Poe's Law

Recently I come across a YouTube video “New Proof of a Flat Earth: Distance to the Sun? Shorter Version” Sasha appears in the video and talks about measuring angles from the horizon to the sun and how you can make an equilateral triangle which is a triangle in which all three sides and all three internal angles are equal. The video and all Sasha’s videos are done in such a way that its hard to take them or her seriously, I can only assume this is intentional as her channel is labelled as comedy but on the other hand she stats that she is passionate about flat-earth, so is she mocking it? I invoke Poe's Law on this and I will assume she is serious and believes the earth is flat.

On a flat plain, the equilateral triangle would work well, it will give you the exact distance to the sun and as stated in the video it would disprove the sun’s 93 million mile distance. Sasha gives us two places on earth that are the correct distance apart to see the sun at 60° at the same point in time. One is Macapá in Brazil and the other is Malabo in Equatorial Guinea. She also gives us the distance between them as 4134.6 miles, a quick look on google earth and I got 4144.7 miles so a 10.1 mile difference? I’ll use her figures as 10 miles will make little to no difference and is well within any reasonable margin of error. If you work this out using a triangle solver app like this one you will find it gives you a sun height of 3580.67 miles, a long way short of 93 million. Yep that's it the globe earth is done for the world can only be flat…. “Sparkle Sound” 

But wait, wouldn't the curvature of the earth change the angles? Yes and no, from the people standing into two cities measuring the suns altitude there should be no difference but the horizons they are measuring from will be vastly different in relation to each other. Sasha has thought about this and demonstrates how the curvature of the earth will affect the angles. As you can see in the image to the left (grabbed from her video) the 60° measurements taken from our two cities will diverge and simply get further apart the further you are from earth. one thing bugs me about this drawing, where is the working out? how do we know the drawing is correct? I know its hand drawn and it doesn't need to be exact or to scale, but it would be nice to have a little more than just two points marked with time and angle as our only verification. For a start would 4134.6 miles create that much curvature? I had to check this out for my self, time to do some maths!

Ok to test this for my self and demonstrate my finding I decided to test out a new app I found, its called GeoGebra (if you like maths and/or geometry you seriously need to check this app out) and I was pleasantly surprised how easy it was to use. At first I simply tried to replicate the above drawing and found I could manipulate positions of the two cities and the curvature of the earth at will and get the lines diverging, parallel or converging with out changing the 60° angle. this was no good at all I needed to know I was getting all the angles and distances right, so I made a circle and set its circumference to 24901 miles, the same as earth equatorial circumference. Next I set points “A” and “B” to slide along the circumference and the green line that lets us know the distance between the two. I then created another point and placed it 93 million miles away in the direction of the red arrow, this I joined to points “A” and “B” with lines and had the angle between them and the imaginary observers local horizon displayed in green. as you can see in the drawing to the right I used the distance Sasha gave us (within a tenth of a mile) and as you can see the altitude of the sun from our two cities would be incredibly close to 60° and converging on a single point. Infect if we want to get it to exactly 60° we only have to move our observers 2.8 miles closer together giving us a distance of 4131.8 miles.

So in short, if you had two people on a globe earth stand 4131.8 miles apart along the equator, one could watch the sun rise to 60° at the same time as the other watches it set to 60° with the sun that is 93 million miles away. “New Proof of a Flat Earth”? New Proof of Bad Geometry more like.

I’m going to leave you with the worksheet I created in GeoGebra. You should be able to have a little play with it.

Great ball of fire

The sun provides us all with light, warmth and food, keeping our home not just liveable but comfortable, all the sun gods throughout history should be no surprise. But we're not here to talk about gods, today's topic is the sun or more specifically the way it moves or appears to.

As we all know the globe earth spins making a stationary sun appear to move across the sky. At the end of the day, the sun appears to drop below the western horizon, travelling under our feet before risings from the eastern horizon the following morning. On flat earth things are very different, the sun and moon float about 4000 miles above the ground moving in a circle that follows the equator. both shine there light down on earth like spotlights only illuminating half (approximately) of the world at any one time.

I have spent quite some time looking in to this and at first I assumed that the sun on a flat earth would orbit over the top and then underneath much like in the Discworld series of book. If you don’t know about the Discworld books I highly recommend you read one or two and I’m shore you will want to read the rest. The Discworld its self is a flat world ten thousand miles across which rests on the back of 4 elephant's which in turn stand on the back of the giant star turtle Great A'tuin. But let's get back on track. The flat earth is very different from the Discworld, for a start the ocean's don't flow off the edge as the Antarctic stretches all the way around the rim, it also doesn't rest on the back of any animals but its exact nature is subject to debate. Some say its a flat disc, others say its a never-ending flat plain, furthermore some say the earth is covered by a dome called the firmament and then it all starts to get complicated with every theory splitting off into two or three others. For my own sanity, I’m going to ignore some of the more outlandish theory's and stick to the basics that most flatheads can agree on. 

