# Posts from the ‘geometry’ Category

A quickie: here’s an interesting game from Canada where users have to find various interesting geometrical properties by eye and are assessed programmatically on their accuracy:

My score as about 3.03, having frustratingly crepty above an accuracy score of 3 with a shocking 9 in my final problem.

I was recently invited to do an IQ test. One of the questions was as follows: “You walk five miles north, five miles west, then five miles south. How far are you from where you started?” The answer that they were looking for was 5 miles.

Perhaps our use of maps convince us of this logic; a logic based on Cartesian 2-dimensional geometry. Unfortunately, we do not live on a Cartesian plane!

How should we understand this IQ test question when we correctly consider that we live on a sphere? A good way to consider this is to think about what happens at the two poles!

### The South Pole

Start at the south pole. Travel five miles north. Travelling west is to travel parallel to the equator, so then when travelling five miles west, you get no further from the pole. Then travel five miles south. You arrive where you started, with no distance between where you started and where you are now.

### The North Pole

The north pole example is more difficult to imagine, and some may think there’s a trick here. If you start five miles and a bit south of the north pole, move five miles north then five miles west in many tight circles around the pole ending up exactly opposite to where you were when you originally arrived near the pole. Then move five miles south. More or less, you are now ten miles away from where you started.

*The trick here is the ‘bit’ which ensures two things: firstly that you can in fact travel west; secondly that having travelled five miles west you finish up exactly opposite where you were when you started moving west. It might be argued that if you ended up exactly at the pole then you would be unable to move west at all. The ‘bit’ ensures that there is a trivially small circle around the pole that you can travel west around. It is also necessary to end up exactly opposite where you started to maximise the resultant difference between the two starting positions. If you imagine the bit as a radius of a circle around the pole, then it can be calculated as any r such that 5 miles = (2n+1)*pi*r where n is a positive integer.*

### Summary

Thinking about a problem often involves thinking around its extremes or limits. When thinking about compass bearings on a sphere, the poles offer places where their odd relationship to each other are most apparent. The IQ problem assumes that we live somewhere where the relationship between the compass bearings closely resembles the relationship between Cartesian axes. At the poles this similarity breaks down most markedly. By thinking about moving to and and from the poles, it transpires that if you move five miles north, five miles west, and five miles south, you may end up a distance of x miles from where you started, where x is such that: 0 miles <= x < 10 miles.

Now, if the IQ test was testing for this as an answer, I would have been suitably impressed!

### Philosophical Footnote

The earliest known argument against the earth being flat comes from Aristotle, who argued that the shadow that the Earth casts onto the moon during lunar eclipses is always circular. The only object which casts a circular shadow irrespective of its orientation is a sphere, and since night and day convince us that the earth does not have a constant orientation with respect to the moon and the sun, the Earth must be spherical. (Aristotle *De Caelo*, 297b31-298a10)

Our natural instincts about the world are perhaps that it exists on a plane that is looped; the outside of a cylinder with the poles at the top and bottom of the cylinder. This is a practical simplified model of the earth because until you get into the arctic and antarctic more or less, two people moving north are moving more or less parallel to each other, and the consequences of longitude and latitude working on quite separate principles need not be considered.

We could of course change the way that north and south work, and make them akin to east and west. Perhaps the great circle through Grenwich could be the East-West equator, as in some senses it is, and we could therefore define an east and west pole, somewhere on the equator! It is an interesting thought experiment. Would our concepts be more easily understandable if we did this? Why did North and South become defined as it is?

## Sketchpad Files

Geometer’s Sketchpad is one of my favourite programs but extremely annoying to author in. Here I have created a zipped archive of some of my past Sketchpad efforts for anyone who wants them. I have included a brief description of each file below.

If you don’t have Geometer’s Sketchpad then these will not be very useful to you. You can get hold of a copy from The Curriculum Press in the UK.

#### ConstructingAnEllipse

An extremely simple file exhibiting the construction of an ellipse defined by two centers and a point. The problem is that the “Locus Constructions” that are revealled don’t tell the whole story; how did I arrive at choosing those two circles to construct the circle with? This requires thought about the definition of an ellipse and the nature of construction.

#### ConeFromCircle

If you cut a sector from a circle (the bright-red sector) you can join the two radial sides together to form a cone. What is that cone’s volume? How do you maximise the volume of the cone? This sketch allows an exploration of the structure of the problem prior to doing the maths on it. It is a good exploration for construction of graphs at lower levels, or using calculus at higher levels.

#### Projectile Motion

A useful sketch for teaching calculus, start with the initial position of the projectile on the ground. Some of the ‘features’ on this sketch are odd and may require some exploration before they make sense!

#### ProjectiveCuboid

An example of how to construct a cube using two-point perspective

#### DoQuadrilateralsTessellate

I believe that this sketch comprises a proof that all quadrilaterals do in fact tessellate. It is an interesting question to whether it does in fact constitute a proof (what is a proof anyway?!) Choose your quadrilateral and click the buttons in order to follow the argument steps.

#### HexagonalSlideTessellation & HexagonalRotationalTessellation

Dynamic examples of how to distort tessellating hexagons in such a way to ensure that the result also tessellates.

#### Egg

An interesting construction of an egg using four arcs of four circles, it questions students to consider the relationship between mathematics and aesthetics.

A plat diviseur is a plate designed to make portioning cakes easy. Click here to see one. It works by putting dots at appropriate angles around the rim of the plate, so that when someone wants to cut, for example, five portions they merely cut from the center of the plate to wherever there is a five. For neatness one cut is always common, and tends to have just the number 0!

Displayed on screen this image could be used to motivate an activity in which students practice finding angles by creating their own plat diviseurs. It can be quite a fun activity, especially if you buy paper plates for them to use (though I recommend that they practice on paper first!).

They could also be asked which numbers are absent; they hopefully notice that 2,4 and 8 are missing. Discussing why can get into thinking about halving and fractions.

The reciprocity of 3 and 1/3, 5 and 1/5 can also be brought up, by discussing how often the image of a divided circle is often produced as a visualisation of proper fractions.

Since 5 and 7 are not factors of 360 this is also an interesting discussion point. While discussing factors, they should note that points for 3,6 and 9 are coincident at two points, and that if 2,4 and 8 are added what other numbers would have coincident points. I remember asking whether 10 and 12 would have coincident points if we had them as divisions too, which caused some thought.

Note that there are very few good images of plat diviseurs on the internet. I originally got the idea from Problem Pictures.