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Geometry Expressions Newsletter
Geometry Expressions Newsletter
September 2013   
24 Whimsical Clocks
Each chapter of this new eBook for your iPad starts with an unusual clock inspired by a piece of mathematics.

The mathematics is then expanded upon with creative use of interactive diagrams. 

From kaleidoscopes to cubic splines, binary abacuses to Fermat Toricelli Points, Archimedes Trammels to Paucellier's linkages, the clocks illustrate familiar and unfamiliar concepts in a unique and surprising way.

If you liked Hilbert & Cohn-Vossen's Geometry and the Imagination, you will love this innovative eBook, available from iTunes.

How to... create a clock app in Geometry Expressions
simple clock Clocks, whimsical and otherwise, can be created from Geometry Expressions using the timer style for variables in an HTML5/JavaScript app.  Here's how to create a basic clock.
  1. Show the axes. 
  2. Draw a circle centered at the origin to represent the clock face.
  3. Draw three lines from the center to the circumference to represent the hands.  (Put these in the first quadrant for now).
  4. Constrain the angle between each hand and the y axis.  Use the variable h, m, s for the hour, minute and second hands.
  5. Use File/Export/HTML5/JavaScript App to create your app.
  6. Set variable h to have UI Type Timer and Timer Style 12 hour period continuous.
  7. Set variable m to have UI Type Timer and Timer Style 1 hour period continuous.
  8. Set variable s to have UI Type Timer and Timer Style 1 minute period continuous.
  9. Hide the axes and the points, and make the hour hand thick. 

The result can be seen here.  

Problem of the Month
circle isotomic
Circle isotomic
  

Given a circle, and a point P, the isotomic of the circle with respect to P is the locus of the reflection of P in the tangents to the circle.

 

Let A be a point on the circle, and let B and C be points on the line through A and P a fixed distance d from A.

 

Then the locus of B and C as A runs round the circle is a Circle Conchoid.


For what locations of P and values for d is the conchoid the same as the isotomic?
circle conchoid
Circle conchoid

   

An app which lets you explore this problem is here.



August Problem Solution
  
This one involves a little physics ...

Equal masses B and C are joined by equal length rods AB and BC of negligible mass. 

A is fixed and C is constrained to lie on an inclined plane.  Gravity acts downwards.

For a given angle of the inclined plane, what is the angle of AB?

Clearly if the plane is horizontal or vertical, the answer is that AB lies vertically.  How about in between?

   

An app which lets you explore this problem is here.

First, the physics: in static potential energyequilibrium, the system will have minimal potential energy.  As the two masses are identical, then their combined potential energy is proportional to the sum of their heights. To minimize the potential energy, then we need to minimize the sum of their heights. 

A Geometry Expressions model yields values for the heights of the masses in terms of the angle of the inclined plane and the angle between the first rod and the plane.

These may be comath in maplepied into, for example, Maple, added, differentiated and solved.

We can see from this that the tan of the angle between the pendulum and the inclined plane is 1/3 the tan of the angle between the inclined plane and the vertical.
eBooks

Our Euclid's Elements now covers books 1-6 and contains over 180 interactive diagrams.
Available on iTunes
common core book cover
Each group of apps consists of simple, short, and focused interactive demonstrations and explorations of a single standard.
Available on iTunes
van's book
L Van Warren has combined Geometry Expressions apps with the Kindle in his new eBook available from amazon.com here
11 Geometry Expressions  eBooks are now available as a bundle for the recession-friendly price of $85.95

Saltire author stands for NCTM board
Saltire author and teacher Larry Ottman is standing for election to the NCTM board.
 
If you are a member of the NCTM, you can vote here.

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