John Derbyshire set for himself a daunting task in writting Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics: to bring the complex world of math down from its ivory tower and present a glimpse of its magic to the laymen reader. For his challenge Derbyshire picked a riddle, the Riemann Hypothesis (RH), that has been tantalizing mathematicians for over a century; a conundrum that Derbyshire labels as the “greatest unsolved problem in mathematics.†In Prime Obsession, he not only attempts to tell the story of Bernhard Riemann and his famous hypothesis but to communicate the complex and high level math involved down in such a way that a laymen reader might glimpse its meaning. It is to his immense credit that Derbyshire makes this interwoven tale of math and history both interesting and illuminating.
A description of the RH should reveal the challenge Derbyshire faced:
All non-trivial zeroes of the zeta function have real part one-half.
Huh? As I said, it is a daunting task to attempt to unpack this dense subject so that someone outside a graduate program in math can understand it. Obviously, the author was well aware of this challenge when he set out:
To an ordinary reader, even a well-educated one, who has had no advanced mathematical training, this is probably quite incomprehensible. It might as well be written in Old Church Slavonic. In this book, as well as describing the history of the Hypothesis, and some of the personalities who have been involved with it, I have attempted to bring this deep and mysterious result within the understanding of a general readership, giving just as much mathematics as is needed to understand it . . . I think I have pitched my book to the level of a person who finished high school math satisfactorily and perhaps went on to a couple of college courses.
I can speak to this issue with some personal experience. After all, the last math class I had was as a sophomore in high school and, as I recall, I didn’t do that well. Despite a medium to high level of math phobia, I was able to work my way through (i.e. read some sections whole, skim others) the math and follow the arc of the story and the math. I won’t pretend I “understand†the RH but I have a greater appreciation for the concepts and, in fact, know a lot more about math in general as a result. All of this highlights the fact that the author is a patient, witty, and gentle teacher. If he hadn’t been, I would’ve given up on the book very early.
To spread the pain, Derbyshire alternates between math heavy and history focused chapters throughout the book. The result is a unique blend of careful calculation and logical building blocks with the history of mathematics during the last 150 years. In its basic form the book tries to build up the reader’s foundational knowledge to the point where he can understand the “who, what, and where†of the RH. The “who†is Bernhard Riemann and a number of other key mathematicians of the last century and a half. Derbyshire tries to give the reader a glimpse into the life and mind of this quite, shy, and pious visionary not just to provide the back drop to his famous hypothesis but also, I think, to extol his virtues as a man and a scholar. Part of this backdrop includes Riemann’s mentors, contemporaries, and the men who took up the challenge he discarded almost casually in his 1859 paper. Derbyshire touches on a host of interesting men and relationships: Carl Friedrich Gauss and his relationship with the Duke of Brunswick; Leonhard Euler – author of what Derbyshire calls the “Golden Key†– and of Russia under Peter the Great; the interestingly named Pafnuty Lvovich Chebyshev and “Chebyshev’s first Resultâ€; Jacques Hadamard and Charles de la Vallee Poussin and their simultaneous but separate work proving the Prime Number Theory; and the inenigmatic David Hilbert who spoke the inspirational words: “We must know, we shall know†(in German). In other words, after having read this book you will be familiar with most of the important mathematicians of the last 150 years and their contributions to the field.
The “what†is of course the RH and the concepts that are needed to understand it. As noted above, Derbyshire outlines these concepts with wit and pace despite the heavy subject. The chapter titles give a hint of his tone: Nine Zulu Queens Ruled China, The Argument Ant and the Value Ant, Big Oh and Mobius Mu, and the aptly titled A Little Algebra. Those whose better math days are behind them – or who never had better math days – will enjoy Derbyshire’s brevity and straightforwardness. Those with a deeper knowledge and love of math will surely enjoy his guided tour through the increasingly complex topics as he leads you toward the payoff.
The “where†is the history in which all this takes place. What makes Prime Obsession so interesting is the way it blends the math with the history so that one complements the other. Luckily for the reader, Derbyshire is just as skillful at historical sketches as he is at unpacking advanced math. It takes a skillful hand to provide historical background and important details without becoming tedious or distracting the reader from the main story. Having a background in history myself, I was impressed with his ability to not only set the stage but also bring the period to life.
As I hope I have made clear above, Prime Obsession is a unique work. The math involved is serious and difficult but presented as cleanly and as simply as is possible. The history is fascinating and neatly tied into the ever-progressing math. I would recommend this work to anyone with a love of math. The larger your knowledge base in the subject the greater ease with which you can work your way through the math; and likely the greater enjoyment you will receive. For those with less knowledge, the history and personalities remain fascinating but the math will be a challenge. Like most things in life, however, you get what you put into it.
Be sure to check out my interview with John here.
Derbyshire interview
I have posted an interview with John Derbyshire over at Collected Miscellany that might be of interest. In the interview
Derbyshire interview
I have posted an interview with John Derbyshire over at Collected Miscellany that might be of interest. In the interview
Winter Wonder Land
Just got back from Denison University where we heard John Derbyshire speak about his book Prime Obsession (my review here,
Winter Wonder Land
Just got back from Denison University where we heard John Derbyshire speak about his book Prime Obsession (my review here,
To:
copy: to anyone, who can comprehend the subject.
From: Alexei ARSENTYEV, P.O. Box 93112, NL-1090 BC, Amsterdam, The Netherlands.
“perpendicular ‘reality’ and artefact of Pi”
An extract from the article “Balled Vision” by Alexei ARSENTYEV
1) “And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his
height was five cubits: and a line of thirty cubits did compass it round about” – 1 Kings 7:23 and again
in 2 Chronicles 4:2. [1]
2) “Ubi materia – ubi geometria” – Johannes Kepler.
3) “Thus, the task is, not so much to see what no one has yet seen; but to think what nobody has yet thought,
about that which everybody sees” Erwin Schrodinger
4) “This is not a spoon what bends, it is only yourself” – bald wiz-kid from “The Matrix” movie.
Pi is one of the most fundamental constants we know. It is by definition the ratio between the length
of a circumference and its diameter. It doesn’t matter how big or small any circle is – the ratio stands.
Because of this universality Pi is used in calculations of all scales: from formula’s describing celestial
bodies’ orbital movements in astronomy to determination of electronic orbits in atoms in quantum physics
and chemistry. From early school days we remember that Pi is irrational number and it is approximately
3,14159265358 … ad infinitum. Exact value of Pi is theoretically incalculable so modern mathematicians
are competing for more accuracy which on practice means obtaining ever more digits after the comma.
Noticeably Mr. Yasumasa Kanada at Computer Centre from Tokyo University [2] has managed to score
about half of existing records on Pi-calculation field. The current official record is way above couple of
hundred billion digits. Unofficial records go even higher, to few trillion. It seems the more computer power
mathematicians have got the deeper they intend to dig after the comma. Is there gonna be a point in all this?
Well, there has been one distinguished challenge to Pi=3,14159265358…ad infinitum and it comes
from the quite prominent source. Attentive Bible readers have been pointing at the text in 1 Kings 7:23
and 2 Chronicles 4:2 [1] which if taken literally gives us value for Pi as 3 sharp [3]. Pretty comfortable
for such an important constant, isn’t it? Temptation is obviously there. Assuming Pi=3,00000000…etc then
“with this Pi we can do calculations quite a few”. So shell we take this episode of the Biblical text seriously
or ignore it as a minor glitch, an inaccuracy or even a copyist’s personal note slipped into the Holy Book?
First of all anyone can argue that for quite wide audience The Bible (or Torah) is neither a study book of
geometry nor it is Holy to other religions outside Judaism or Christianity. However not so many people
will try to deny the fact that The Bible is at least an important historical document. For whatever reason
we don t have that many other documents that old, worshiped that much and considered as Holy. Besides
there is a phenomenon known as “The Bible code” and it only strengths this view.
