The end

Finally I have come to the end of the route. I have not answered many questions during abseiling, but I have at least brought you some tools for understanding the world. It has been nice blogging in english, which is not my mother tongue. I will not quit blogging. I will just change to a more convenient garment.


Maybe a norwegian sweater, and maybe writing in norsk. With the aid of Bable fish, everything is possible.



Thank you for your kind attention!

Where is the cube?

In a former blog I showed a relation between the tetrahedron and the cube. If you try to build it from marbles, you will probably have no success. Rather you get something like this:

Stella Octangula

This is not a cube. The reason it is not, would I explain like this:

If the side of the tetrahedron is a, then the cube is ½*a*sqrt(2), which is an irrational number.

diamond.

Still there must be a rational relation, since the tetrahedron is called "cubic". The cube looks like this:
The unit cube. Scary! I should say that this cube is among the last things you discover, when looking at the model. Probably predilection to ortogonal coordinate system brought the term "cubic" to a system that seems to be something else. When you look up "chrystallography", you will most likely get to know, that it is an experimental science. It can also be handled geometrically. The spatial capacity of human mind is often limited, even when concerning professors'. The cure is building models of what you can't imagine. Hand and mind together brings the most out of it. Certainly you know that a coin on the table could be closely surrounded by six equally sized coins. But did you know that a marble in space will have 12, and this can be accomplished in more than one way? Assuming there is a fourth dimension, modelling is not very easy. So this is my problem to you: How many is it in 4D-space? Due to crisis in my economy, I can not give you a prize this time, but I will publish your name and photo.

Dedekind

At last I found a definition of real numbers. Richard Dedekind defined them as "cuts" of rational numbers. You will find his text translated to english here: http://www.gutenberg.org/etext/21016

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It is not important, but I was curious after having read many textbooks starting with "definition" of real numbers. Newton did not have the definition. Yet he accomplished a lot. Another thing quite interesting is that there seems to be a definition of "line". Not the straight line, but a kind of curve, that splits the plane in two halves. The text was far too complex for me to understand. I use the term "line" frequently, dispite my lack of knowledge about the definition. Hope I have got it right.

regular system

Seems like my last actvities brought me too much headache. Maybe I should read a book, for a change.
The cuboctahedron. The "coordination polyhedron" in the fcc. It contains one central point and 12 ligands. The "elementary cubic cell" of the system holds 14 points and has no central point. I find it hard to connect these basic concepts.


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Face centered cubic. Only idiots like me fail to see the cube.

ccp (continued)

Another imaging of 'close packed cubic', or "face centered cubic", as it is also called, is this:

You start with a layer, that is not closepacked, but rather the conventional rectangular. The second and third layers are dislocated as shown in the picture. It may be hard to see, but each point has 12 neighbours or "ligands". The coordinating polyhedron is a cuboctahedron, which is not "regular". In the simple cubic system each point has 8 ligands, and the octahedron is the coordinating figure. Suppose we are talking about integers! But if we talk about "real" numbers, a paradox will reveal.
I have a cube of points (a;b:c). a b c belongs to the set of real numbers in a certain interval. If I dislocate the layers like said above, and hence changing the structure from simple cubic to face centered cubic, the volume of the cube must decrease. Is there anyone out there who can explain this? The best would be, if you catch me with having made a logical mistake.
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I did a mistake. Hope you see it, too.

A lie is an approximate truth

A segment of a line contains infintely many points. For example the points of the interval [a;a+d] contains alef-null points for any non-vanishing d. Thus: You can't make a line by adding points. Accordingly the plane is not generated by adding lines, and so forth. The point, the line and the plane are sovereign enteties. But the integer correspondances are different. Integer means there are elements. I saw the surrealistic equation on the web:


1+1=3 , for large values of 1


It seems like nonsence, but actually your hand-held calculator may reckon like that. Here is the display (and also its memory):

It uses in fact integers. In this case ten digits for the mantissa and two for the characteristica. What is presented in the display is an interval - not a real number. After the "9" in the mantissa could follow infinitely many digits. These are truncated. When processing long series of calculation, one may get the surrealistic answer. This is only a result of how the machine works - not of mathematics. The Homer Simpson falsification of FLT is an example of false calculation. You can see it is wrong. On the left side there is an odd number and on the right side is an even. How can engineers build things that work, when they are using this false-calculating device? Because "real numbers" are not real. For every-day reckoning ten digits is satisfactory. No one has much use of numbers containing millions of decimals. Math, on the other hand, demands exact calculation. The mathematician wants to know if two enteties are equal, not if they are approximately equal. Hence the "transcendental" numbers. We can stick to the integers 1,2,3.... for a while. Not that I think it will be simple, but it will be "integral".

The sum of 1+2+3... is a "tiangle number"

The sum of triangle numbers 1+3+6... is a "tetrahedral number"

The sum of two consecutive triangle numbers is a "quadratic number" (The quadrate is not elementary in this weird math).

