111, 333, 666, and many other curious digits. What are Demlo numbers?

The Earth map used in the UN flag: 33 sectors

A grid, probably representing parallels and meridians, divides the Earth in 33 sectors. Another special number is 11, 33 being a multiple of it. You probably wonder what  it means and what relation have these numbers with the Earth.

33 appears immediately as a particular number, being a palindrome. Moreover 3×3=9;  33×3=99… another palindromic digit I like is the number 12321. If you sum all digits, you’ll obtain 9 again.

    Numbers  and the Bible

Thinking to these numbers  it could be easy to establish a link with the cubit of ancient Hebrews. The cubit value can be understood when we think that the Siloam inscription says that the water gallery built by the king Ezekias was 1200 cubits long. The gallery actually measures 533m, revealing that a cubit was 44,4 centimeters.

The ark of the covenant  in the Bible  is described having dimensions 2.5 x 1.5 x 1.5 cubits that are 1110 x 666 x 666 mm. All these numbers are obviously multiples of 111. A lot of numbers that describe the Earth are actually more understandable when you think to the cubit, multiples of 111, of 666, or maybe 33 and so on.

     The speed of light

Let’s consider the particular case of the speed of light. This speed is known to be 299792,458 Km/s in the vacuum. When we want to calculate the speed in the air (that is what actually interests us, since we live in the air) we have to use the formula        where v is the speed of light in the air, c is the speed of light in vacuum, εr is the dielectric constant of the air in relation to the vacuum. In relation  to the air , the root of  εr is 1.0003 that gives a speed of  light in the air v=299700Km/s. When you want to express this speed in cubits per second you obtain an incredible result: v=675000000 cubit/sec.

299700 is  a multiple of 111 too, being 299700=2700x111. The energy behind this speed is proportional to the square of the speed. (Einstein postulated the formula E=mc^2). So 299700^2 =89820090000. This square number is a multiple of the palindrome 12321, in fact we have:


12321 is the square of 111. But  when you sum up the digits of 111^2=12321 you obtain 3^2=9 being 3=1+1+1 and 9=1+2+3+2+1. These are Demlo numbers.

   Demlo numbers

These are the squares of multi-unit numbers. The first 9 Demlo are palindromes:

  • 12= 1;
  • 112= 121;
  • 1112= 12321;
  • 11112= 1234321;
  • 111112= 123454321;
  • 1111112= 12345654321;
  • 11111112= 1234567654321;
  • 111111112= 123456787654321;
  • 1111111112= 12345678987654321.

The sum of the single digits of these numbers is a square. This is the series of the Demlo squares:  1, 4, 9, 16, 25, 36, 49, 64, 81.  All numbers when multiplied  by a multi-unit number great enough become a Demlo number.

Other examples of Demlo numbers

In the globular Earth to each degree of latitude corresponds  111  Km.

The tilt of Earth axis is 23,4° that leaves, as a complementary angle, the terrible 66.6°. These numbers have been used to hide the truth in plain sight. Actually Earth dimensions are often multiples of 111 and proportional to Demlo numbers. So, to give an example, the radius of the Earth can be expressed as 19980 km=180x111. Interesting enough is the fact that the earth radius is proportional to 180 (180° is half a circle) while the diameter is 39960Km = 360*111 (360° is an entire circle). The surface is obviously proportional to the square of radius through the formula S=πxradius^2 . We have:

19980^2=39920400=32400x12321 Demlo applied!

Let’s consider now the trajectory of  the Sun.

We know that it covers a cone trajectory  following these data:

radius height
Tropic of Cancer 6660 6660
Tropic of Capricorn 13320 3330

We easily obtain:




6660×6660=44355600=3600x12321. Demlo applied!

13320×3330=44355600=3600x12321. Does 3600 mean anything to you? Aren’t the seconds in one hour?

Let’s  apply this criterion to another astronomical aspect: the precession. After this example you will probably understand and will start to apply  Demlo to the Earth.

The precession of the equinoxes

As you surely know, in one year the sun travels the entire zodiac but year after year it acquires a little delay in respect to the dome. This delay completes an entire circle, 360 degrees, in about 26000 years, as believed by many astronomers. Is it possible to  define precisely the period of this time by applying Demlo?

You probably know  the moon changes the slope of its cone around the ecliptic every 18.5 years (moon nodal precession). This cycle is contained in the sun precession cycle.Thus we have  to find a number around 26000 that is a multiple of 18,5 but also of 111  and apply Demlo. This number is 26640 years.

26640/18.5=1440 moon cycles in one sun precession cycle.  (1440 are also the minutes in a day, coincidence?);


But the more astonishing results is this:

26640×18.5=492840=40x12321.  Demlo verified  here too!

This convinces me that 26640 years is the correct number for the precession.

Applying Demlo to the Earth is a game : an endless number of applications are possible. Try!

A short note about the Indian mathematician who studied the Demlo numbers.

Kaprekar was an Indian recreational mathematician. He described several classes of natural numbers: the Kaprekar, Harshad and self numbers and he discovered the Kaprekar constant. He also studied the Demlo numbers, named after a train station 30 miles from Bombay, where he had  the idea of studying them. These are the numbers 1, 121, 12321…which are the squares of the repunits (repeated units) 1, 11, 111.

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