It has being claimed by co-researchers Dr. Eliyahu Rips, Doron Witztum who were later joined by Yoav
Rosenberg that the Torah in general and the Book of Genesis in specific contains a wealth of information
in cryptic form. This information is encoded in a number of different ways. One of the ways mentioned is
ELS (Equidistant Letters Sequences) which is commonly referred as “The Bible Code” [4]. Since we live
in the World were for every proponent there is at least one opponent no wonder that skeptical voice is
adequately loud. Without going into discussion in this particular article whether “The Sign of God” is there
or not let’s discuss Biblical ‘Pi-hint’ from geometrical point of view. The explanation expressed below
might interest You for the variety of reasons. I would like to call it “Balled Vision”.
Everything in The Universe is built on principle of rotation. The Planets in our Solar system rotate
around the Sun while Moon rotates around the Earth together with men-made satellites. The whole Solar
system rotates around the center of our Galaxy; electrons rotate around the nucleus in atoms etc. The only
difference is in scale and in speed. Most of us are aware of that fact. Yet we all have been taught at school
on geometry lessons the postulates of so called “Euclidean geometry” as undoubtedly most simplistic and
quite obvious (then and it seems today as well) point of view. Any 2-Dimensional figures (circles, squares,
triangles etc) a priori belong to ‘truly’ flat not even slightly bent Euclidean plane. Perception of 3D-version
of the Euclidean space effectively means cubic-style structures manifesting itself via ever perpendicular axis
X,Y,Z of Descartes coordinates. It is widely assumed today that at the dawn of our civilization people thought
our Planet was flat and resembled a coin or even (according to some believe) a square. With the time we have
learned that the Earth looked like a gigantic ball rather then a coin. And this “balled perception” of the Planet
has become widely accepted today.
But it seems that some flatness in our minds has managed to remain untouched. This is perhaps due to the fact
that almost everything hand-made around us appears to be flat, square and cubic. In continues effort to
comprehend The Universe we use flat geographical maps, look upon 2 Dimensional images on flat TV’s and
PC’s screens, build up and live in cubic buildings while peeping out through flat and mainly square windows.
And good old cylinder-shaped bottle of milk has been recently replaced by far more practical “Tetra Brik”.
It is practical no doubt about it. But such ‘practicality’ also shapes our mind. Every second, day by day, year
by year cubic mind-frame is being passed from generation to generation. Using the terms such as “square root”,
“square” and “cubic meter” in both scientific literature and every day language we only fooling ourselves by
subconsciously visualizing square or a cube to measure area of some surface or volume of a vessel. No wonder
that many people envisage the structure of the Universe to have that characteristic 3 Dimensional perpendicular
structure. We have failed to upgrade all levels of our mind to the understanding that perpendicular ‘reality’ with
its absolute straight lines, perfectly square angles and its ideally flat surfaces exist only as OUR IMAGINATION
or on limited scale. As ‘room temperature’ exist only inside a room perpendicular ‘reality’ is product of our
trapped imagination. I can foresee that most of you, my Dear Readers, are passionately arguing at this point that
he/she never thought about Our Planet as huge cube. And she/he was always convinced that the closest star
(the Sun) isn’t a pyramid and the Moon on clear nights doesn’t appear rhombic-shaped. And almost everybody
from scientific world would tell us that we all should remember that the Euclidean geometry is only a
mathematical model and as in any model we don’t need a reference to the ‘real’ world, gravitational forces of
the Universe or the shape of the Earth or another planets. Having that all in mind tell me then are we living in
the ‘real’ world or inside some cubic space where all lines of forces (gravitational, electromagnetic etc) are
truly flat? PLEASE EXPLAIN ME HOW COME WE HAVE BEEN USING FLAT value of Pi for every day
‘REAL’ LIFE CALCULATIONS for so long ??? Specially in astronomy! Let’s not forget that “room temperature”
exists only inside a room providing all doors and windows are closed. And on limited Earth scale we have
the temperature range roughly from minus fifty to plus fifty Celsius only because there is the atmoSPHERE
around our Planet. Take the edge of the Solar system and it is pretty cold out there. Even on the Moon.
We should be considering our Universe and space around us to have spherical structure. Any points,
lines, circles, triangles, squares and other geometrical figures ought to be envisaged as laying on (imaginary
/invisible) spherical surfaces of various diameters. One obvious example of such sphere (although visible
and quite big) is the surface of our own Planet. Another is the surface of an ordinary apple. What was Sir
Isaac Newton thinking about when the famous apple hit him? Oh ya, gravitation,not geometry. How about
Adam and Eve? What was their thoughts about? Obviously an apple (as fruit) have played an important
part in human history. When I am looking at an apple fruit I think Apple should start making apple-shaped
computers with spherical monitors. M-m-m, what a good idea. Shell I write them a letter?
Coming back to geometry. Thus diameter of a non-Euclidean circle (not circumference!) is not a straight
segment belonging to ‘truly’ flat plane containing circumference. When we talk about “a circle” we have to
include into consideration the surface contained, locked within by circumference in question. Let’s be
accurate with terms and distinguish between “a circle” and “a circumference”.
“A circumference” is 1 Dimensional creature. It is an infinite loop. Undoubtedly the most perfect of them
all but merely a ring, a hoop, a coil, a thin (blue?) line marking the edge of a circle. While any circle
(Euclidean or not) is 2 Dimensional figure with a surface. Euclidean circle resembles a CD-ROM,
non-Euclidean looks like contact lens. Keeping that in mind we will have no problem of understanding
that diameter of non-Euclidean circle is slightly bend segment which belongs to some ‘invisible’ sphere of
‘unknown’ diameter. Naturally, if we make a circle (again, not circumference!) on the ground with let’s say
3 meter in diameter then degree of curvature of such surface is trifling considering the size of our Planet.
But if we imagine a circle as big as my native Russia; draw 3 meter circle on the wall in the famous round
Chamber “La Salle Pi” (Palais de la Decouverte in Paris); paint one on the wall of the Oval Office in the
White House or merely outline a coin on any apple fruit then in all those cases surface’s curvature will
become obvious. I expect that geographers and astronomers will be among the first one who understand
what I mean. Perhaps the only case when we can speak about surface with truly ideal flatness arises
if we make a section of a ball by surface going through its center. Considering this scenario let me ask:
“Are we still living on a flat pizza-like Planet in the center of the Universe?” Then viva la Ptolemy!
To those who doubt the idea of spherical Universe being expressed here I would offer to explain the
following facts using ever prevailing ‘cubic’ point of view. Let me repeat: “cubic point”. A pixel? Just
look at us. What have we come to! So far we have agreed upon the following facts.
1) we live on a surface of a big ball (not a cube or a pyramid) rotating around much bigger one we call
“the Sun”.
2) As we observe drops of liquid in free fall we always see that those drops form balls and not cubes,
cylinders, pyramids etc.
3) When astronauts on space-station occasionally or otherwise spill some liquid it doesn’t form little
pyramids, cylinders, dodecahedrons etc.
4) Throw a little stone into a lake on a quite day and waves on a water surface will always be shaped as
concentric circles, not triangles or squares. Have anyone ever seen concentric square waves on a water
surface? Are you convinced yet? No? OK, let me continue. Why do we prefer round stadiums or concert
halls to those which are square? Because the seats in round football arena’s are spread in much more ‘justice’
manner: any seat in the same row gives the same view of the stage. Ancient Coliseum in Rome is pretty
graphic example. And to make all the rows equal we have to place the stage in the center of a … sphere.
Would you prefer to seat or to stand in a corner? Do you remember this cruel punishment from the childhood?
This is where it all begins. Starting from cubic toys including “Game Cube”.
