Euclid's Muse

Two pentagons are nested within a third, regular pentagon (ABCDE). They are each defined by a point on each side of ABCDE; these points are proportional along their sides at functions of t. Determine the function of t that represents the proportionality of each point. Notice that both pentagons are congruent to ABCDE when t = 0 and t…

A simple Pythagorean Theorem calculator that accepts inputs for either two sides or one side and the hypotenuse, then calculates the remaining side using the Pythagorean Theorem. Both exact and approximate solutions are provided, as well as the complete Pythagorean Theorem version.

These flowers are constructed with polar rose functions, play with n and d to change the flower’s shape. Download the .gx source to check out the polar function.

A more user-oriented version that lets you change the color of the flower and download the image can be found here: http://celloexpressions.com/misc/make-flowers/.

Four points are located proportionally around a circle, according to four different functions of t. A figure connecting the four points is traced through t. What are the four functions? Look at the .gx source for the answer.

Tip: press "go" to animate t at a constant rate from 0 to ∏ and back, looped.

A triangle, defined by three points that are located proportionally around a circle by functions of t, is traced as t varies from 0 to 2Π. What are the functions of t, f(t), g(t), and h(t), that define the points D, E, and F, respectively?

Hint: one of the functions is _(t) = t.

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…[Read more]

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 da…[Read more]

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…[Read more]

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 anima…[Read more]

Demo for video

AB and AC are vectors. Vector AF is defined by t(AB) + s(AC) where t and s are scalars. Drag E and D to change the scalars and see how using the scalars creates vectors in the plane defined by AB and AC.

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 has promising applications for rooftop usage, but its high cost has kept it unfeasible for widespread adoption. The Twisted Savonius Geometric Modeling project explored the geometric properties of the turbine’s shape, and proposed a more efficient method of construction and geometric design as a result.

This…[Read more]

The three primary types of wind turbine, the Savonius VAWT, the Modern HAWT, and the Darrieus VAWT, are animated as if in operation.

I first created this model in Geometry Expressions two years ago and after sharing it on Wikipedia it became quite popular and has been reused in many places. I decided to create an updated version, which…[Read more]

The three primary types of wind turbine, the Savonius VAWT, the Modern HAWT, and the Darrieus VAWT, are animated as if in operation.

I first created this model in Geometry Expressions two years ago and after sharing it on Wikipedia it became quite popular and has been reused in many places. I decided to create an updated version, which…[Read more]

The chord angle theorem states that in an inscribed triangle (ABC) where A is the center of the circle and BC is a chord, and BDC is an inscribed triangle on the same chord, angle BDC must equal one half of angle BAC. Try changing the angle and moving point D and observe the theorem’s truth.

Note: the measure of angle BDC is being constantly r…[Read more]

The Cyclic Quadrilateral Theorem states that for a quadrilateral inscribed in a circle, the measures of opposite angles must add to 180 degrees. Drag the points and observe the angle measures to see how this theorem holds true.

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 Savon…

Drag points A, B, and C to change the size and shape of the blue triangle, and its white counterpart that is similar (constrained by proportional SAS). Drag the Red point D to change the ratio in sizes.

Observe the multitude of calculated output lengths and angles, and how they match the proportion value, proving similarity, regardless of the…[Read more]

You’ll want to start out with the heptagon and work your way up. This one’s the same as all the others, just with a 13-sided regular polygon. Observe the tangencies to diagonals of circles centered at intersections of diagonals, when the circles are resized (by dragging).

This is a smaller version that works well on most monitors (zoom in with…[Read more]