A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.^[1] The centered octagonal numbers are the same as the odd square numbers.^[2] Thus, the nth centered octagonal number is given by the formula

The first few centered octagonal numbers are^[2]

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089.

Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.^[2]

See also[edit]

• Octagonal number

References[edit]

The New Jerusalem Diagram – Introduction

“The New Jerusalem Diagram” is the name given by the scholar of ancient lore, John Michell [The original explanations in John Michell's books, especially "The Dimensions of Paradise", Thames and Hudson, 1986] to ageometrical construction that allows the “squaring of the circle” and the division of the circle to 28 equalsectors (a zodiac wheel) and the ceremonial positioning of the twelve tribes (or constellations) in a pattern of four camps at the four sides of the square.

John Michell has reconstructed the geometric pattern of the ‘heavenly city’ which is the template of the New Jerusalem of Revelation 21. 

He discovered that this diagram served as the design (blueprint) for many sacred sites, including theStonehenge, The Great Pyramid, St. Mary’s Chapel at  Glastonbury and the City of Magnesia in Plato’s The Laws.

The Great Pyramid, Earth and Moon fit the New Jerusalem Diagram

It turns out that also the Nazca glyph called “Mandala” aka “Sun-Star-Cross” perfectly matches the New Jerusalem Diagram.

 Superimposed image of NJ Diagram and Mandala glyph near Nazca, Peru

It is no surprise that the NJ diagram matches blueprint for the The Dome of the Rock, a shrine located on the Temple Mount in the Old City of Jerusalem.

The wheel of 28 divisions is common to various esoteric systems and serves as representation for the schematic month: 4 weeks = 28 days. The solar calendar of the Dead Sea scrolls sect was of 364 days, which is a multiple of28 ( 28×13 = 364). In many Sufi diagrams, including in the ceiling of the dome at the El-Aqsa Mosque* (not the Dome of the Rock) at the Temple Mount, there appears the motif of the wheel made of 28 divisions.

Al-Aqsa. The resplendent ceiling of the Al-Aqsa Mosque on the Temple Mount in Jerusalem. The mosque, one of Islam’s most holy sites, sits near the Dome of the Rock and above the Marwani Mosque, aka Solomon’s Stables.
Image Source/Credit.

The NJ diagram is also hinted and enfolded (according to Micheell), in the description of the holy city, The New Jerusalem, whose descent from heaven concludes the Christian Bible.

 Below we present 3-D version of the New Jerusalem Diagram.

Is it possible the 3D NJ diagram was used as a blueprint for the Crown Jewels?… Perhaps this is a case of  a “secret hidden in plain site”?

The collective term Crown Jewels denotes the regalia and vestments worn by the sovereign of the United Kingdom during the coronation ceremony and at other state functions. The term refers to the following objects: the crowns, sceptres (with either the cross or the dove), orbs, swords, rings, spurs, colobium sindonis, dalmatic, armill, and the royal robe or pall, as well as several other objects connected with the ceremony itself. The Crown Jewels have a religious and sacral connotation. Their hidden  symbolism seems to be related to the sacred geometry of the New Jerusalem Diagram.

The Construction of the New Jerusalem Diagram

The construction of this diagram starts with the attempt to “square the circle” by drawing a circle and a squarethat have the same circumference. 

• So in building this sacred diagram, one starts with a circle with a diameter of 22 units, and blocks it with asquare of 22 x 22 units (like the number of the Hebrew letters, with which supposedly “heaven and earth were made”).
• Then one draws from the same centre an additional circle whose diameter is 28 units – “the perfect number” that is already the number of the letters in the first verse of genesis and marks the circle of the seasons. The circle with diameter 28 has circumference of Pi x 28, which is as noted 22/7 x 28 = 88 – just the circumference of the square with edges of 22 and circumference thus 2 x 22 = 88.

The circle and the square have the same circumference, and they intersect in 8 points, where the square juts out of the circle in 4 vertices.

•  The building of the complete pattern continues with the drawing of 12 circles with diameter of 6 units – the number of letters in the first word of Genesis – Bereshit – as the first “perfect number”. The centers of these twelve circles are situated on the outer big circle, whose diameter is 28, and they touch (are tangent to) theinner circle, whose diameter is 22. Four of these circles are situated according to the points were the square is tangent to the inner circle, and the other 8 circles are placed according to the 8 points of intersection between the square and the outer circle.

There are two amazing consequences of this construction:

• This construction squares the circle
• It divides the circle into 28 equal segments
• It reflects ratio 22:7 which is good approximation of the “Pi”
• It reflects golden ratio 1.618
• It reflects the ratio of 22:6, which issues theoretical-geometrical considerations is also the exact real ratio between the diameter of the earth and the diameter of the moon! 

There are many more features of the NJ diagram which will be presented in our upcoming articles (e.g. amazing connection with Teotihuacan).