So do we see cubic apple fruits growing on threes with square boles under sunlight blazing from the rhombic
Sun? When was the last time you saw the rainbow in full glory? Was it perfectly straight line? I think I have
made the point.
Why it is so and not otherwise? It is merely a feature of our space and it originates from the principle
of rotation. Even if we rotate any cube, pyramid or even a cylinder we will end up with a ball anyway
but with noticeable gradient of density. Ball is the most perfect shape in 3D-world we know just because
ball has infinite number of planes and axles of symmetry. After all who would dare to say that an atom
shaped like a little brick? Have electrons orbits become square or cubic? Halloo, Mr. Schrodinger!
How are you? Could you foresee this with your Psi-function? Whom are we kidding by ignoring the obvious?
The whole Universe is built on the same principle of rotation. The only difference is scale and speed.
The most adequate model of matter structuring should be the most natural one. An onion-like structure where
bigger sphere contains smaller one. So called “matrёshka”. Back to the USSR! Mathematically it can be
described by tensors using Riemman (not Lobachevsky!) mathematical spaces, containing one another.
It easily combines with the quite popular “sandwich-theory” of the Universe but “sandwich” layers aren’t
flat at all! Meaning: on the scale of the Universe light travels on bent and not on absolutely straight trajectories.
So despite using appropriate mathematical tool (tensors) the one of the postulates of the famous
Relativity Theory is only relatively correct. Sorry, Mr. Einstein. But Happy Birthday anyway!
Size of such Universal layers may very. On smaller scale onion-like model would perfectly
describe a single atom with electronic clouds having different energy levels. Below you will find
the famous 3 (among others) axioms of incidence for the flat space and one additional axiom formulated by
yours truly for the spherical state of mind. Here are the 3 “Axioms of Incidence” for the Euclidean,
truly flat, cubic-like space:
* via 1 point we can draw infinite number of straight lines through which we are able to draw infinite
number of planes. (So far so good from spherical point of view).
**For every two points A and B there exits a straight line that contains each of the points A, B.
Meaning: via any 2 points we can draw only 1 straight line (but plenty of curved ones!) through
which we are able to draw infinite number of planes.
*** There exist at least 3 points that do not lie on a line. Otherwise: via any 3 points (in general case)
we can not draw any straight line (but at least one circle!) and one and only one plane. Or in more
traditional reading: For any three points A, B, C that do not lie on one and the same straight line [but they
can belong to the same circle!] there exists no more than one plane that contains each of the three points A, B, C.
****When it comes to 4 points it is postulated: “There exist at least four points which do not lie in a plane”.
Usually we learn this stuff at school but we don t really think about it at the later stage. Only when I have
grown up it struck me one day: they may belong to a sphere! For that particular case I have formulated an
additional axiom for the spherical space (and mind!) which should read as following: Through 4 points
(in general case) we cannot draw any straight line or any plane, but one and only sphere. So hereby I am
expressing my “Balled Vision” by stating that currently Pi=3. Why currently? Because continuing to use
the abovementioned logic we should come to the conclusion that for any hemi-sphere Pi (as ratio) between
its diameter and length of circumference) equals 2. In case of hemi-sphere its diameter is the line (once
again: it is not straight segment!) connecting 0 and 180 degree points of the Equator going through the Pole.
And Pi (again, as ratio) of complete sphere is equal to one! “The Ratio” of a point or complete sphere =1.
So we have got the full spectrum of Pi quantities where sharp values Pi=3, Pi=2 and Pi=1 are representing
a step, a “quantum jump” from one state of mind (and therefore matter) into another. This is a perfect example
of so called “artefact”. In this case the value of mathematical constant is changing in correlation with the
state of experimenter s mind and the shape and size of surface in question. On a physical level the current
shift from flat Pi= 3,14159265358… to 3 sharp can be interpreted as change of the space curvature around
us according to gravitational gradient. That is another question “what is causing it?” It also could mean that
the Universe has made at least 1/6 of the complete rotation and some of us already feel it by expressing as Pi=3.
One particularly interesting case is someone’s car registration plate in Maryland, USA reading “pi is 3”.
I wonder who the owner is. In order to distinguish the different sharp values of the famous ratio from the
actual flat value of pi=3,1415…which has already become very chrestomathy I would like to name the
Biblical conditions under which length of a circumference contains exactly 3 diameters as “equilibrium”.
So let me letter the ratio between the length of a circumference and its diameter in equilibrium as E.
Then equality will look like E=3. And You, Dear Reader, are very welcome to make your suggestions in
naming the famous ratio for hemi-spherical and spherical conditions. Let me just emphasize that the point of
view where the ratio equals to 1 is the point of complete spherical state of mind. It represents the vision
of human or other being (call such being God if you wish) who covers and feels the whole Universe at once,
by one thought. Don’t look at me that way I don t feel the whole Universe just yet. But by adopting “Balled Vision”
we all will feel and think as OneBeing effectively getting closer to “The Day we all can glean an inkling of
God’s mean”. Until (and beyond) of such Magic Day I remain Yours Truly Alexei ARSENTYEV.
By the way: flat Pi^3 =31,0062766803…Surprise but I have never seen any references to this “cubic root from 31”
as ancient way to calculate Flat Pi in any Pi-historic articles. Here are few sources which helped me to shape my mind.
Quite literally.
[1] The Holy Bible, The Books of the Old and New Testament, 1 Kings 7:23 ; 2 Chronicles 4:2.
[2] Mr Y.Kanada and his record calculations at
[3] “Does the Bible say pi equals 3.0?” Russell M. Grigg
[4] “The Bible Code” explained at
As you may have already guessed English is not my native language. So I apologize for occasional mixture
of articles (a/the), confusing past tenses with present perfect ones and using present continuous tense instead
of present simple and any other possible grammar mistakes. You are very welcome to send your comments
(preferably in traditional mail) to: Alexei ARSENTYEV, P.O. Box 93112, NL-1090 BC, Amsterdam,
The Netherlands or via contemporary means of communication such as e-mail: .
For those who wish to comment on non-grammatical mistakes I have typed a short epilogue.
Epilogue.
Michael Stifel, German mathematician (1487- 1567) said back in 1544 : “Flustra laborant quotquot se
calculationibus fatigant pro inventione quadraturae circuli”. In English it will be close to: “Futile is the
labor of those who fatigue themselves with calculations to square the circle”. I only can add to this wise
saying that: ” IT IS DIFFICULT”. I know it from my personal experience. No, not to imagine matreshka-like
Universe with spherical onion layers. Not that. It is difficult to admit that despite of so many obvious hints
surrounding us all the time we have been blind (or blinded?) for so long. It is difficult to admit of giving
lack of proper consideration to such an important ratio which resulted in something what looks like the
oldest mistake in the human history. There is a difference between “pride” and “arrogance”. Synonymous
to “arrogance” is “stubborn”. Antonymous to “pride” is “shame”. It is no shame in making a mistake.
Nobody is exempt from it since nobody is perfect on this Planet. There is just a degree of perfection.
From black to white, from pride via arrogance passing stubbornness to blunt stupidity. So what we call
today “a mistake” yesterday was just a step on The Evolutionary Ladder. And anyone should be proud of
being able to upgrade itself to the higher degree of understanding where (looking back from the next level)
there is no shame to say: “that was just another step, the previous version”. But is it arrogant not to look
around, it is stubborn to hold on outdated visions despite plenty of facts being available and it is dangerously
stupid to be that stubborn and arrogant to deny the obvious. It holds the process of evolution by adding
“another brick in The Wall” .
There is a difference between “pride” and “arrogance”. Probably the same as between “circle” and “circumference”.
And for some of us that difference is not visible while for others it is clearly there despite being thin. Very thin.
Deadly thin. Thin as any ‘straight’ or ‘bent’ line; ‘truly flat’ or curved surface in Geometry. But it is there.