When looking at how the sun would work I kept getting the same thoughts and I decided to do some maths to see if things would look the same as I have witnessed time after time from my own back garden. If you look into the distances involved you will come across some quite contradictory figures, most flat heads believe the sun is somewhere between three and four thousand miles high, but I’ve come across as little as 35 miles. To keep things simple I'm going to use the figures that seem to be the most common and to simplify things further I will be calculating the suns movements as if it's at the equinox. This will give me a sun that is 3000 miles high and 32 miles across. Assuming the equator is half way between the north pole and the rim and that the equators circumference is the same as that of a globe earth the earth's diameter must be 25000 miles.

So just to recap the figures I'm going to use are as follow

• Sun’s diameter : 32 miles
• Sun’s height : 3000 miles
• Earth's diameter : 25000 miles
• The Equator from North pole : 6250 miles

I used MS Excel to do the maths involved, mainly because I’m familiar with using formula in Excel but also it would be easy to send the file to any one who wishes to look at my workings. The following is the work sheet I created to show the flat earth model working, the graph at the top shows a representation of the flat earth as viewed from above with the north pole in the centre. positions “A”, “B” and “C” are lined up along the Greenwich prime meridian. the box on the bottom left displays the figures as well as the time. the box on the bottom right gives information about the sun from each of the points labelled.

Let's have a closer look at position “A” the rest will then be self-explanatory. Position “A” is 1250 miles from the north pole and at this time its 5438 miles from the point which the sun is directly over. Using this distance and the suns height (3000 miles) we can work out the angle above the horizon the sun would appear to be, at this time from “A” it would be 28.88° (I’m not going to go in to the maths but if you want to check it you can do so here). The last bit “Sun Angular Size” is simply how small the sun would look with the naked eye. (not that I recommend looking at the sun with the naked eye as it can do some serious damage however if you feel the urge to, please don't let me stop you)  The sun always stays the same size but due to it being further away it will look smaller, so to work this out all I need is the distance to the sun its self (which works out to be 6210 miles) and its diameter, if you assume its a right-angled triangle you can use the same maths as before.

As you advance time on the worksheet you will see the distances and angles change in accordance with how things would appear if you were stood at any one of the three positions marked on the graph.

the following are examples of different times of the day

The first chart shows you the figures for 6pm this should be sunset, but if you look the lowest angle as seen from “C” the sun is still 13.12º above the horizon. Worse still is the apparent size of the sun, if you've ever paid attention to the sun as it passes over your head it always look roughly the same size and yet the maths says that on a flat earth the time of day will drastically affect how big it looks. As you can see the graph on the far right is set to 12 o'clock midnight yet the sun is still above the horizon, infect using these figures the sun will never go below 9° and will always be visible (unless its obscured by something like a building or a big hill) and it appears to have halved its size compared to 3pm. Point “B” sees the sun seemingly accelerating towards you before slowing down as it moves off into the distance more like what you would expect if you watched a passenger jet fly over. The chart to the right shows the angular size and altitude every hour for point “B”, these numbers fly in the face of anything I would see if I was stood at the equator observing the sun.

All my charts and figures are very well and good but not very visual and unfortunately I'm not all that good at animation, however as I was looking into this I stumbled across the following gif file that displayed exactly what I was trying to. I don't know what distances are used so the numbers are a little different but the results are more or less the same. the software used to create it can be found here. Ill let you scrutinise the picture your self as its mostly self-explanatory

After carefully calculating all the angles and distances on a flat earth the results just don't fit with real-world observations. The sun does does not grow visibly as it approaches noon before shrinking off into the distance. Some flatheads may show you photos or videos of the sun shrinking as it setts but this is an artefact of the lens or the sensor being flooded with light in one spot and struggling to deal with the information, any good photographer will know this and should be able to suggest a way or two to fix it. (an ND filter or small aperture, even the use of HDR may help).

In the real world the sun appears to move across the sky at a consistent speed never slowing as it reaches the horizon, but on the flat earth the sun would appear to slow as it approaches the horizon, a horizon it will never actually meet or dip below. but flatheads have this covered as well, saying perspective will make the sun “set” as it travels beyond the vanishing point, however, a vanishing point is simply an imaginary point on a picture or graphical projection to create the correct visual perspective and so making it seem more realistic. Perspective will only make the sun look smaller it will not make it move out of sight as if setting below the horizon.

in conclusion, the sun from our point of view can not move round in a circle 4000 miles above us, it must stay the same distance from us and we spin creating the illusion of the sun moving across the sky. At 93 million miles any change in apparent size due to earth's spin would be unnoticeable with the naked eye and hard to measure with good equipment. The observable sun does not support the flat earth it sits squarely in the globe earth model.