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Geometry Expressions Newsletter
Geometry Expressions Newsletter
January 2014 
Mechanical Expressions Beta Extended
Mechanical Expressions is a new program which allows you to create explicit mathematical models of mechanical phenomena.

Based on feedback from beta testers, we have added a number of features, including Matlab support, export of differential equations, forces aligned with vectors and additional capabilities for our HTML5 JavaScript export.  

We have built these into a new Beta which is available for download here.  The Beta period is extended to the end of March 2014.


How to... embed a Euclid's Muse model in a web page
People have asked us about the simulations which appear on the Mechanical Expressions web pages. For example, the double pendulum on the features page.

Yes, this is a simulation created with Mechanical Expressions and it is embedded using an iFrame.

Now these simulations live on our server, but it is possible to create an iFrame which links to a simulation on Euclid's Muse.  
A double pendulum simulation, for example, is here.

There are 2 methods of embedding it in an iFrame, but before you do, check the licensing for the simulation and make sure your use conforms. Most models on Euclid's Muse are licensed as Creative Commons Attribution, which means you can do what you want with them, so long as you attribute the author.  The license type is shown in a small icon beside the app's author. Click on it to get a description of the license.

Method 1: Click the Embed button and copy the iframe html into your webpage.  This will create an iFrame linking to the Euclid's Muse version.

Method 2: Click the Download App button.  This will give you the full html for the simulation.  You can put this on your website and link to it from within an iFrame in your document.
Problem of the Month
 
This month, a geometry problem which dates back to Archimedes.

CD is the diameter of a circle, and FB is perpendicular to CD.  Tangents to the circle at D and B intersect at E.

Show that CE bisects BF.

Is there an analogous theorem for ellipses?  How about hyperbolas and parabolas?

   

An app which lets you explore this problem is here.

A Solution to December's Problem
 
Points A, B and C lie on a smooth closed curve such that triangle ABC has sides with maximal sum of squares.  Show that the medians of ABC are normal to the curve.

   

An app which lets you explore this problem is here.


We'll use a trick from Mark Levi's Mathematical Mechanic and Mechanical Expressions to solve this problem.

The trick is this, if we insert springs of equal stiffness between AB, BC and CA, then the potential energy of the springs is proportional to the sum of squares of their extension.  If we give our springs 0 natural length, then the potential energy is proportional to the sum of squares of their lengths.  When the mechanical system is in equilibrium, the potential energy is a minimum (or maximum for an unstable equilibrium).  Hence if we can use physical arguments to determine an equilibrium position, we will have solved our optimization problem.

We can make a model in Mechanical Expressions.  We use a generic parametric curve (f(T),g(T)) and put three points at parametric locations s, t and u on the curve.  We add springs with 0 natural length and stiffness k.  Mechanical Expressions will compute the reaction force in the parameters s,t and u.  At equilibrium the forces should all be 0.

Examining the reaction in s (above) we see that it is proportional to the dot product of the tangent at B with the sum of the vectors BA and BC. Hence the median of the triangle (which is (BA+BC)/2) is perpendicular to the tangent.

This is clearly also an equilibrium position for the same set of springs attached to the sides of the tangent triangle.

The problem of finding the point with minimum sum of squares to the sides of a triangle is solved by the Symmedian Point, the reflection of the medians in the angle bisectors.  
eBooks

A breakthrough in the use of interactive diagrams for gaining new insight into mathematics.
Available on iTunes

Our Euclid's Elements now covers books 1-6 and contains over 180 interactive diagrams.
Available on iTunes

L Van Warren has combined Geometry Expressions apps with the Kindle in his latest eBook available from amazon.com here
11 Geometry Expressions  eBooks are now available as a bundle for the recession-friendly price of $85.95

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