To those who wish to free their mind I would strongly recommend to watch “The Matrix” movie (part one).
There is an episode where Morpheus is saying to Neo: “I didn’t say it would be easy ” and later
“We have a rule: we never free a mind once it’s reached the certain age, the mind has a trouble if letting it go”.
So let “it” go while your mind is flexible enough. I know it is difficult. Difficult to shake off the cliches and difficult to admit.
Difficult to be taught and preached by ‘a stranger’ what to do. Our arrogance makes it difficult.
That is why it was proclaimed as “deadly sin”. Because those who don’t except the words from a ‘stranger’ marking ANY
unsolicit information as “spam” are walking on thin ice. There are no strangers. We are all God’s creatures. We just being
played against each other by those who benefit from it. With 100% chance that one day The Creator will be kind enough to
explain it to them. Personally. You do know what it means. Don’t let it happen to you. A.A.
To:
copy: to anyone, who can comprehend the subject.
From: Alexei ARSENTYEV, P.O. Box 93112, NL-1090 BC, Amsterdam, The Netherlands.
“perpendicular ‘reality’ and artefact of Pi”
An extract from the article “Balled Vision” by Alexei ARSENTYEV
1) “And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his
height was five cubits: and a line of thirty cubits did compass it round about” – 1 Kings 7:23 and again
in 2 Chronicles 4:2. [1]
2) “Ubi materia – ubi geometria” – Johannes Kepler.
3) “Thus, the task is, not so much to see what no one has yet seen; but to think what nobody has yet thought,
about that which everybody sees” Erwin Schrodinger
4) “This is not a spoon what bends, it is only yourself” – bald wiz-kid from “The Matrix” movie.
Pi is one of the most fundamental constants we know. It is by definition the ratio between the length
of a circumference and its diameter. It doesn’t matter how big or small any circle is – the ratio stands.
Because of this universality Pi is used in calculations of all scales: from formula’s describing celestial
bodies’ orbital movements in astronomy to determination of electronic orbits in atoms in quantum physics
and chemistry. From early school days we remember that Pi is irrational number and it is approximately
3,14159265358 … ad infinitum. Exact value of Pi is theoretically incalculable so modern mathematicians
are competing for more accuracy which on practice means obtaining ever more digits after the comma.
Noticeably Mr. Yasumasa Kanada at Computer Centre from Tokyo University [2] has managed to score
about half of existing records on Pi-calculation field. The current official record is way above couple of
hundred billion digits. Unofficial records go even higher, to few trillion. It seems the more computer power
mathematicians have got the deeper they intend to dig after the comma. Is there gonna be a point in all this?
Well, there has been one distinguished challenge to Pi=3,14159265358…ad infinitum and it comes
from the quite prominent source. Attentive Bible readers have been pointing at the text in 1 Kings 7:23
and 2 Chronicles 4:2 [1] which if taken literally gives us value for Pi as 3 sharp [3]. Pretty comfortable
for such an important constant, isn’t it? Temptation is obviously there. Assuming Pi=3,00000000…etc then
“with this Pi we can do calculations quite a few”. So shell we take this episode of the Biblical text seriously
or ignore it as a minor glitch, an inaccuracy or even a copyist’s personal note slipped into the Holy Book?
First of all anyone can argue that for quite wide audience The Bible (or Torah) is neither a study book of
geometry nor it is Holy to other religions outside Judaism or Christianity. However not so many people
will try to deny the fact that The Bible is at least an important historical document. For whatever reason
we don t have that many other documents that old, worshiped that much and considered as Holy. Besides
there is a phenomenon known as “The Bible code” and it only strengths this view.
It has being claimed by co-researchers Dr. Eliyahu Rips, Doron Witztum who were later joined by Yoav
Rosenberg that the Torah in general and the Book of Genesis in specific contains a wealth of information
in cryptic form. This information is encoded in a number of different ways. One of the ways mentioned is
ELS (Equidistant Letters Sequences) which is commonly referred as “The Bible Code” [4]. Since we live
in the World were for every proponent there is at least one opponent no wonder that skeptical voice is
adequately loud. Without going into discussion in this particular article whether “The Sign of God” is there
or not let’s discuss Biblical ‘Pi-hint’ from geometrical point of view. The explanation expressed below
might interest You for the variety of reasons. I would like to call it “Balled Vision”.
Everything in The Universe is built on principle of rotation. The Planets in our Solar system rotate
around the Sun while Moon rotates around the Earth together with men-made satellites. The whole Solar
system rotates around the center of our Galaxy; electrons rotate around the nucleus in atoms etc. The only
difference is in scale and in speed. Most of us are aware of that fact. Yet we all have been taught at school
on geometry lessons the postulates of so called “Euclidean geometry” as undoubtedly most simplistic and
quite obvious (then and it seems today as well) point of view. Any 2-Dimensional figures (circles, squares,
triangles etc) a priori belong to ‘truly’ flat not even slightly bent Euclidean plane. Perception of 3D-version
of the Euclidean space effectively means cubic-style structures manifesting itself via ever perpendicular axis
X,Y,Z of Descartes coordinates. It is widely assumed today that at the dawn of our civilization people thought
our Planet was flat and resembled a coin or even (according to some believe) a square. With the time we have
learned that the Earth looked like a gigantic ball rather then a coin. And this “balled perception” of the Planet
has become widely accepted today.
But it seems that some flatness in our minds has managed to remain untouched. This is perhaps due to the fact
that almost everything hand-made around us appears to be flat, square and cubic. In continues effort to
comprehend The Universe we use flat geographical maps, look upon 2 Dimensional images on flat TV’s and
PC’s screens, build up and live in cubic buildings while peeping out through flat and mainly square windows.
And good old cylinder-shaped bottle of milk has been recently replaced by far more practical “Tetra Brik”.
It is practical no doubt about it. But such ‘practicality’ also shapes our mind. Every second, day by day, year
by year cubic mind-frame is being passed from generation to generation. Using the terms such as “square root”,
“square” and “cubic meter” in both scientific literature and every day language we only fooling ourselves by
subconsciously visualizing square or a cube to measure area of some surface or volume of a vessel. No wonder
that many people envisage the structure of the Universe to have that characteristic 3 Dimensional perpendicular
structure. We have failed to upgrade all levels of our mind to the understanding that perpendicular ‘reality’ with
its absolute straight lines, perfectly square angles and its ideally flat surfaces exist only as OUR IMAGINATION
or on limited scale. As ‘room temperature’ exist only inside a room perpendicular ‘reality’ is product of our
trapped imagination. I can foresee that most of you, my Dear Readers, are passionately arguing at this point that
he/she never thought about Our Planet as huge cube. And she/he was always convinced that the closest star
(the Sun) isn’t a pyramid and the Moon on clear nights doesn’t appear rhombic-shaped. And almost everybody
from scientific world would tell us that we all should remember that the Euclidean geometry is only a
mathematical model and as in any model we don’t need a reference to the ‘real’ world, gravitational forces of
the Universe or the shape of the Earth or another planets. Having that all in mind tell me then are we living in
the ‘real’ world or inside some cubic space where all lines of forces (gravitational, electromagnetic etc) are
truly flat? PLEASE EXPLAIN ME HOW COME WE HAVE BEEN USING FLAT value of Pi for every day
‘REAL’ LIFE CALCULATIONS for so long ??? Specially in astronomy! Let’s not forget that “room temperature”
exists only inside a room providing all doors and windows are closed. And on limited Earth scale we have
the temperature range roughly from minus fifty to plus fifty Celsius only because there is the atmoSPHERE
around our Planet. Take the edge of the Solar system and it is pretty cold out there. Even on the Moon.