If there is something you can call a "cubic number", I have not found out yet.

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When looking at Wolfram, I gained some self-confidence. This is the first layer in a closed packed system. The second layer must have one point at "X". The third can have a point at eighter A or B. They correspond hexagonal (A) and cubic (B) close packing. Of course, this is seen from one specific angle. There are other angels making it easier to grasp. Building models is the best way to learn.

Tetrahedron

The tetrahedron is
Ta = a*(a+1)*(a+2)/6 ; where 'a' is an integer.

Lemma :
A tetrahedron may be the sum of two tetrahedrons*

Peculiar! A cube is never a sum of two cubes, according to FLT. Seems I don't know the properties of the integers. Do I understand Pythagoras' theorem? Probably not.

This is one relation between the tetrahedron and the cube. There are others, but I have difficulties drawing them. Some other day, maybe.
Today I discovered the OpenOffice programs from Sun. They're free, and contain a lot of facilities. I feel an easy-to-understand proof should be graphical. I do not want to create optical illusions, since I am most serious right now.

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* I have a beautiful proof for this lemma, but I don't want to write it right now.

October

This month has a latin name, meaning the '8:th month'. Historically the year begun at the vernal equinox. The year in the russian orthodox calender had 365,25 days, which is the reason the 'october revolution' actually happened in november. Of all metrological mess, chronology is the worst. Time keeping differs from culture to culture, and from one country to another. The day begins at noon, midnight, sunrise or sunset, all depending on culture. Adding to that the silly DST the time keeping is most confusing. The "standard pacific time + 1 hr, except for native americans" is the most ridiculous concept. USA likes registering ethnicity. If your grandmother is native american and your grandfather is a jew, then you should not practise 'daylight saving time', but if you're black and your grandmother is hispanic, then you should apply it.

Congruency

This is an interlocking jigsaw puzzle. The two pieces A and B are congruent according to Euclid, but yet they are not identical. Only one of them fits.
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Pasteur made great acheivements in chirality. Some tried to make fun of his idea that molecules could exist in two forms: A left (S) and and right (R) one. Since Euclid did not mention this, it must be wrong, according to the fundamentalists. Dogma can be right, but nature itself is always right.

FLT

"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." I don't think he had a general proof. He proved validity for third and fourth degree. If he had the general proof this would be unnecessary. However, we have the proof today. Personally I think the problem is queer: "It is not like this - prove it!" There must be millions of statements that are not true.

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The purpose of this blog is not to insult Homer Simpson or any other of your idols.

1782^12 + 1841^12 is not equal to 1922^12 . To prove the statement, don't use your hand-held calculator! Use your head!
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Summan av ett jämnt och ett udda tal kan inte vara ett jämnt.

Friction

When going up it is necessary - when going down it is useful.

complex numbers

The equation x^2+1=0 has two solutions: x=i ; x=-i. Every equation

P(x)=0 (2)

has at least one solution. The solutions are complex numbers. They are written as an ordered pair a+ib. Another possibilty is to call some solutions "imaginary". I dont see why we should do that. We have other complex enteties like triangulae and points, that we don't call "imaginary". Anyhow, the roots of (2) that can be written "a+i*0" are called "real". These have the characteristic that they can be ordered. Two numbers have the relation 'a is less than b', if they are not identical. Other complex numbers do not have this characteristics. They can be treated in calculus anyhow. Here are some rules:

(a+ib)+(c+id)=((a+c)+i(b+d)

(a+ib)*(c+id)=((ac-bd)+i(ac+bd))

Light

Light is an electromagnetic wave, just like broadcasting radio. The difference is, light can be seen by human eye. Like matter, light consists of tiny "particles", called photons". Each of them has an energy of 'frequency multiplied by planck's constant'. The reason light is measured in candela and not watts, is an human aspect of measurement. Human eye sees light. Other frequences that the human eye does not catch, are often negligible. The candela refers to the obvious.

Darkness

Wilhelm Reich invented, among many other silly things, the "orgone". This is a kind of energy or power that flows into your balls, when you are horny. He was able to measure orgone by a special device. His proof of existence should satisfy any physicist. Or? No, there has been too many similar obscure "proofs". On a logical level, it is impossible to prove the non-existence of anything, so I say: I doubt the existense of orgone, at least I hope not to have any of that kind in my apartment.

This kind of bullshit tends to come from certain "sciences". It does not make much damage to the world. The proprietor will be quite satisfied living with his humbug, I have been told. This implies falsehood and truth are equally good. There is a danger accepting this thought. It will yield us bridges that collapse and medicines that kill the patients.
I could easily make up some new material. For example: darkness consists of obscurones. The more obscurones, the darker it is. A black hole holds at least 98% obscurones, I should say. Just a joke!

