The Helicone: Toy Or Mathematical Oddity?

January 7, 2025 by Al Williams 7 Comments

We always enjoy videos from the [Mathologer], but we especially liked the recent video on the Helicone, a toy with a surprising connection to mathematics. The toy is cool all by itself, but the video shows how a sufficiently large heliocone models many “natural numbers” and acts, as [Mathologer] puts it, acts as “microscope to probe the nature of numbers.”

The chief number of interest is the so-called golden ratio. A virtual model of the toy allows easy experimentation and even some things that aren’t easily possible in the real world. The virtual helicone also allows you to make a crazy number of layers, which can show certain mathematical ideas that would be hard to do in a 3D print or a wooden toy.

Apparently, the helicone was [John Edmark’s] sculpture inspired by DNA spirals, so it is no surprise it closely models nature. You can 3D print a real one.

Of course, the constant π makes an appearance. Like fractals, you can dive into the math or just enjoy the pretty patterns. We won’t judge either way.

We’ve seen math sequences in clocks that remind us of [Piet Mondrian]. In fact, we’ve seen more than one of those.

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Fibonacci Clock Is Hard To Read, Looks Good

May 7, 2015 by Bryan Cockfield 18 Comments

Artists have been incorporating the golden ratio in their work for many hundreds of years, and it is thought that when proportions are in line with this ratio, it tends to be more aesthetically pleasing. With that in mind, the clock that [Philippe] created must mathematically be the best looking clock we’ve ever featured, even if it is somewhat difficult to tell time from it.

The clock is made up of squares which represent the first five numbers of the Fibonacci sequence. The squares are backlit with LEDs, which will illuminate red for the hour, green for the minute, and blue representing the overlap of hours and minutes. Simply add up the red and blue squares to get the hour, and add the green and blue squares to get the minutes. The minutes are displayed in 5 minute increments since there aren’t enough blocks though, so you’ll also have to multiply. Confused yet? If not, it turns out that there are several ways to display certain times using this method, any of which can be randomly selected by the clock. [Philippe] reports that there are 16 different ways to represent 6:30, for example.

The clock is driven by an ATmega328P and is housed in a wooden case. There are schematics and code available on [Philippe]’s site if you want to build your own, there are detailed descriptions of how to tell time with this clock. You’ll probably need those. If you like getting confused by clocks, you might also like this one as well.

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Animate Your Artichoke With A Lathe And Camera

January 14, 2015 by Sarah Petkus 13 Comments

Spirals, fractals, and even bone length proportions whisper of a consistent ratio woven into the universe. Math is hidden in the fabric of things, and when this fact is observed in art, magic happens. Professor, artist, and inventor [John Edmark] draws inspiration from geometric patterns found in nature and builds sculptures using the golden ratio as a standard for design. In this project, he expresses these characteristics through animated biomorphic zoetropes.

[John] modeled several 3D sculptures in Rhino containing similar geometric properties to those found in pinecones and palm tree fronds. As the segments grow from those objects in nature, they do so in approximately 137.5 degree intervals. This spacing produces a particular spiral appearance which [John] was aiming to recreate. To do so, he used a Python script which calculated a web of quads stretched over the surface of a sphere. From each of the divisions, stalk-like protrusions extend from the top center outward. Once these figures were 3D printed, they were mounted one at a time to the center of a spinning base and set to rotate at 550 RPM. A camera then films the shape as it’s in motion at a 1/2000 sec frame rate which captures stills of the object in just the right set of positions to produce the illusion that the tendrils are blooming from the top and pouring down the sides. The same effect could also be achieved with a strobe light instead of a camera.

[John] has more information on his instructables page. He also provides a video of this trick working with an actual artichoke; one of the living examples of the golden ratio which this project was inspired by. Thank you, [Charlie Nordstrom] for helping him document these awesome sculptures and for telling us about them!

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