We should be considering our Universe and space around us to have spherical structure. Any points,
lines, circles, triangles, squares and other geometrical figures ought to be envisaged as laying on (imaginary
/invisible) spherical surfaces of various diameters. One obvious example of such sphere (although visible
and quite big) is the surface of our own Planet. Another is the surface of an ordinary apple. What was Sir
Isaac Newton thinking about when the famous apple hit him? Oh ya, gravitation,not geometry. How about
Adam and Eve? What was their thoughts about? Obviously an apple (as fruit) have played an important
part in human history. When I am looking at an apple fruit I think Apple should start making apple-shaped
computers with spherical monitors. M-m-m, what a good idea. Shell I write them a letter?
Coming back to geometry. Thus diameter of a non-Euclidean circle (not circumference!) is not a straight
segment belonging to ‘truly’ flat plane containing circumference. When we talk about “a circle” we have to
include into consideration the surface contained, locked within by circumference in question. Let’s be
accurate with terms and distinguish between “a circle” and “a circumference”.
“A circumference” is 1 Dimensional creature. It is an infinite loop. Undoubtedly the most perfect of them
all but merely a ring, a hoop, a coil, a thin (blue?) line marking the edge of a circle. While any circle
(Euclidean or not) is 2 Dimensional figure with a surface. Euclidean circle resembles a CD-ROM,
non-Euclidean looks like contact lens. Keeping that in mind we will have no problem of understanding
that diameter of non-Euclidean circle is slightly bend segment which belongs to some ‘invisible’ sphere of
‘unknown’ diameter. Naturally, if we make a circle (again, not circumference!) on the ground with let’s say
3 meter in diameter then degree of curvature of such surface is trifling considering the size of our Planet.
But if we imagine a circle as big as my native Russia; draw 3 meter circle on the wall in the famous round
Chamber “La Salle Pi” (Palais de la Decouverte in Paris); paint one on the wall of the Oval Office in the
White House or merely outline a coin on any apple fruit then in all those cases surface’s curvature will
become obvious. I expect that geographers and astronomers will be among the first one who understand
what I mean. Perhaps the only case when we can speak about surface with truly ideal flatness arises
if we make a section of a ball by surface going through its center. Considering this scenario let me ask:
“Are we still living on a flat pizza-like Planet in the center of the Universe?” Then viva la Ptolemy!
To those who doubt the idea of spherical Universe being expressed here I would offer to explain the
following facts using ever prevailing ‘cubic’ point of view. Let me repeat: “cubic point”. A pixel? Just
look at us. What have we come to! So far we have agreed upon the following facts.
1) we live on a surface of a big ball (not a cube or a pyramid) rotating around much bigger one we call
“the Sun”.
2) As we observe drops of liquid in free fall we always see that those drops form balls and not cubes,
cylinders, pyramids etc.
3) When astronauts on space-station occasionally or otherwise spill some liquid it doesn’t form little
pyramids, cylinders, dodecahedrons etc.
4) Throw a little stone into a lake on a quite day and waves on a water surface will always be shaped as
concentric circles, not triangles or squares. Have anyone ever seen concentric square waves on a water
surface? Are you convinced yet? No? OK, let me continue. Why do we prefer round stadiums or concert
halls to those which are square? Because the seats in round football arena’s are spread in much more ‘justice’
manner: any seat in the same row gives the same view of the stage. Ancient Coliseum in Rome is pretty
graphic example. And to make all the rows equal we have to place the stage in the center of a … sphere.
Would you prefer to seat or to stand in a corner? Do you remember this cruel punishment from the childhood?
This is where it all begins. Starting from cubic toys including “Game Cube”.
So do we see cubic apple fruits growing on threes with square boles under sunlight blazing from the rhombic
Sun? When was the last time you saw the rainbow in full glory? Was it perfectly straight line? I think I have
made the point.
Why it is so and not otherwise? It is merely a feature of our space and it originates from the principle
of rotation. Even if we rotate any cube, pyramid or even a cylinder we will end up with a ball anyway
but with noticeable gradient of density. Ball is the most perfect shape in 3D-world we know just because
ball has infinite number of planes and axles of symmetry. After all who would dare to say that an atom
shaped like a little brick? Have electrons orbits become square or cubic? Halloo, Mr. Schrodinger!
How are you? Could you foresee this with your Psi-function? Whom are we kidding by ignoring the obvious?
The whole Universe is built on the same principle of rotation. The only difference is scale and speed.
The most adequate model of matter structuring should be the most natural one. An onion-like structure where
bigger sphere contains smaller one. So called “matrёshka”. Back to the USSR! Mathematically it can be
described by tensors using Riemman (not Lobachevsky!) mathematical spaces, containing one another.
It easily combines with the quite popular “sandwich-theory” of the Universe but “sandwich” layers aren’t
flat at all! Meaning: on the scale of the Universe light travels on bent and not on absolutely straight trajectories.
So despite using appropriate mathematical tool (tensors) the one of the postulates of the famous
Relativity Theory is only relatively correct. Sorry, Mr. Einstein. But Happy Birthday anyway!
Size of such Universal layers may very. On smaller scale onion-like model would perfectly
describe a single atom with electronic clouds having different energy levels. Below you will find
the famous 3 (among others) axioms of incidence for the flat space and one additional axiom formulated by
yours truly for the spherical state of mind. Here are the 3 “Axioms of Incidence” for the Euclidean,
truly flat, cubic-like space:
* via 1 point we can draw infinite number of straight lines through which we are able to draw infinite
number of planes. (So far so good from spherical point of view).
**For every two points A and B there exits a straight line that contains each of the points A, B.
Meaning: via any 2 points we can draw only 1 straight line (but plenty of curved ones!) through
which we are able to draw infinite number of planes.
*** There exist at least 3 points that do not lie on a line. Otherwise: via any 3 points (in general case)
we can not draw any straight line (but at least one circle!) and one and only one plane. Or in more
traditional reading: For any three points A, B, C that do not lie on one and the same straight line [but they
can belong to the same circle!] there exists no more than one plane that contains each of the three points A, B, C.
****When it comes to 4 points it is postulated: “There exist at least four points which do not lie in a plane”.
Usually we learn this stuff at school but we don t really think about it at the later stage. Only when I have
grown up it struck me one day: they may belong to a sphere! For that particular case I have formulated an
additional axiom for the spherical space (and mind!) which should read as following: Through 4 points
(in general case) we cannot draw any straight line or any plane, but one and only sphere. So hereby I am
expressing my “Balled Vision” by stating that currently Pi=3. Why currently? Because continuing to use
the abovementioned logic we should come to the conclusion that for any hemi-sphere Pi (as ratio) between
its diameter and length of circumference) equals 2. In case of hemi-sphere its diameter is the line (once
again: it is not straight segment!) connecting 0 and 180 degree points of the Equator going through the Pole.
And Pi (again, as ratio) of complete sphere is equal to one! “The Ratio” of a point or complete sphere =1.
So we have got the full spectrum of Pi quantities where sharp values Pi=3, Pi=2 and Pi=1 are representing
a step, a “quantum jump” from one state of mind (and therefore matter) into another. This is a perfect example
of so called “artefact”. In this case the value of mathematical constant is changing in correlation with the
state of experimenter s mind and the shape and size of surface in question. On a physical level the current
shift from flat Pi= 3,14159265358… to 3 sharp can be interpreted as change of the space curvature around
us according to gravitational gradient. That is another question “what is causing it?” It also could mean that
the Universe has made at least 1/6 of the complete rotation and some of us already feel it by expressing as Pi=3.
One particularly interesting case is someone’s car registration plate in Maryland, USA reading “pi is 3”.
I wonder who the owner is. In order to distinguish the different sharp values of the famous ratio from the
actual flat value of pi=3,1415…which has already become very chrestomathy I would like to name the
Biblical conditions under which length of a circumference contains exactly 3 diameters as “equilibrium”.
So let me letter the ratio between the length of a circumference and its diameter in equilibrium as E.
