recycled materials around your house & try turning them into spinning toys.

The basic elements of a top are: body, axle, and weight. Depending on what you choose for those elements, the shape and size of the top will vary greatly. We think this selection serves as a good starting point to quickly build a simple top and then experiment further.

It is often helpful to keep your first project simple. Making a straightforward top will allow you to experience what it needs to stay balanced, and give you the opportunity to practice spinning it a few times.

Here are some starting materials and elements to make your first top and exeriment with variables and styles. Think outside the box and consider how many different types of top you might be able to make!

Ready to take a deep dive into the world of top making? Check out our Spinning Tops Project page for lots more tips, tricks, techniques, and inspiration!

Spinning can be a way to turn static images into apparent motion! There are several variations on this theme that you could explore, from using salad spinners or CDs to create fast motion, to relying on your hands. You can also experiment with the frame rate of video cameras to create interesting motion effects.

Zoetropes rely on slits in a vertical cylinder to work, while phenakistoscopes usually have a short sequence of frames arranged radially on a round disc; a strobing light then creates the illusion. There are several apps for both iOS and Android that allow you to turn your phone into a stroboscope and turn a CD spinning on a marble into an animation device! A thaumatrope is even simpler: just a piece of paper or cardboard with a picture on each side, which gets twirled rapidly to blend the two images together! Pro tip: a drill makes for an excellent way to spin a thaumatrope and experiment with how speed affects the illusion of motion.

Spin art takes advantage of rotational motion to create beautiful arrangements of colors, lines, and shapes. It's very easy to do and can be really messy, or quite tidy depending on the materials used. The basic idea is quite simple: find a way to spin a surface - canvas, paper, cardboard, or foamcore will all do - and then add pigment: you can drip paint or watercolors on it, or use markers, pencils, or even chalk to contain or eliminate the mess. If you're brave or don't mind some spillage, you can also turn the equation around: find a way to spin a marker or brush dipped in some paint, and catch the colors on a surface!

Can you think of things in nature that spin, and use them for inspiration to explore spinning by building and experimenting with various materials?

We are inspired by this exploration of helicopter seeds and what properties makes them spin or fly straight down leading into an exploration of whether it is possible to reproduce that same behavior with craft materials. Can you think of ways to make things that spin when dropped from a height?

Spinning with Technology

Credits:

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Tinkering Exploration: Spinning | Exploratorium

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Tops are one of the oldest toys known to humankind. They are easy to make with common household materials like yogurt cups, bottle caps, pencils, cardboard cutouts and more! Making a top sparks a learner's curiosity and opens up many tinkering opportunities. Gather recycled materials around your house & try turning them into spinning toys.

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Our community of users is continually discovering new ways to use Desmos as a canvas and outlet for creativity. From geometric patterns to architectural scenes and recreations of famous paintings, Desmos art comes in various forms, all created from mathematical expressions and geometric objects.

Let's take a look at how we might graph a boat in the Desmos Graphing Calculator:

The point-and-click interface of the Desmos Geometry Tool enables you to quickly start creating. Using the circle, arc, and polygon tools, you can construct shapes in just a few clicks. Additionally, incorporating transformations enables you to translate, reflect, dilate, and rotate your shapes, allowing you to build on your creation.

Let's take a look at how we might create a fish:

Desmos 3D adds another dimension for creativity. Many of the same strategies used in 2D, such as inequalities, restrictions, custom colors and more can be applied in 3D. A great starting place for art in 3D is utilizing the built in function 'sphere.' You can try out this function and start creating with the Build a Snowman quest.
Getting Started: Desmos Geometry Getting Started: Desmos 3D Custom Colors Extending from 2D to 3D Sliders and Movable Points

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Introduction to Math Art Our community of users is continually discovering new ways to use Desmos as a canvas and outlet for creativity. From geometric patterns to architectural scenes and recrea...

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HSV

The expression c=hsv(230, 0.52, 0.7) defines a new color with a hue of 230°, 52% saturation, and a value of 70%.

The hue of a color is a number between 0 and 360, representing an angle on the chromatic circle. Note that by convention, this angle is always measured in degrees. Example: Hue gradient

The saturation of a color is a number between 0 and 1, describing how grey or "washed out" the color is. Example: Saturation gradient

Finally, the value of a color works together with hue and saturation to influence how bright the color appears: from 0 for black up to 1 for full brightness. Example: Value gradient

RGB

The expression c=rgb(86, 101, 179) defines a new color that is a mixture of 86 units red, 101 units green, and 179 units blue.

