Fixed Area, Infinite Perimeter

The Koch Snowflake (named after its inventor, the Swedish mathematician Helge von Koch) is a fractal with a number of interesting properties.


The first four generations of the Koch Snowflake

As the number of generations increases, the area of the snowflake increases, but it increases towards a limit: eight-fifths of the size of the first (triangular) generation. This is because each additional generation adds three triangles which are one-ninth the size of the triangle added in the previous generation (for a total increase of one-third), and the additional of increasingly small triangles has a lesser and lesser effect on the overall area as the number of generations increases.

But as the number of generations increases, the perimeter of the shape continues to grow, without approaching a limit: each generation of the Snowflake has a perimeter which is four-thirds of the previous generation’s perimeter. This is because the additional perimeter added each time is four times one-third of the length of each side, for a total increase of four-thirds, as opposed to the one-third increase in area. Because four-thirds is greater than one, the perimeter tends to infinity, whereas the area (which at one-third is less than one) does not.


As you can see from the graph above, the area approaches its limit very quickly, whereas the perimeter grows very quickly (which is why it has to be shown on a different axis).

Veblen and Giffen Goods

There is a fundamental law of economics that says that as the price of a good or service increases, the demand for that product decreases. This is the Law of Demand: if prices are high, people cannot buy as much. But there are some products for which this is not the case.

Veblen Goods do not obey the Law of Demand: as their price increases, so does demand for them. This is a case of conspicuous consumption, people show off and demonstrate their status by buying increasingly expensive products. A £10000 gold Rolex watch does not really tell time any better than a £10 Casio, but owning a £10000 gold Rolex demonstrates that you are so wealthy that you can afford to spend £10000 on a watch. People do not want to buy a £10 Rolex, but they do want to buy a £10000 Rolex, to show off how much money they have.

There are some goods to which the Law of Demand does not apply, and which are not Veblen Goods. These are called Giffen Goods, and on the face of it, they seem to disobey all rational economic rules: demand for them increases even when their price increases, despite the fact that they cannot be used to demonstrate status via conspicuous consumption.

But imagine this situation: You need to feed your family. You normally buy a mixture of expensive (tasty) and inexpensive (staple) products to provide enough nutrition, but also some variety. But one day the price of rice quadruples: now you can’t afford the more expensive products and the rice, and you still need the staple rice to provide basic nutrition. Thus you buy more rice, even though its price has increased, and the rice is a Giffen Good.

Types of Perpetual Motion Machines

Perpetual motion machines (which don’t exist) come in many forms.

Perpetual motion machines of the first kind violate the First Law of Thermodynamics. They produce mechanical work (i.e. movement) without any energy being input; this violates the principle of the conservation of energy.

Perpetual motion machines of the second kind violate the Second Law of Thermodynamics. They convert thermal energy directly into mechanical work, with no exhaust heat being emitted; this violates the rule of the production of entropy, that entropy in a system must always increase.

Perpetual motion machines of the third kind do not, as their name suggests, violate the Third Law of Thermodynamics. Rather they claim to maintain motion (once started) forever, and do so by assuming that frictional forces can be eliminated completely (often through operating in a perfect vacuum, which is also not possible).


Altitude and Flight Level

The term elevation refers to the position of a point or an object on the ground above a fixed reference point, usually mean sea level. The term altitude refers to the position of a point or object in the air above a fixed reference point.

But defining altitude can be difficult, and so when altitude is referred to, the reference point must always be explicitly defined. Altitude is normally measured above mean sea level (AMSL) or above ground level (AGL). For example, if an aeroplane flew over the peak of Mount Everest, its altitude could be referred to as either 38 000 ft AMSL or 9000 ft AGL, because the peak of Everest has an elevation of 29 000 ft.

For aircraft it is difficult to measure altitude. You might think that GPS could provide this information, but GPS is designed for positioning on the surface of the Earth and isn’t very good at measuring altitude.

Pilots have always used atmospheric pressure to measure altitude. As an aircraft moves further upwards into the air, there is less air above it pushing down on it and the pressure decreases. This decrease is predictable and easy to calculate:

p_h = p_0 \left( 1- \frac{\gamma h}{T_0} \right)^{\frac{gM}{R\gamma}}

where p_h is the pressure at height hp_0 is the standard atmospheric pressure of 1013.25 hPa\gamma is the rate at which temperature decreases with altitude (the temperature lapse rate); T_0 is the standard temperature; g is the gravitational field strength; M is the molar mass of dry air; and R is the molar gas constant.


Aeroplanes do not fly at a set altitude. Rather they fly at a given flight level, which – although it sounds like a height – is actually a pressure. When a pilot flies at a flight level of 32 000 ft (FL320) they are actually flying at a constant pressure of 275 hPa, and may actually be far above or far below this altitude, depending on the local weather (and therefore pressure) conditions. If they enter an area of particularly high or low pressure they will have to ascend or descend correspondingly.

Using flight levels helps to prevent collisions between aircraft: in the UK aircraft flying on headings between 000° and 089° (north to east) they flight at odd numbered flight levels (FL310, FL330, etc); flying between 090° and 179° (east to south) at odd numbered flight levels plus 500 ft (FL315, FL335); flying between 180° and 269° (south to west) at even numbered flight levels (FL320, FL340); and flying between 270° and 359° (west to north) at even numbered flight levels plus 500 ft (FL325, FL 345). In other parts of the world different flight level rules are used.

2014-2015 Review

Mr Reid travelling salesman problem (2000 nodes)

I had a very busy timetable at work this year, so I didn’t blog here as much as I would have liked. Hopefully the 2015-2016 Review will be longer than this one.

I enjoyed writing about rotational kinetic energy in September, and the difference between centre of mass and centre of gravity was something people don’t often think about.

A post on how latitude affects the length of the day was my favourite post from October.

November was one of my favourite months: I particularly liked the posts about fail-safes, citrus fruit and analemmas.

I only made two posts in December, about the colour of diamonds and the possibility of building a vacuum airship, but I liked them both.

January was also a good month. My favourites were two posts about ranking things properly, and the different types of microscope. I also wrote about different naming systems across the world, but I’m not sure I got my facts 100% correct.

A post about the thermal expansion of petrol was easily my favourite post from February.

I liked all the posts I made in March, particularly the posts about colour mixing (why is light red, green and blue, but ink cyan, magenta and yellow?), the order of adjectives in English, and alternatives to GPS.

April was also a good month: I wrote about the nuclear “double flash” and worked out a much better system of coinage for the UK.

A post about how to look at the back of your head using a black hole was my favourite post from May.

I didn’t make any posts in June. :(

I particularly like my post about why some particles don’t decay from July. I also wrote about the different type of multiplication and the world’s longest golf shot.

August was another good month (school holidays!). I wrote about the US’s “soft bomb”longitude, latitude and precision and the Kp index for aurora spotting.

Here’s to 2015-2016. I hope you’ll keep reading.