Then equality will look like E=3. And You, Dear Reader, are very welcome to make your suggestions in
naming the famous ratio for hemi-spherical and spherical conditions. Let me just emphasize that the point of
view where the ratio equals to 1 is the point of complete spherical state of mind. It represents the vision
of human or other being (call such being God if you wish) who covers and feels the whole Universe at once,
by one thought. Don’t look at me that way I don t feel the whole Universe just yet. But by adopting “Balled Vision”
we all will feel and think as OneBeing effectively getting closer to “The Day we all can glean an inkling of
God’s mean”. Until (and beyond) of such Magic Day I remain Yours Truly Alexei ARSENTYEV.
By the way: flat Pi^3 =31,0062766803…Surprise but I have never seen any references to this “cubic root from 31”
as ancient way to calculate Flat Pi in any Pi-historic articles. Here are few sources which helped me to shape my mind.
Quite literally.
[1] The Holy Bible, The Books of the Old and New Testament, 1 Kings 7:23 ; 2 Chronicles 4:2.
[2] Mr Y.Kanada and his record calculations at
[3] “Does the Bible say pi equals 3.0?” Russell M. Grigg
[4] “The Bible Code” explained at
As you may have already guessed English is not my native language. So I apologize for occasional mixture
of articles (a/the), confusing past tenses with present perfect ones and using present continuous tense instead
of present simple and any other possible grammar mistakes. You are very welcome to send your comments
(preferably in traditional mail) to: Alexei ARSENTYEV, P.O. Box 93112, NL-1090 BC, Amsterdam,
The Netherlands or via contemporary means of communication such as e-mail: .
For those who wish to comment on non-grammatical mistakes I have typed a short epilogue.
Epilogue.
Michael Stifel, German mathematician (1487- 1567) said back in 1544 : “Flustra laborant quotquot se
calculationibus fatigant pro inventione quadraturae circuli”. In English it will be close to: “Futile is the
labor of those who fatigue themselves with calculations to square the circle”. I only can add to this wise
saying that: ” IT IS DIFFICULT”. I know it from my personal experience. No, not to imagine matreshka-like
Universe with spherical onion layers. Not that. It is difficult to admit that despite of so many obvious hints
surrounding us all the time we have been blind (or blinded?) for so long. It is difficult to admit of giving
lack of proper consideration to such an important ratio which resulted in something what looks like the
oldest mistake in the human history. There is a difference between “pride” and “arrogance”. Synonymous
to “arrogance” is “stubborn”. Antonymous to “pride” is “shame”. It is no shame in making a mistake.
Nobody is exempt from it since nobody is perfect on this Planet. There is just a degree of perfection.
From black to white, from pride via arrogance passing stubbornness to blunt stupidity. So what we call
today “a mistake” yesterday was just a step on The Evolutionary Ladder. And anyone should be proud of
being able to upgrade itself to the higher degree of understanding where (looking back from the next level)
there is no shame to say: “that was just another step, the previous version”. But is it arrogant not to look
around, it is stubborn to hold on outdated visions despite plenty of facts being available and it is dangerously
stupid to be that stubborn and arrogant to deny the obvious. It holds the process of evolution by adding
“another brick in The Wall” .
There is a difference between “pride” and “arrogance”. Probably the same as between “circle” and “circumference”.
And for some of us that difference is not visible while for others it is clearly there despite being thin. Very thin.
Deadly thin. Thin as any ‘straight’ or ‘bent’ line; ‘truly flat’ or curved surface in Geometry. But it is there.
To those who wish to free their mind I would strongly recommend to watch “The Matrix” movie (part one).
There is an episode where Morpheus is saying to Neo: “I didn’t say it would be easy ” and later
“We have a rule: we never free a mind once it’s reached the certain age, the mind has a trouble if letting it go”.
So let “it” go while your mind is flexible enough. I know it is difficult. Difficult to shake off the cliches and difficult to admit.
Difficult to be taught and preached by ‘a stranger’ what to do. Our arrogance makes it difficult.
That is why it was proclaimed as “deadly sin”. Because those who don’t except the words from a ‘stranger’ marking ANY
unsolicit information as “spam” are walking on thin ice. There are no strangers. We are all God’s creatures. We just being
played against each other by those who benefit from it. With 100% chance that one day The Creator will be kind enough to
explain it to them. Personally. You do know what it means. Don’t let it happen to you. A.A.
To: John Derbyshire
copy: to anyone, who can comprehend the subject.
From: Alexei ARSENTYEV, P.O. Box 93112, NL-1090 BC, Amsterdam, The Netherlands.
“perpendicular ‘reality’ and artefact of Pi”
An extract from the article “Balled Vision” by Alexei ARSENTYEV
1) “And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his
height was five cubits: and a line of thirty cubits did compass it round about” – 1 Kings 7:23 and again
in 2 Chronicles 4:2. [1]
2) “Ubi materia – ubi geometria” – Johannes Kepler.
3) “Thus, the task is, not so much to see what no one has yet seen; but to think what nobody has yet thought,
about that which everybody sees” Erwin Schrodinger
4) “This is not a spoon what bends, it is only yourself” – bald wiz-kid from “The Matrix” movie.
Pi is one of the most fundamental constants we know. It is by definition the ratio between the length
of a circumference and its diameter. It doesn’t matter how big or small any circle is – the ratio stands.
Because of this universality Pi is used in calculations of all scales: from formula’s describing celestial
bodies’ orbital movements in astronomy to determination of electronic orbits in atoms in quantum physics
and chemistry. From early school days we remember that Pi is irrational number and it is approximately
3,14159265358 … ad infinitum. Exact value of Pi is theoretically incalculable so modern mathematicians
are competing for more accuracy which on practice means obtaining ever more digits after the comma.
Noticeably Mr. Yasumasa Kanada at Computer Centre from Tokyo University [2] has managed to score
about half of existing records on Pi-calculation field. The current official record is way above couple of
hundred billion digits. Unofficial records go even higher, to few trillion. It seems the more computer power
mathematicians have got the deeper they intend to dig after the comma. Is there gonna be a point in all this?
Well, there has been one distinguished challenge to Pi=3,14159265358…ad infinitum and it comes
from the quite prominent source. Attentive Bible readers have been pointing at the text in 1 Kings 7:23
and 2 Chronicles 4:2 [1] which if taken literally gives us value for Pi as 3 sharp [3]. Pretty comfortable
for such an important constant, isn’t it? Temptation is obviously there. Assuming Pi=3,00000000…etc then
“with this Pi we can do calculations quite a few”. So shell we take this episode of the Biblical text seriously
or ignore it as a minor glitch, an inaccuracy or even a copyist’s personal note slipped into the Holy Book?
First of all anyone can argue that for quite wide audience The Bible (or Torah) is neither a study book of
geometry nor it is Holy to other religions outside Judaism or Christianity. However not so many people
will try to deny the fact that The Bible is at least an important historical document. For whatever reason
we don t have that many other documents that old, worshiped that much and considered as Holy. Besides
there is a phenomenon known as “The Bible code” and it only strengths this view.
It has being claimed by co-researchers Dr. Eliyahu Rips, Doron Witztum who were later joined by Yoav
Rosenberg that the Torah in general and the Book of Genesis in specific contains a wealth of information
in cryptic form. This information is encoded in a number of different ways. One of the ways mentioned is
ELS (Equidistant Letters Sequences) which is commonly referred as “The Bible Code” [4]. Since we live
in the World were for every proponent there is at least one opponent no wonder that skeptical voice is
adequately loud. Without going into discussion in this particular article whether “The Sign of God” is there
or not let’s discuss Biblical ‘Pi-hint’ from geometrical point of view. The explanation expressed below
might interest You for the variety of reasons. I would like to call it “Balled Vision”.
