# Euclid's Muse

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##### Circle Radius and Proportional Position Puzzler
Five circles are centered proportionally around an ellipse according to functions of the variable r. Each circle’s radius is defined by the same function as its position, of r. Can you figure out the five functions of r which define the five circles? Download the .gx source for the answer. Tip: if you press the ”go” button, r will animate at a constant rate and the ellipse will stay a constant size. If your drag the slider, note that the ellipse is not changing size or position, but the app is zooming in and out in order to display everything within its window most efficiently. Hint: one circle’s radius and position are defined by f(r) = r.

By Duncan
The radius of the incircle of a 3,4,5 triangle is 1. How about other Pythagorean triangles?

By Faith
##### Soda Cantastrophy

Tags: circle, radius, circumfrance, rate-of-change, related-rates, draggable

##### Polar Proportional Point and Circle Puzzler 3
A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, u(t), and is located at the current value of t. Adjust t, or animate it by pressing ”go”. What is u(t)? Look at the .gx source for the answer. Hint: if you look closely, u(t) can be seen in dark gold. It is a particular ”polar flower”.

Tags: Polar-function, proportional-point, circle, radius, flower, puzzler

##### Polar Proportional Point and Circle Puzzler 2
A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, s(t), and is located at the current value of t. Adjust t, or animate it by pressing ”go”. What is s(t)? Look at the .gx source for the answer.   Hint: if you look closely, s(t) can be seen in dark purple. It is a particular ”polar flower”.

Tags: Polar-function, proportional-point, circle, radius, flower, puzzler

##### Polar Proportional Point and Circle Puzzler
A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, r(t), and is located at the current value of t. Adjust t, or animate it by pressing ”go”. What is r(t)? Look at the .gx source for the answer. Hint: if you look closely, r(t) can be seen in dark blue. It is a particular ”polar flower”.

Tags: Polar-function, proportional-point, circle, radius, flower, puzzler

##### Squeezing Twisted Savonius Wind Turbine Model
This model demonstrates that the surface of the Twisted Savonius wind turbine's blades are geometrically squeezed as the twist angle is increased and the parametric position is moved up and down the turbine. Learn more about the squeeze. Learn more about the Geometry of the Twisted Savonius Wind Turbine project. Note: the calculated radius in this particular example cannot be accurate because the model is a 2d geometric approximation of the real 3d shape. Accurate calculations are made from the top view model, which is visually more difficult to comprehend. The calculation here still varies accurately as the twist angle is changed and the position is moved up and down the turbine, but it also varies as the rotation is changed (which shouldn't happen).

Tags: Twisted-Savonius, Wind-Turbine, Pseudo-3d, Model, Squeeze, Geometric, Real-World, Ellipses, Arcs, Loci, Parametric/Proportional

##### Geometric Top View Model of the Twisted Savonius Wind Turbine (Interactive)
This app models the top view of the Twisted Savonius Vertical Axis Wind Turbine (VAWT). The various inputs and draggable points allow you to see how the model can trace the blades' surfaces. You can also control the twist angle, radius, and rotation - which makes the whole thing spin! Learn more about the Twisted Savonius Modeling Project here.

Tags: Twisted-Savonius, Top-View, Model, Geometric, Real-World, Circles, Arcs, Loci