Slope

When walking in the mountains there is not only one slope. I told you ' is the derivative. This is true only in a twodimensional coordinate system.

An introduction to calculus

This is a staircase. Each stair is twice as big as the preceding one. The curve is the derivative and the sum of the boxes is called the Riemann integral. The same reasoning holds for rational numbers and algebraic numbers. If you are keen on mumbo-jumbo, you can say:

The derivative equals the function f=2*x (1)
If you make a similar staircase with rational numbers, and you want (1) to be sound, you have to replace "2" by "e". e is not a rational number, but a "transcendent" one.

Extending the staircase to algebraic or real numbers, won't give you very much new information. That is what I call 'Ockhams razor'. Stop when have got enough information! But I am just a metrologist, and not a "pure" mathematician.

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example:
1+1+½+1/2*3+1/2*3*4+.... will quickly converge to 2,7182818..., which is the logarithmic base e. As said afore, (e^x)'=e^x, that is quite handy. (e^îx)'=i*e^ix and e^ix=cos x+i*sin x. Thus we get a simple relation between e and pi. We can easily conclude why it is possible to construct a regular 17-gon, but not a 19-gon. It is so very easy that a child can do it.
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the sign ' means the derivative, that is the slope of the curve, but I am sure you knew that already

Pro life

The german cockroach, blatta germanica, has come to our beautiful country. Some people want to kill off those golden bugs, but our party, the FRA, wants to save them. They are creations of God, and man has no right exterminating them.

Golden bugs forever!

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* Friends of Rare Animals

Cardinal numbers

Kobe

If there is a one-one relation between the elements in two sets, the sets have the same cardinal number. The sets do not have to be finite. The cardinal number of the set of algebraic number is alef-null, for example. There is not an one-to-one relation between the elements in the set of non-algebraic numbers and the set of natural numbers. Accordingly, the cardinal number of the first set, the transcendentals, must be greater. This might surprise those of us, who know only two: pi and e. The transfinite cardinal numbers in every-day sense. They are infinite. Adding numbers to infinity leaves us with infinity, which is not very interesting. What set theory showed was relations between various infinte sets. There are different infinities, depending of the starting point, which might surprise some. "There is nothing infinite in the real world, except the phantasy of an idiot". I am not sure, if this medium is a part of the real world. Let's suppose it is not, or rather say I am an idiot. Georg Cantor's achievements are dubious. Was it mathematics he dealt with, or was it logic? Maybe we should look into the logic of arithmetics. I should say it is about answering the question: are the two enteties a and b equal? How do we prove equality? If it is not possible to prove equality, we can not be sure of the opposite, inequality, eighter. This was not fairly understood until the 20th century.

The surreal numbers

My latest post did not mention surreal numbers, which were invented 1974. I am sorry, but I did not know about them until now. My studies in calculus terminated in 1969. I am not confused though. If we define something "as the rest", there will be problems. Linnaeus classified plants by looking at their flowers and seeds. However, a lot of plants lack genital organs. He classified these as "cryptogams", and I don't think he liked them, especially not litchens. To a modern paleontologist, the cryptogams may be more interesting than the sexual plants. This is the consequence of leaving a rest when you classify.

If surrealistic numbers will be the future of mathematics, I can't tell, The inventor of SN is a famous computer scientist, whom I first met in MAD magazine in 1957. It was an article on metrology. The man has phantasy, humour and a sound distance to science. By distance I mean like a painter beholding his piece of art from some steps away.Knuth. The inventor of surreal numbers.

Continuum

Real numbers


In my local chinese restaurant I found a book in calculus*. It was quite extensive, having 999 pages of text. I was just able to start reading it during my lunch. The first issue was defining the real numbers. This affected my digestion negatively. The definition was incomplete, and there is no need to define the concept. Thousands of years of mathematics have yielded lots of achievements without that definition. Calculus does not aquire a definition eighter. Numbers are basic pieces of arithmetics. They can be manipulated according to the rules. I think, not even Newton know the definition of real numbers. Checking some encyclopediae on the web I found incorrect or incomplete definitions of the subject. What metrology concerns: we never use real numbers. We use rational numbers, that are easy to define. If one measures the circumference of a circle, one will never get "pi" as a result. It is only in pure mathematics this mysterious constant exists. Written as a decimal fraction, no integer is a function of the preceeding ones. That kind of numbers are called "transcendental". They are many, but very hard to find.

• The natural numbers are 1, and its followers ad infinitem
• The rational numbers are x; a*x=b
• The algebraic numbers are x; P(x)=0, where P(x) is a polynom
• The non-algebraic number are the rest. Pi is one of them

Let's calculate!

x^3=-1 **

One root is real and two are "imaginary". Imaginary numbers have no order. You can't tell which is bigger, i or -i. Real numbers are ordered. Given two different numbers, one must be the least. As you can see, real numbers is not the fundament of arithmetics. They are just a part. The fundament is natural numbers. They are used by your computer and all other instruments. "1" can be interpreted as one gallon or one apple or one nanogramme and so forth. That is all that is needed for practical reckoning.