Everything in The Universe is built on principle of rotation. The Planets in our Solar system rotate
around the Sun while Moon rotates around the Earth together with men-made satellites. The whole Solar
system rotates around the center of our Galaxy; electrons rotate around the nucleus in atoms etc. The only
difference is in scale and in speed. Most of us are aware of that fact. Yet we all have been taught at school
on geometry lessons the postulates of so called “Euclidean geometry” as undoubtedly most simplistic and
quite obvious (then and it seems today as well) point of view. Any 2-Dimensional figures (circles, squares,
triangles etc) a priori belong to ‘truly’ flat not even slightly bent Euclidean plane. Perception of 3D-version
of the Euclidean space effectively means cubic-style structures manifesting itself via ever perpendicular axis
X,Y,Z of Descartes coordinates. It is widely assumed today that at the dawn of our civilization people thought
our Planet was flat and resembled a coin or even (according to some believe) a square. With the time we have
learned that the Earth looked like a gigantic ball rather then a coin. And this “balled perception” of the Planet
has become widely accepted today.
But it seems that some flatness in our minds has managed to remain untouched. This is perhaps due to the fact
that almost everything hand-made around us appears to be flat, square and cubic. In continues effort to
comprehend The Universe we use flat geographical maps, look upon 2 Dimensional images on flat TV’s and
PC’s screens, build up and live in cubic buildings while peeping out through flat and mainly square windows.
And good old cylinder-shaped bottle of milk has been recently replaced by far more practical “Tetra Brik”.
It is practical no doubt about it. But such ‘practicality’ also shapes our mind. Every second, day by day, year
by year cubic mind-frame is being passed from generation to generation. Using the terms such as “square root”,
“square” and “cubic meter” in both scientific literature and every day language we only fooling ourselves by
subconsciously visualizing square or a cube to measure area of some surface or volume of a vessel. No wonder
that many people envisage the structure of the Universe to have that characteristic 3 Dimensional perpendicular
structure. We have failed to upgrade all levels of our mind to the understanding that perpendicular ‘reality’ with
its absolute straight lines, perfectly square angles and its ideally flat surfaces exist only as OUR IMAGINATION
or on limited scale. As ‘room temperature’ exist only inside a room perpendicular ‘reality’ is product of our
trapped imagination. I can foresee that most of you, my Dear Readers, are passionately arguing at this point that
he/she never thought about Our Planet as huge cube. And she/he was always convinced that the closest star
(the Sun) isn’t a pyramid and the Moon on clear nights doesn’t appear rhombic-shaped. And almost everybody
from scientific world would tell us that we all should remember that the Euclidean geometry is only a
mathematical model and as in any model we don’t need a reference to the ‘real’ world, gravitational forces of
the Universe or the shape of the Earth or another planets. Having that all in mind tell me then are we living in
the ‘real’ world or inside some cubic space where all lines of forces (gravitational, electromagnetic etc) are
truly flat? PLEASE EXPLAIN ME HOW COME WE HAVE BEEN USING FLAT value of Pi for every day
‘REAL’ LIFE CALCULATIONS for so long ??? Specially in astronomy! Let’s not forget that “room temperature”
exists only inside a room providing all doors and windows are closed. And on limited Earth scale we have
the temperature range roughly from minus fifty to plus fifty Celsius only because there is the atmoSPHERE
around our Planet. Take the edge of the Solar system and it is pretty cold out there. Even on the Moon.
We should be considering our Universe and space around us to have spherical structure. Any points,
lines, circles, triangles, squares and other geometrical figures ought to be envisaged as laying on (imaginary
/invisible) spherical surfaces of various diameters. One obvious example of such sphere (although visible
and quite big) is the surface of our own Planet. Another is the surface of an ordinary apple. What was Sir
Isaac Newton thinking about when the famous apple hit him? Oh ya, gravitation,not geometry. How about
Adam and Eve? What was their thoughts about? Obviously an apple (as fruit) have played an important
part in human history. When I am looking at an apple fruit I think Apple should start making apple-shaped
computers with spherical monitors. M-m-m, what a good idea. Shell I write them a letter?
Coming back to geometry. Thus diameter of a non-Euclidean circle (not circumference!) is not a straight
segment belonging to ‘truly’ flat plane containing circumference. When we talk about “a circle” we have to
include into consideration the surface contained, locked within by circumference in question. Let’s be
accurate with terms and distinguish between “a circle” and “a circumference”.
“A circumference” is 1 Dimensional creature. It is an infinite loop. Undoubtedly the most perfect of them
all but merely a ring, a hoop, a coil, a thin (blue?) line marking the edge of a circle. While any circle
(Euclidean or not) is 2 Dimensional figure with a surface. Euclidean circle resembles a CD-ROM,
non-Euclidean looks like contact lens. Keeping that in mind we will have no problem of understanding
that diameter of non-Euclidean circle is slightly bend segment which belongs to some ‘invisible’ sphere of
‘unknown’ diameter. Naturally, if we make a circle (again, not circumference!) on the ground with let’s say
3 meter in diameter then degree of curvature of such surface is trifling considering the size of our Planet.
But if we imagine a circle as big as my native Russia; draw 3 meter circle on the wall in the famous round
Chamber “La Salle Pi” (Palais de la Decouverte in Paris); paint one on the wall of the Oval Office in the
White House or merely outline a coin on any apple fruit then in all those cases surface’s curvature will
become obvious. I expect that geographers and astronomers will be among the first one who understand
what I mean. Perhaps the only case when we can speak about surface with truly ideal flatness arises
if we make a section of a ball by surface going through its center. Considering this scenario let me ask:
“Are we still living on a flat pizza-like Planet in the center of the Universe?” Then viva la Ptolemy!
To those who doubt the idea of spherical Universe being expressed here I would offer to explain the
following facts using ever prevailing ‘cubic’ point of view. Let me repeat: “cubic point”. A pixel? Just
look at us. What have we come to! So far we have agreed upon the following facts.
1) we live on a surface of a big ball (not a cube or a pyramid) rotating around much bigger one we call
“the Sun”.
2) As we observe drops of liquid in free fall we always see that those drops form balls and not cubes,
cylinders, pyramids etc.
3) When astronauts on space-station occasionally or otherwise spill some liquid it doesn’t form little
pyramids, cylinders, dodecahedrons etc.
4) Throw a little stone into a lake on a quite day and waves on a water surface will always be shaped as
concentric circles, not triangles or squares. Have anyone ever seen concentric square waves on a water
surface? Are you convinced yet? No? OK, let me continue. Why do we prefer round stadiums or concert
halls to those which are square? Because the seats in round football arena’s are spread in much more ‘justice’
manner: any seat in the same row gives the same view of the stage. Ancient Coliseum in Rome is pretty
graphic example. And to make all the rows equal we have to place the stage in the center of a … sphere.
Would you prefer to seat or to stand in a corner? Do you remember this cruel punishment from the childhood?
This is where it all begins. Starting from cubic toys including “Game Cube”.
So do we see cubic apple fruits growing on threes with square boles under sunlight blazing from the rhombic
Sun? When was the last time you saw the rainbow in full glory? Was it perfectly straight line? I think I have
made the point.
Why it is so and not otherwise? It is merely a feature of our space and it originates from the principle
of rotation. Even if we rotate any cube, pyramid or even a cylinder we will end up with a ball anyway
but with noticeable gradient of density. Ball is the most perfect shape in 3D-world we know just because
ball has infinite number of planes and axles of symmetry. After all who would dare to say that an atom
shaped like a little brick? Have electrons orbits become square or cubic? Halloo, Mr. Schrodinger!
How are you? Could you foresee this with your Psi-function? Whom are we kidding by ignoring the obvious?
The whole Universe is built on the same principle of rotation. The only difference is scale and speed.
