It’s pretty hard to get to grips with the scale of the solar system when Venus, the closest planet, feels really far already. But we’re practically squished together compared to Uranus and Neptune — they are a really, really, really long way away.

James O'Donoghue has made some brilliant videos about the solar system that explain this and more.

Solar system planets — distances to scale

For almost all mammals that’s more or less how much you got. Remarkably, from the teeniest, tiniest shrew weighing just a few grams, to the largest whale — passing through animals as different as robins, monkeys, sea lions, and giraffes in between — each mammal has broadly in the region of 1.5 billion heartbeats. You can see the striking regularity in the logarithmic chart in the sketch.

What’s more, other aspects of mammals can be predicted to a surprising accuracy simply from their weight. A tiny animal will have a heart that goes like crazy, live only briefly, and reach maturity quickly. A large animal, in contrast, will have a slow heart rate, a long life, and take its time to adulthood.

The biggest outlier is us, largely because in recent years we’ve managed to significantly extend our lifetime from what would be expected from Nature.

This gem I learned once again from Scale by Geoffrey West.

1.5 billion heartbeats in a lifetime

This is the fascinating observation that it’s not straightforward to say how long a coastline is. If you were to measure the coastline of a country by using a ruler on a globe, you would come out with a vastly different number than if you were to pace around the edge. The closer you look, the more wiggles and squiggliness you come across and instead of converging on a more accurate length, the coastline just keeps getting longer. The smaller your ruler, the longer it gets.

This was originally spotted, incredibly, in the 1950s, by an Englishman, Lewis Richardson, when trying to check a theory he had that the likelihood of war between countries depended on the length of their shared borders. Remarkably, he found that the quoted lengths of borders varied significantly. While measuring on maps at different scales he saw that the smaller scale map he used, or the smaller the width of his calipers he was measuring with, the length systematically increased. When looking at coastlines, instead of borders, some countries had wigglier coasts and so the length increased at a faster rate with the scale — for instance, Norway’s coastline, with it’s crinkly fjords, increases faster than Britain’s, which in turn increases faster than South Africa’s, as he zoomed in. The rate of this increase later became known as its fractal dimension.

Long after Richardson’s research, Benoit Mandelbrot published a paper How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension that discussed how the wiggliness of something like a coastline at one scale can be repeated at smaller and smaller scales. The work led to the later term fractals. Many other things exhibit fractal-like behaviour such as river networks, borders, brains, frequencies, lightning or even the stock market.

More mindbending mapping, and the difference between Great Britain, the UK and the British Isles.

There’s a super section on this in Scale, by Geoffrey West.

The coastline paradox

This is the idea that people across history budget on average about an hour a day for traveling. Originally observed by Yacov Zahavi, it came from the observation that even as our means of transport sped up, instead of traveling less each day we would just travel further. As in the sketch, this leads to a conclusion that the size of our cities increases partly according to how efficiently we can travel in and out—they generally increase to a size that maintains about an hour round-trip journey.

Cesare Marchetti observed that Greek villages and ancient walking cities, for example, as marked by ancient city walls, tend to share a radius of about 2.5km. Walking at a typical 5km/hr that makes an hour round trip journey. He also pointed out that: “Even people in prison for a life sentence, having nothing to do and nowhere to go, walk around for one hour a day, in the open.”

Unfortunately, if you live in London, the average round-trip seems to be an exception of 1h30.

This is also known as the constant travel time budget hypothesis.

Also see isochrones. It might be interesting to look at these according to transport method…

Image adapted from: Marchetti, C., 1994: Anthropological Invariants in Travel Behavior, Technological Forecasting and Social Change , 47 :75–88, Internal Publication, International Institute for Applied Systems Analysis.

Marchetti’s constant

I assumed fun is all the same. Fun is fun. But then, when I first heard about Type I and Type II fun, I had to admit there was some validity in there being different types of fun.

Type 1 fun is like eating ice cream, or skiing with ski lifts to whisk you back up to the top — you love it while you’re doing it, and when you think back on it you remember how good it was. Straightforward— sounds like fun.

Type 2 fun, on the other hand, is the sort of enjoyment you get from something like running a marathon or climibing a mountain. You often have to admit that it wasn’t necessarily all fun at the time. In fact, it could have been quite hard and might have hurt a bit (like the swimrun event I did last weekend). But when you look back on it, you think of it as often more fun and fulfilling than the Type I fun.

Type 3 fun is usually type 1 or 2 fun that went wrong. Though it was supposed to be fun, things actually got a little hairy – like you got lost, or it got dark, and it got cold, or you had to be rescued — but…it sure makes a great story when you make it back.

HT: my brother shared this with me having learned it from his mountain guide doing classic Type II fun.

The fun scale

For years, I’ve wondered at how as a kid I found the monkey bars in a playground (often a jungle gym in the US) so much easier to swing on than I do as an adult. And I see our kids swinging around easily on them, while now it’s hard for me just to hang off them. The reason why is partly to do with the square-cube law.

The square-cube law is the deceptively simple observation that as you scale up dimensions — say, as you get taller — then area of the object increases proportional to the square of the length, while volume increases proportional to the cube of length. This fundamental relationship has enormous significance in the sizes and shapes of animals and the limits of how big trees can get and how big we can build our bridges and buildings.

Because the strength of our bones, muscles or wooden beams is proportional to their cross-sectional area, this means that while the strength of a larger object, assuming the same proportions as the smaller one, increases according to the square of a length, the weight (from the volume it takes up) increases according to the cube of a length. So, relative strength decreases as size increases.

Some corollaries are therefore that children’s small size relative to the area of their muscles and bones makes them relatively stronger than adults and so they can more easily support their weight as they swing on the monkey bars. In the same way, press ups are harder for bigger people than they are for smaller. And while an ant has tiny spindly legs, those legs could easily support 50 other ants of its size, whereas for an elephant to support its own weight it needs chunky legs — not spindly ones — and there’s no chance it could get close to carrying 50 other elephants on its back.

It’s also part of the reason short trees can be spindly, but big sequoias have relatively chunkier trunks, and we need increasingly strong materials to build bigger and bigger skyscrapers compared to the shed in your garden.

Galileo discussed this back in 1638 and it still requires some getting your head around today.

For this and other fascinating insights of scaling check out Scale, by Geoffrey West.

The Square-Cube law

The clarity of the night sky. A low number means clear, black and amazing stars. Higher numbers mean gradually more urban environments with increasing light pollution. At 9 you’ll barely spot a star being pretty much in Piccadilly Circus or Times Square.

More detail needed? John Bortle, Light Pollution And Astronomy: The Bortle Dark-Sky Scale, Sky and Telescope, July 18 2006.

Sketchplanations tagged with scale