Iovandrake & the First Basic Law of Shapes

Well, I recently had a fun conversation with a flat earth proponent – aren’t they always fun?

It began with the usual heaping on of the furtive fallacy, bizarre and unfounded ad hominem attacks, appeals to motive and the egregious claim that no good debunkings of the flat Earth theories exist (especially those espoused by that great modern philosopher, Rory Cooper – see his mind-effluent here:

After a long process of providing argument after argument and evidence after evidence, the veritable titan of wisdom (that is, Iovandrake), produced the most humourous line I’ve ever read:Image

“The distance between any place on the earth is the same regardless of whether it is a flat circle or a sphere.”


Now, I shouldn’t have to work too hard to point out the major flaws in this argument. But let’s do it anyway….


Anybody whose understanding of shapes, when they were a child, went beyond just trying to put the funny looking blocks into their mouths and instead led to them trying to fit cubes into square holes, should have a simple grasp of geometry.

Indeed, anyone with a basic grasp of geometry should have learned an important fact, which we can happily call “The First Basic Law Of Shapes”: In order for the distances between ALL points on any 2 objects to be the same, those objects have to be congruent – that is to say, they have to be geometrically identical, which means the exact same size and shape. Don’t confuse this with genetically identical, which is what I’m betting Iovandrake’s parents were….

After pointing this out, and explaining that he must then believe circles and spheres are the exact same thing, I was told that this wasn’t what he meant…. before he went on to make the exact same argument. Repeatedly.






In fact, what Iovandrake was proposing was that he’d found a way to do something that the entire human race has found to be impossible.

The example is simple. Look at a flat map of the world. Now look at a globe. You should be able to see a big difference, other than one being flat and the other being spherical. The land masses and the seas are not quite the same.

Cartographers and navigators have known about this for a long time. It is impossible to make a true flat representation of a spheroid.

Anybody who uses hiking maps will know about a little key on those maps, pointing to grid north, true north and magnetic north – and they will know that these maps differ in how big the angle between grid north and true north is. This is because, even though these are the closest representations of the surface of the Earth, if you stitched together every map of the same scale from around the world (if you could get ones for the oceans as well as lands, that is), you wouldn’t get a true representation of the surface of the Earth – or even of the distances as portrayed from one map to the next.

But Iovandrake thinks it is possible to take a sphere and a circle (hell, I’ll even give him just a hemisphere and a circle if he likes, it doesn’t change a thing) and place 2 maps upon them that perfectly agree with each other.

Iovandrake’s conjecture seems to be a common one among flat Earth proponents, eager to insist that whatever measurements we can make to supposedly prove a round Earth, can also fit a flat Earth. In this regard, they are not unlike Geocentrists, who also like to claim that any astronomical predictions made using a heliocentric model would be the same using a geocentric model, without backing that claim up by showing their workings.


What can I say?

Well, some basic geometry may help.

To begin with, we can demonstrate that on a sphere you can have 2 circumnavigational distances of the same length. You can take one circuit around the Earth tracing all points equidistant from one pole, and find it has the same length as a circuit around the Earth tracing all points the exact same distance from the other pole.

On a flat earth no 2 circumnavigational routes (tracing all points equidistant from the center) can be the same length. Simply put, anyone traveling around the world in the Southern hemisphere will find it takes them less time than traveling around the earth at the equator, and yet they can find a line of latitude in the northern hemisphere where their route takes the same amount of time. This is impossible on a flat Earth.

Now, some flat Earthers try to claim that nobody has traveled such routes, so how could we know – and some even seem to claim that we couldn’t tell if the antarctic actually lies at what we would call the equator, blocking our passage to what we would like to think of as the southern hemisphere.

As I will show below, none of that matters in order to prove Iovandrake’s conjecture to be wrong.

In order for the distances between all points on any 2 objects to be the same, they must necessarily have the same area. NB, we’re only talking about points on the surface of the shapes here, so volume doesn’t matter. Though it’d be worse if it did, because flat shapes have no volume by definition. They must also have the same dimensions – all their angles and edges must be equal, which in this case means that the circumference of the hemisphere and the circumference of the circle must be equal.

So, if we just assume a hemisphere for a moment (spheres get even more messy for Iovandrake’s conjecture), then we shall need to calculate the quarter circumference, which would be the same as the distance between any point on the hemisphere’s rim and its top. This will be analogous to the radius of the flat circle. The circumference is 2 x pi x r, just like a circle, so the quarter circumference will be 1/2 x pi x r.

Now, the area of a circle is pi x r^2 – as I hope we all remember.

The area of a hemisphere (not including it’s base), is naturally half the area of a circle (4 x pi x r^2), which makes it 2 x pi x r^2.

Hmmm, I can spot a problem coming up already. But let’s go with it.

Let’s give the Hemisphere a radius of 100, nice and simple (units don’t matter here, we could be talking about miles or millimeters for all we care). That gives it a circumference of 628.32.