The most adequate model of matter structuring should be the most natural one. An onion-like structure where
bigger sphere contains smaller one. So called “matrёshka”. Back to the USSR! Mathematically it can be
described by tensors using Riemman (not Lobachevsky!) mathematical spaces, containing one another.
It easily combines with the quite popular “sandwich-theory” of the Universe but “sandwich” layers aren’t
flat at all! Meaning: on the scale of the Universe light travels on bent and not on absolutely straight trajectories.
So despite using appropriate mathematical tool (tensors) the one of the postulates of the famous
Relativity Theory is only relatively correct. Sorry, Mr. Einstein. But Happy Birthday anyway!
Size of such Universal layers may very. On smaller scale onion-like model would perfectly
describe a single atom with electronic clouds having different energy levels. Below you will find
the famous 3 (among others) axioms of incidence for the flat space and one additional axiom formulated by
yours truly for the spherical state of mind. Here are the 3 “Axioms of Incidence” for the Euclidean,
truly flat, cubic-like space:
* via 1 point we can draw infinite number of straight lines through which we are able to draw infinite
number of planes. (So far so good from spherical point of view).
**For every two points A and B there exits a straight line that contains each of the points A, B.
Meaning: via any 2 points we can draw only 1 straight line (but plenty of curved ones!) through
which we are able to draw infinite number of planes.
*** There exist at least 3 points that do not lie on a line. Otherwise: via any 3 points (in general case)
we can not draw any straight line (but at least one circle!) and one and only one plane. Or in more
traditional reading: For any three points A, B, C that do not lie on one and the same straight line [but they
can belong to the same circle!] there exists no more than one plane that contains each of the three points A, B, C.
****When it comes to 4 points it is postulated: “There exist at least four points which do not lie in a plane”.
Usually we learn this stuff at school but we don t really think about it at the later stage. Only when I have
grown up it struck me one day: they may belong to a sphere! For that particular case I have formulated an
additional axiom for the spherical space (and mind!) which should read as following: Through 4 points
(in general case) we cannot draw any straight line or any plane, but one and only sphere. So hereby I am
expressing my “Balled Vision” by stating that currently Pi=3. Why currently? Because continuing to use
the abovementioned logic we should come to the conclusion that for any hemi-sphere Pi (as ratio) between
its diameter and length of circumference) equals 2. In case of hemi-sphere its diameter is the line (once
again: it is not straight segment!) connecting 0 and 180 degree points of the Equator going through the Pole.
And Pi (again, as ratio) of complete sphere is equal to one! “The Ratio” of a point or complete sphere =1.
So we have got the full spectrum of Pi quantities where sharp values Pi=3, Pi=2 and Pi=1 are representing
a step, a “quantum jump” from one state of mind (and therefore matter) into another. This is a perfect example
of so called “artefact”. In this case the value of mathematical constant is changing in correlation with the
state of experimenter s mind and the shape and size of surface in question. On a physical level the current
shift from flat Pi= 3,14159265358… to 3 sharp can be interpreted as change of the space curvature around
us according to gravitational gradient. That is another question “what is causing it?” It also could mean that
the Universe has made at least 1/6 of the complete rotation and some of us already feel it by expressing as Pi=3.
One particularly interesting case is someone’s car registration plate in Maryland, USA reading “pi is 3”.
I wonder who the owner is. In order to distinguish the different sharp values of the famous ratio from the
actual flat value of pi=3,1415…which has already become very chrestomathy I would like to name the
Biblical conditions under which length of a circumference contains exactly 3 diameters as “equilibrium”.
So let me letter the ratio between the length of a circumference and its diameter in equilibrium as E.
Then equality will look like E=3. And You, Dear Reader, are very welcome to make your suggestions in
naming the famous ratio for hemi-spherical and spherical conditions. Let me just emphasize that the point of
view where the ratio equals to 1 is the point of complete spherical state of mind. It represents the vision
of human or other being (call such being God if you wish) who covers and feels the whole Universe at once,
by one thought. Don’t look at me that way I don t feel the whole Universe just yet. But by adopting “Balled Vision”
we all will feel and think as OneBeing effectively getting closer to “The Day we all can glean an inkling of
God’s mean”. Until (and beyond) of such Magic Day I remain Yours Truly Alexei ARSENTYEV.
By the way: flat Pi^3 =31,0062766803…Surprise but I have never seen any references to this “cubic root from 31”
as ancient way to calculate Flat Pi in any Pi-historic articles. Here are few sources which helped me to shape my mind.
Quite literally.
[1] The Holy Bible, The Books of the Old and New Testament, 1 Kings 7:23 ; 2 Chronicles 4:2.
[2] Mr Y.Kanada and his record calculations at
[3] “Does the Bible say pi equals 3.0?” Russell M. Grigg
[4] “The Bible Code” explained at
As you may have already guessed English is not my native language. So I apologize for occasional mixture
of articles (a/the), confusing past tenses with present perfect ones and using present continuous tense instead
of present simple and any other possible grammar mistakes. You are very welcome to send your comments
(preferably in traditional mail) to: Alexei ARSENTYEV, P.O. Box 93112, NL-1090 BC, Amsterdam,
The Netherlands or via contemporary means of communication such as e-mail: .
For those who wish to comment on non-grammatical mistakes I have typed a short epilogue.
Epilogue.
Michael Stifel, German mathematician (1487- 1567) said back in 1544 : “Flustra laborant quotquot se
calculationibus fatigant pro inventione quadraturae circuli”. In English it will be close to: “Futile is the
labor of those who fatigue themselves with calculations to square the circle”. I only can add to this wise
saying that: ” IT IS DIFFICULT”. I know it from my personal experience. No, not to imagine matreshka-like
Universe with spherical onion layers. Not that. It is difficult to admit that despite of so many obvious hints
surrounding us all the time we have been blind (or blinded?) for so long. It is difficult to admit of giving
lack of proper consideration to such an important ratio which resulted in something what looks like the
oldest mistake in the human history. There is a difference between “pride” and “arrogance”. Synonymous
to “arrogance” is “stubborn”. Antonymous to “pride” is “shame”. It is no shame in making a mistake.
Nobody is exempt from it since nobody is perfect on this Planet. There is just a degree of perfection.
From black to white, from pride via arrogance passing stubbornness to blunt stupidity. So what we call
today “a mistake” yesterday was just a step on The Evolutionary Ladder. And anyone should be proud of
being able to upgrade itself to the higher degree of understanding where (looking back from the next level)
there is no shame to say: “that was just another step, the previous version”. But is it arrogant not to look
around, it is stubborn to hold on outdated visions despite plenty of facts being available and it is dangerously
stupid to be that stubborn and arrogant to deny the obvious. It holds the process of evolution by adding
“another brick in The Wall” .
There is a difference between “pride” and “arrogance”. Probably the same as between “circle” and “circumference”.
And for some of us that difference is not visible while for others it is clearly there despite being thin. Very thin.
Deadly thin. Thin as any ‘straight’ or ‘bent’ line; ‘truly flat’ or curved surface in Geometry. But it is there.
To those who wish to free their mind I would strongly recommend to watch “The Matrix” movie (part one).
There is an episode where Morpheus is saying to Neo: “I didn’t say it would be easy ” and later
“We have a rule: we never free a mind once it’s reached the certain age, the mind has a trouble if letting it go”.
So let “it” go while your mind is flexible enough. I know it is difficult. Difficult to shake off the cliches and difficult to admit.
Difficult to be taught and preached by ‘a stranger’ what to do. Our arrogance makes it difficult.
That is why it was proclaimed as “deadly sin”. Because those who don’t except the words from a ‘stranger’ marking ANY
unsolicit information as “spam” are walking on thin ice. There are no strangers. We are all God’s creatures. We just being
played against each other by those who benefit from it. With 100% chance that one day The Creator will be kind enough to
explain it to them. Personally. You do know what it means. Don’t let it happen to you. A.A.