Each input to rgb() should be a number between 0 and 255. The red, green, and blue components act like colored lights: rgb(0,0,0) produces a black color, while rgb(255,255,255) will produce white.

Note that opacity can also influence the actual color that appears in the graph. (Example: Opacity and Color.)

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Use custom colors to go beyond the default Desmos palette and add a personal touch to your graphs! You can define colors using hue, saturation, and value parameters (HSV); or by specifying the amou...

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How to Make a Tessellation - Rotation Style

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This is a video I made to teach my elementary art students how to make a rotation style tessellation. Students start with a square paper which they will use to create a stencil. Next they cut off one side of the square, rotate it around the corner and tape it onto another side.

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How to Make a Tessellation - Translation Style

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What is Tessellation? | by M. C. Escher inspired Tessellation Art

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Reading time: ~25 min Reveal all steps

Polygons appear everywhere in nature. They are especially useful if you want to tile a large area, because you can fit polygons together without any gaps or overlaps. Patterns like that are called tessellations.

Humans have copied many of these natural patterns in art, architecture and technology - from ancient Rome to the present. Here are a few examples:

Here you can create your own tessellations using regular polygons. Simply drag new shapes from the sidebar onto the canvas. Which shapes tessellate well? Are there any shapes that don't tessellate at all? Try to create interesting patterns!

Examples of other students' tessellations

Tessellations from regular polygons

You might have noticed that some regular polygons (like squares pentagons) tessellate very easily, while others (like pentagons triangles hexagons) don't seem to tessellate at all.

This has to do with the size of their At every internal angles, which we learned to calculate before. vertex in the tessellation, the internal angles of multiple different polygons meet. We need all of these angles to add up to °, otherwise there will either be a gap or an overlap.

You can similarly check that, just like pentagons, any regular polygon with 7 or more sides doesn't tessellate. This means that the only regular polygons that tessellate are triangles, squares and hexagons!

Of course you could combine different kinds of regular polygons in a tessellation, provided that their internal angles can add up to 360°:

Tessellations from irregular polygons

We can also try making tessellations out of irregular polygons - as long as we are careful when rotating and arranging them.

It turns out that you can tessellate not just equilateral triangles, but any triangle! Try moving the vertices in this diagram.

The sum of the internal angles in a triangle is °. If we use each angle twice once three times at every vertex in the tessellation, we get 360°:

More surprisingly, Their internal angle sum is °, so if we use each angle any quadrilateral also tessellates! once twice three times at every vertex in the tessellation, we we get 360°.

Pentagons are a bit trickier. We already saw that regular pentagons don't tessellate tessellate, but what about non-regular ones?

Here are three different examples of tessellations with pentagons. They are not regular, but they are perfectly valid 5-sided polygons.

So far, mathematicians have only found 15 different kinds of tessellations with (convex) pentagons - the most recent of which was discovered in 2015. No one knows if there are any others, or if these 15 are the only ones...

Tessellations in Art

Many artists, architects and designers use tessellations in their work. One of the most famous examples is the Dutch artist M. C. Escher. His works contain strange, mutating creatures, patterns and landscapes:

These artworks often look fun and effortless, but the underlying mathematical principles are the same as before: angles, rotations, translations and polygons. If the maths isn't right, the tessellation is not going to work!

Penrose Tilings

All the tessellations we saw so far have one thing in common: they are periodic. That means they consist of a regular pattern that is repeated again and again. They can continue forever in all directions and they will look the same everywhere.

In the 1970s, the British mathematician and physicist These are called Roger Penrose discovered Penrose tilings, and you only need a few different kinds of polygons to create one: non-periodic tessellations - they still continue infinitely in all directions, but never look exactly the same.

Penrose was exploring tessellations purely for fun, but it turns out that the internal structure of some real materials (like aluminium) follow a similar pattern. The pattern was even used on toilet paper, because the manufacturers noticed that a non-periodic pattern can be rolled up without any bulges.

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Tessellations - Polygons and Polyhedra - Mathigon

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Reading time: ~25 min Reveal all steps Polygons appear everywhere in nature. They are especially useful if you want to tile a large area, because you can fit polygons together without any gaps or overlaps. Patterns like that are called tessellations.