The sun over the flat Earth

Earth is flat and certainly you already have a good number of proofs. But, if the Earth is flat and motionless, you will wonder how the sun can move over it to perform what your senses really perceive.

To get an idea you have to deal with trigonometry.

The best way is to buy a sextant, as the one in the image, and make many measurements of the height of the sun in many days of the year and in many different places of the globe.

Learning to use a sextant and correct the altitude from refraction, aberration and parallax can be very interesting…if you have time and money for it.

But what about when you have not?

You can try to do like me and download an application that gives the height of the sun for every place in the world. So, I’ve downloaded Sun surveyor from Google play.

However, you should be careful since these applications are intended for the use relatively to a globe Earth geometry and data are calculated according to a globe geometry  that has never been actually tested. Anyway, when you consider values relating to a single place in the northern hemisphere, experience shows that heights can be used for a first approximated calculation.

Let’s suppose that we are in Rome. Rome has a latitude of 41°54’. You have to consider that in globe geometry each degree of latitude corresponds to a 111Km arc.

Rome is distant 41°54’-23°27’=18°27’ (2050km) from the summer tropic and 41°54’+23°27’=65° 21’ (7250km) from the winter tropic.

Sun surveyor gives these two eights angles for the solstices at Rome:

Let’s now calculate, with these two values, the height of the sun in winter and in summer.

  H=tg71.5°*2050= 6127km
    H=tg24.7°*7250= 3334km

When you consider some other place in the northern hemisphere you will find estimates from slightly different  to very different values due to the original  geometry setting  the software is based upon. Nevertheless, you can get a first impressive idea of the behavior of the sun which is running a conical spiral between the two tropics.

The above value of 6127km is slightl

y different in relation to the one obtained by Eratosthenes and to the radius of the Earth of 6356 km. Since we are dealing with the sun along a meridian we should take into consideration the polar Earth radius of 6356,752 km. The difference from our calculation can be an approximation and maybe a wrong altitude angle of the sun given by the software. The idea is, however, that on a flat Earth, an

experiment in the Eratosthenes’ style reaches as a result the sun’s height and not the Earth radius.


Day and night alternate in this model. This is not because the sun is hidden behind the other side of the Earth, but because it departs so much that it becomes invisible due to perspective.

Objection: if the sun doesn’t set on the other part of the Earth you should see the sun also during the night.

Answer: the sun disappears over the horizon due to perspective. It becomes a little smaller as it goes farther and it goes down till it becomes hidden beyond the horizon.

We have already discussed about perspective and about the one-point perspective. Lines that are perpendicular to the direction of sight converge to a point on the horizon as we can see in this picture. The lights get smaller and smaller as they get farther, till they completely disappear at the horizon. The sun gets only a little smaller because atmosphere refraction acts as a lens. When the sun, in fact, is low on the horizon, a thicker layer of the atmosphere is between the observer and the sun, that is thus seen as bigger.

Someone, looking at this image, made me a smart objection: “Let’s say that the space between two lamps there, is a 5 meter distance. We can see about 20 lamps before they get blurred, that means a distance of 100 meters. If a lamp is 5 meter high, this means that the light disappears with a distance that is about 20 times its height. Now let’s suppose the sun is at 6000km height, it should disappear at a distance of 6000*20=120000kms, that is clearly impossible”.


I can easily answer to this objection considering the perspective rules. Consider a corridor seen from a person 2 meter tall. He will see something like this:

This is however the same corridor seen by a child 1 meter tall.

What you will clearly understand is that, the higher is something moving upon you, the faster it gets above the horizon, and with a greater inclination. The sun is at least 6000kms height: it comes to the horizon faster and with a greater inclination than a street lamp that is only 5 meters high.

Can we establish with a greater precision the height of the sun? Yes, we can build an improved model by using the math we have introduced in the previous chapter. Watch the following table; we will try to explain it.


Height [km] Radius [km] Circumference [km]
Cancer tropic 6660 6660 39960
Capricorn tropic 3330 13320 79920

In the previous chapter I have dealt about irrational numbers and fractals. We can’t represent nature perfectly as it is, but we have always to rationalize, i.e. to adjust the description we make. At the same time we have to cut in the better possible place to have however a first good description of reality. We have also explained that reality can be described with fractals that have the property to show that parts are similar to the total. If we are able to find the description of the first fractal, the bigger one, we are able to have a good description of the whole, because it will contain an infinite number of repetitions of smaller fractals similar to the first. Here is the reason why, for example, in our description, we will cut pi to the value of 3.

Sun’s height= 6660-333+33.3-3.33+0.333…=6357,…= 6660-6660/20+6660/200-6660/2000…

It seems to be a good fractal description, with a good numerical result. Isn’t it?
As a consequence we can describe the height of the sun at Capricorn tropic with the height 3330km that is 6660/2. This way we have 6660 and the series of 2 at denominator.

I love this description that remembers me the fact that in the math of the Earth numbers with repeated digits, in Demlo style, frequently appear.

Someone can ask why 6660 and not, for instance, 6666 that is a repeated digit number as well. You certainly remember that Demlo numbers, that are digits that are fit to well describe many natural phenomena, have a deep relation with the repeated digit 1.  The number 111 for example is used to describe the globular Earth since one degree of latitude corresponds to 111 km on the meridian.

We know however that the Earth is flat and that Eratosthenes’ radius of the Earth is in reality the height of the sun. So if we divide 6660 for 111 we obtain as a result the integer 60. On the other hand, 6666/111 gives instead a result of 60.54, which is not fit to describe the bigger fractal. But we could discuss about this topic even deeper and find analogies with ϕ, the golden number I have examined in the previous chapter.

Consider the circumference of the two tropics we have obtained by using 6660 and divide that for 111.

Cancer tropic 39960/111=360
Capricorn tropic 79920/111=720


Can you notice the neat precision of the above calculations? Doing this way you can obtain that one degree on the tropic of cancer corresponds to 111km while one degree on the Capricorn tropic is 222 km.

The above 720 quotient we have just obtained is ten times 72, that is the value we have already found when talking about ϕ. 72° is the internal angle of the golden triangle of the pentagon, it is the fifth part of the circle angle and, moreover,  we can state 72=44,4*ϕ. Our description is a circle, a big circle, with an astonishing beautiful math inside.


This remembers me of the fact that everything around us, inside our cosmos, is perfect beauty. In the book of Job I found a description of Leviathan given by God Himself. Here are the words: “Whatever is under the heavens is mine. I will not be silent about its limbs. About its mightiness and its well formed body.” And this is a wonderful description of the universe we all live inside.

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