In the previous article of the series “Parallelograms and Parallelograms”, I discussed some interesting things about using a Parallelogram in mathematics. In this one we will move on to more advanced uses of the Parallelogram. Let us say that you are doing a problem such as the sorting of two arrays of numbers. You use your favorite graphing calculator, plot your function on a graph, plot the array onto your canvas, and then zoom in on the chosen array. Without even drawing the actual function or plot of your choice, you have already transformed your problem into a Parallelogram.
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A parallelogram is a type of quadratic equation. It has two quadrants, with each quadrant representing a different color. Using the quadratic formula you can find the area between any two parallelograms. Using a Geometry Parallelogram Worksheet you can find the solutions for all the quadratic equations involving only rectangular arrays of numbers. The best thing about the quadratic formula is that the solution can be applied to any geometric shape.
Using a Geometry Parallelogram Worksheet you can make sure that your graphing problem is solved correctly and analytically. For example, if you set up your worksheet like a puzzle, then each cell of the matrix is a different color. Each cell can be solved independently, by using the quadrilateral formula or some other method. To solve the horizontal and vertical elements separately you just add the corresponding quadratic formula to the appropriate cells. To solve the other cells (and the top row) you must use the Geometry Parallelogram application and solve the problem as normal.
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In a separate article in this series we will look at some more specific properties of the elliptic parallelograms that we studied above. We will look at how they relate to general elliptic equations and other functions. In this article we will not go into detail about the parallax properties, but we can say that the elliptic functions described above are completely fixed.
A typical set of elliptic parallelograms consists of eight parallelograms, each one of which is a paralaxyl (a parallel which is a closed curve). These parallelograms (which is perfectly spherical) contain only fixed points. This property was discovered more than fifty years ago by Ampere, and his theory is well known. It doesn’t, however, solve all the problems of elliptic equations and other quadrilaterals (parallelograms) – as yet there has been no satisfactory reply.
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The quadrilateral proof worksheet is widely used in teaching mathematics, and for that reason it is often included as an addendum to a standard algebra course. But beyond that it is also available separately as a free download. The Geometry Parallelogram Worksheet can be downloaded from the official website for just $9.95. There is also an additional forty-eight-page set of supplementary documents available as a PDF file.
The parallax formula for the plane is also widely used in the aerospace industry. In fact, it forms a key part of the calculation of aerodynamic lift and is especially useful for use in high-wing fighter planes. Many of the designs for these planes – such as the BAA (Baby Airplane) and the F-22 (stealth fighter aircraft) are based on the basic geometrical design of the parallax calculations.
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Included as an option to the Geometry Parallelogram Worksheet is also a selection of practice questions, which can be used in conjunction with the main workbook. A large section of the supplement deals with working through problems by using both the parallelogram and the worksheet. One problem even has a picture to help visualize the shape of a parallelogram. As usual, feedback is encouraged via the website. A blog offers more details.
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