PRESENTATION: Impossible Triangles
An introduction to triangles, with a twist – literally.
One great way to start any topic on triangles, whether that be angles, 2D shapes, types of triangles, constructions,… etc is to start talking about what makes a triangle a triangle. What better way to determine the necessary features of a triangle than by considering shapes that are very similar to triangles, but are most definitely not triangles?
An impossible, or Penrose, triangle is a 3D construct that is based on a 2D triangle. It is formed in such a way that it would be impossible to construct it in real-life. The presentation above introduces the idea of an impossible triangle (and other impossible shapes) before giving step-by-step instructions on how to draw one, which the students can do themselves. It ends by showing where impossible shapes have been used in art; in particular by M. C. Escher.
VIDEO: Pascal’s Triangle
Pascal’s Triangle seems simple from the outset but it has numerous applications to many areas of mathematics and particularly to the binomial expansion. It allows you to expand expressions of the form (ax + b)n with relative ease as the rows of Pascal’s Triangle give you the coefficients needed for the expansion. These coefficients also give us the number of ways of choosing k objects from a possible n.
There are also a number of hidden sequences and patterns inside Pascal’s Triangle which have applications to many different areas. One of which is Sierpinski’s Triangle, which you can find more information about below.
This video from Numberphile does well to explain what Pascal’s Triangle is as well as many of the different patterns that can be found within it.
GEOGEBRA: The Chaos Game
Sierpinski’s Triangle is a fractal formed by taking a triangle, marking the midpoint of each edge, joining them to make an upside-down triangle and then colouring that in. Each following iteration is found by repeating the process with each of the remaining un-filled triangles. The result after infinite iteration is a shape of zero area.
One way of constructing Sierpinski’s Triangle is to draw any triangle and place a point inside it. Assign one of the corners the numbers 1 & 2 (corner A), one of them 3 & 4 (corner B) and the remaining corner 5 & 6 (corner C). Then roll a dice. If it shows a 1 or a 2, move the point halfway between its current position and that of corner A. Then roll the dice again. If it shows a 3 or a 4, for example, move the point halfway between its current position (after already moving towards A) and corner B. If you repeat this many, many times, the resulting shape is that of Sierpinski’s Triangle. What?! How!?
The GeoGebra file does all of the above for you, to make it easier to show to a whole class via the GeoGebra web app. You can move the corners of the triangle to any points you wish or click Regular Triangle to get an equilateral triangle. You can drag the starting point to anywhere you’d like, inside or outside of the triangle. You can either press Random Number to watch the process dice roll by dice roll or press Start to watch the process much more quickly. Use the speed slider to control how fast the process moves.
As mentioned in the Pascal’s Triangle video above, it is possible to find Sierpinski’s Triangle within Pascal’s Triangle. The below video shows a nice animation of moving from one to the other.