Epicycles vs Deltarium

The Epicycles app is a tool for exploring the visualization of complex Fourier series. It allows users to interactively create and visualize epicycles based on sampled parametric curves or predefined Fourier series terms. The app provides a visual representation of the complex Fourier series and demonstrates the approximation of complex-valued functions using epicycles. It can be used to study the properties of Fourier series, understand the concept of epicycles, and explore the visualization of complex functions in a fun and interactive way. • Visualizes Fourier series of 2D functions parameterized by time that are built-in, your own drawn 2D curves, or custom frequency components using the terms editor. • Custom terms can saved by exporting them to 'epi' documents in Files, and then later restored by importing them. • Cyclic animations can be saved to GIFs or snapshots saved to PNGs in the Photos library, with sizes 480x480, 720x720 or 1080x1080. • The main view consists of several functional items: Graphic Menu, Time Slider (t), Number of Fourier series terms slider (N), Current Function Menu. Graphic Menu: In the main view use the menubar in the graphic view to select options to hide or show the following graphic elements of the Fourier series visualization: • Circle: The blue circles that are the paths traced by the epicycles. These illustrate Euler’s formula for the complex valued terms of the Fourier series, expressing each complex exponential term as a pair (r cos(n t), r sin(n t)) that trace a circle of radius r, n times as time t varies in the period of length 2π. • Eye: The green circle that represents the value of the Fourier series at the current time. • Lightning: The red line segment path that consists of the joined radii of the epicycles circles, from the origin to the value of the Fourier series at the current time. • Pencil: The orange path that traces the current 2D function. • Star: The black path that traces the Fourier series 2D approximation of the current 2D function. Additionally in the menubar: • Magnify: Hide some views to make room for the expanded display the graphic. • Share: Save the graphic as an animated GIF or snapshot PNG in Photos library. • Play: Animate the graphic by periodically advancing the current time. • Info: Display internet resources conveniently in the app about Fourier series, epicycles and Euler’s formula. Time Slider (t): Adjust the time slider to see the state of all the graphic elements at any time within the time period [-π,π] on which the current 2D function is defined. Number of Fourier series terms slider (N): Adjust the number of terms included in the Fourier series approximation to the current 2D function. Frequency components in a partial Fourier series range from -N to N. The maximum value is limited to 100. As a guide tap the wand icon to set the number of terms to a value whose corresponding highest frequency can theoretically be reproduced with the given number of samples, based on the concept of Nyquist frequency with uniform sampling. For the built-in sample functions that sample count is fixed. The number of samples of your own drawn curve is variable, and displayed in the drawing view. The custom Fourier series using the term editor has known frequency components, limited to the range -20 to 20. The number of samples generated is sufficient for any selection in that range. Therefore in this case the wand sets N to the highest absolute frequency value of the terms. Current Function Menu: Use the segmented control to select from a variety of built-in 2D parametric curves or select the `?` item. Then you can either draw a 2D curve in the Draw tab view, or edit custom Fourier series terms in the Term tab view. In the latter case the app will numerically generate the Fourier series of a Fourier series, by sampling the summation of the series terms.

Deltarium is a place to walk among worlds, the way a planetarium is a place to walk among stars. Each world is a living system of change. You move a few sliders, press play, and watch what the mathematics does: rabbits and foxes chase each other in cycles, a chaotic attractor traces its butterfly wings, a rocket bends time on its way to another star. Built for kids and curious people first, and for scientists too. There is nothing to sign into, nothing to buy, and no ads. The app runs entirely on your phone and collects nothing. THE 8 WORLDS PREDATOR AND PREY. The classic hare-and-lynx cycle. Set the birth, hunting, and dying rates and watch the populations chase each other around a loop. A guided tour walks through this world one step at a time. THE BUTTERFLY. The Lorenz attractor in 3D, spun with a finger. Turn on the twin path, which starts a hair away from the first, and watch the two separate. The butterfly effect you can see. ESTUARY. A coastal model of nutrients and life that blooms and crashes with the seasons. THE DANCE. Three stars pulling on each other, including the famous figure-eight where they chase one another forever. PARKING IN SPACE. The 5 Lagrange points, the spots where a small craft can hold its place relative to the Earth and Moon. Give all 5 a shared kick and see which ones stay parked and which wander off. RADIOACTIVITY. How a radioactive source fades over time, following the real decay chain from a parent down through its daughters. TIME DILATION. A rocket that holds a steady 1 gravity of thrust all the way to a distant star, flipping over at the midpoint to slow down. The clock on the ship and the clock back on Earth pull apart. BUILD YOUR OWN. Write your own equations, add sliders for the parts you want to tune, and watch the system run. UNDER THE HOOD Every world is a real system of ordinary differential equations, solved on your phone by the same numerical method scientists use. Each screen carries a science card naming the model, its author, and a reference you can look up. The phone's engine is checked against a published Python reference, and all the source code is public at github.com/evanleeturner/deltarium-iphone.