The continuum is "all numbers". We can not decide how big it is. This was proven in 1964. The proof was just a complement to Gödel's theorem, and nothing unexpected. Math is an open theory, that may let certain questions stay unanswered.

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* Adams Robert A. Calculus A complete course, Toronto 2003 , Pearson education, ISBN 0-201-79131-5 "a real number is a number that can be represented by a decimal fraction" That's bullshit rather than a definition. I will demonstrate in a further blog, that definition of real numbers is not necessary for accomplishing calculus.

** It is possible your computer fails to do this operation. This is due to a "bug" in the system. You can certainly work it out in your brain. (r3/6; +-0,5) are what my head gave me as imaginary roots. I can have miscalculated. If so, please tell me! Yes, I was wrong.

Time

Time is measured by the time keeper.

The hands ususally go clockwise. Atoms have a certain frequency, independantly of outer conditions, like temperature and air pressure. A good watch deviates less than a second in a million years. You have no excuse for beeing late for an appointment. If you are asleep, another time scale will rule: your biological clock. You may have dreamt you are falling down. The time moves so slowy, that you are almost still in space. Lewis Carrol wrote about Alice in the Wonderland . The illustration above is from that novel. The white rabbit seems to be in a hurry looking at his watch.

Falling freely

Newton was not the first, as I said earlier. It was Ismaël Bullialdus, who concluded the force is proportional to the square of the distance inverted.

How much one learns from reading! Anyway, my speed must increase if I fall down, since acceleration is proportional to the force.

This is mechanics, a kind of enhanced geometry, where you put in "mass", "force" and "time". Its is formal, and no measurements are needed. That is my definition. It is pure thinking (others claim mechanics is "dirt, sweat and tears"). What Newton studied was "celestial mechanics" rather than "astrophysics". His stars were not glowings bodies in cosmos. but merely points in a coordinate system. The achievements depend on the point of view, I should say. My climbing down will be strictly mechanical.
My friend is stuck. We use nuts and friends in order not to destroy the routes for our fellow climbers, and also protecting the environment. I have been told some routes are crowded with pitons. Others resemble refuse dumps. There are some climbers who use motor drills for attaching the pitons. They are not calculating to be followed by others. "Après nous le déluge" as madame de Pompadour said.

Yesterday

Related to the depth is time measurement. "Time is just an illusion", "Time is just an aspect of time-space" etc. are statements intended to make you shut up. The concept of time is in all our minds. There is a present, there was a past and there will be a future. The present will be past, but past will never be present, though we can sense the past by our memory. Whether this memory is false or sound, does not matter really, since we can not change the past anyway. On the other hand, we can create a future we wish.

That ended the lesson for today. Hope to meet you in the future.

exgen. Nicholas Bourbaki

How deep?

My journey will be finished, as I reach the valley. There are persons that go further. This is called speleology - investigation of caves. Jules Verne wrote "Le voyage au centre de la terre". The centre of the earth is the ultimate point. The lowest level on earth, but there are other bodies in our universe. like "black holes". They have no bottom, as I understand it. According to theory of realtivity there is an infinite way down. Physics demands observation and measurement. One metre is defined as the distance light in vacuum travels in 3,3356409519815 nanoseconds. In a very strong gravitational field, and a state far from vacuum, it will be tricky to measure, but my guess is : there is no bottom in a black hole. The opposite question, the size of the physical universe, has been answered with 'cosmos is finite'. This is an interesting paradox. 'Down' is infinite, but 'up' is finite. Of course this is due to the rigid point of view, that physics has. In the olde days, metrological observation was not necessary. Your feeling, revelations and the text of older scriptures, were the reality. Since individuals differ from each other, there were many realities. Few had accepted the dull world of metrology . Remember that the great Newton, when playing with his mathematical symbols, believed he was investigating God's thoughts. God is greater than that, we gradually discover. Why should he play dice? Nevertheless Newtons achievments are valuable for helping us understand some phenomena, like gravity. I can calculate how fast I will fall, if I lose my grip. This is an obstacle. I have to concentrate. Hope I see you tomorrow.

your descendant

NB

Italian speleologists in the "grotta" - cave.

downwards

So I have come to the top of the mountain. What now remains, is the way back to the valley.


The lodge

This is the trickiest part of my expedition. An elephant can easily walk upwards, but has great troubles descending. A horse has too little brain for this manoeuvre, so he will be stuck in the highlands, if he climbs a steep slope. A man like me, who is related to the apes, might manage. Technical facilities will help, I am sure. And maybe I can get some help from my friends.

repelling