Let’s take a quarter of that, which is 157.08. This is the distance along the surface from the top to any point on the rim. So, this distance must be equal to the distance from the center of the flat circle to any point on its rim, so we’ll make this our flat circle’s radius.

Spot the problem yet? It will become very clear soon.

Now, calculating the area of our flat circle, we find it has an area of 77516.05.

OK, so let’s calculate our hemisphere’s area (again, not counting the base), which will be half the sphere’s area. Remember, its radius is 100. This gives us an area for the sphere of 125663.706, which we halve to get the hemisphere’s area of 62831.853.

Hmmm, we have a problem, don’t we? Yep, the flat circle doesn’t have the same area as the hemisphere, which means that all the distances between all the points on the 2 shapes cannot be the same. In fact, apart from the distances between the centers and any points, none of the distances are the same – and we can prove it.

As we’ve calculated, the hemisphere has a circumference of 628.32, but the flat circle has a circumference of 986.96.

This means that all the distances between all the points are larger on a flat circle than they are on the hemisphere, except for the distances between the center and any points on either shape. In fact, as the circumference of the circle is over 1.5 times that of the hemisphere and the area of the circle is over 20% larger than the area of the hemisphere (and these ratios will be constant no matter what number we wish to begin with), the difference in the distances is going to be obvious and measurable.

Now, some may argue that the Earth is an oblate spheroid. Well, that’s OK, we just calculate for the oval of the polar circumference. The equatorial radius of the Earth is 6378.1 km. The polar radius is 6356.8 km. The more astute among you will realise from this alone that it’s not going to magically make this flat Earther’s claim true.

For one thing, the polar circumference is then 40008 km. Quarter this again for our distance from one pole to a point on the equator, and we get 10002 km – which must be the same as the flat Earth’s radius (the distance from the center to a point on the rim). But we have a problem straight away. The circumference of the flat earth is now 62844.42 km, whereas the circumference of the Earth with an equatorial radius of 6378.1 km is 40075 km.

Hmmm, we have a big problem, don’t we. The distances here are still not the same.

Neither are the areas.

The area of the flat Earth clocks in here at 314284941 km^2.

The given area of the Earth with the above dimensions is roughly 510,072,000km^2, again halving for the hemisphere area to give us 255,036,000 km^2. Still nowhere near.

For those interested, the equation for an oblate spheroid is:

2 x pi x a^2 (1+(1-e^2/e) x tanh^-1(e)), where e^2 = 1 – a^2/b^2.

a is the semimajor axis (the equatorial radius, here) and b is the semiminor axis (the polar radius).

And the equation for the circumference of an oval (the perimeter of an ellipse) is roughly:

pi x [3(a+b) – sqrt((3a+b)(a+3b))]


Now, even more people may complain that I’ve only taken half the earth here. Well, let’s see what happens when we take the accepted distance from the North Pole to Antarctica, and turn that into a hemisphere (as it would seem the flat Earthers want us to do) and also translate it to a flat Earth.

We’ll take a ring 1000 km from the South Pole to be the hemisphere’s and the circle’s rim.

This would give our hemisphere a polar radius of about 12713.6 km a quarter polar circumference of about 19,002 km, an equatorial radius of about 12756.2 km, an equatorial circumference of about 80149.57 km, and an area (excluding the base, again) of around 1,019,000,000 km^2.

Give our flat circle the quarter circumference of 19,002 for its radius (because the distance on the surface from the top of the hemisphere to the rim must equal the radius of the flat circle), and we find it has a circumference of 119,393.08km and an area of 1,134,353,721.55 km^2.


Well, what can we see? There’s still a difference of around 39,ooo km in their circumferences and there’s still a difference of over 115,000,000 km^2 between their areas – that’s an area over 11 times the size of the USA. And we wouldn’t notice that?

Now we can compare this flat Earth to the actual spherical Earth and find that the area is over twice as big – and the longest circumference on the Earth is less than half as long as the circumference of the flat Earth now.

In fact, we can also notice that there is a massive difference between the spherical Earth and the hemisphere. The reason for this is that hemispheres and spheres are not congruent either, thus we would find it easy to spot whether everywhere we’ve explored is the whole of the spherical Earth as opposed to just down to the equator – and it would be just as easy as discerning if we were on a flat or a round Earth.


So, we can safely assert that it is impossible for the statement “The distance between any place on the earth is the same regardless of whether it is a flat circle or a sphere” to be true. We can also safely assert that distances on land masses and the distances between them would be very different, and that we would be able to easily calculate the difference between these distances.  Not only that, but by measuring these distances, we should be able to determine which model is correct, without the hassle of going into space.

I shall endeavour to go further into this in a later post.

Suffice to say, for the moment, that we can easily state that when Iovandrake claims that “there is no difference between REALITY for either a flat earth or a spherical one”, he is clearly demonstrating that he hasn’t got even a child’s grasp of geometry and he has no idea about The First Basic Law Of Shapes.

This, unsurprisingly, seems to be a recurring theme in flat Earth claims.


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