Attempt to visualize higher dimensions

In summary, the conversation discusses the attempt to visualize higher dimensions and the use of the Pascal triangle in identifying structures in spacetime. The formula for the Pascal triangle is N!/k!(N-k)!, where N is the row number and k is the column number. The concept of dimensionality is also explored, with a point representing zero dimension, a line representing one dimension, a triangle representing two dimensions, and a tetrahedron representing three dimensions. This can be represented in a table, showing the relationships between the different simplexes. The conversation also suggests that this process can be continued to find the components of higher dimensional simplices.
  • #1
nightcleaner
This thread is for discussion and presentation of results in the attempt to visualize higher dimensions.

My current work involves identifying stucture in the 4d world of spacetime equivalence. Marcus showed me that structures in spacetime are related to the Pascal triangle, and selfAdjoint helped by pointing out that Pascal's triangle can be calculated from the N-choose-k math. All this is interspersed in Marcus' two-worlds thread on this board.

The Pascal triangle is formed by lining up numbers and then adding the two above on the next line below. The second and subsequent lines also start with the first two numbers, like so:

Code:
            1,1
           1,2,1
          1,3,3,1
         1,4,6,4,1

You see the one and one in the first line add to the two in the second line. In the third line, two plus one is three. In the fourth line, one plus three is four, three plus three is six, and so on.

Marcus and selfAdjoint told us in the Two Worlds thread that you can calculate these numbers by a method known as N-choose-k, in which you have a group of N objects, and you pick them up in as many ways as possible in groups of k. So if you have three N objects, A,B,C, and you pick them up two k ways, obveously you can pick A,B, A,C, or B,C. In this method, A,B is counted as the same group as B,A.

There is a formula for this method, and it involves factorial numbers. These numbers are written as n! where n is a positive integer. 3! is the product of 3 and all the positive integers below it, 1x2x3=6=3!. Another example,
4!=3!x4 is 6x4=24. It is conventional that 0!=1=1!.


The formula for finding the numbers in the Pascal triangle is

[tex]N!/k!(N-k)![/tex]

where N is the row number and k is the column number, counting the first column as zero. For example, in the fourth row of the Pascal Triangle, in the first column, we have 4!/0!(4-0)! is 24/24which equals one. The second number in the fourth row is 4!/1!(4-1)! = 24/3!=24/6=4. And so on.

There is also a tetrahedral form of the Pascal triangle, where there are three ones at the top, and ones around a two in the next layer, and ones around threes around a six in the next layer.

Now to get back to business. There is an idea for thinking of dimensionality in terms of the least structure that can fully represent the dimension in question. For example, as in plane geometry, a point has zero dimension. It has no size that can be measured, and until higher dimensions are invoked, there is no way to talk about where the point is located. The point is called in this system a zero dimensional simplex.

The one dimensional simplex is a line, which has two end points. The only thing you can measure in a line is how long it is, so it is one dimensional. In geometry, a line is thought of as having only one point in cross section, so it is not the same thing as a wire strung between two poles, which has, as well as length, a diameter or thickness. Note also that a line in this sense has to be "straight," because there is no other dimension defined for it to curve into.

The two dimensional simplex is a triangle on a flat plane. It has three apex points, three lines, and one flat surface. We count it as one surface, because it has no thickness, so if you define one side, you have defined the whole thing. From here on, we will be talking about apex points so that the word point can still be general enough to use in dividing higher dimensional line. Surfaces or faces will be exterior faces to the structure.

The three dimensional simplex is a tetrahedron. It has four apex points, six lines, and four surfaces.

Note that each dimension is represented by adding one point to the simplex for the dimension beneath it. Zero dimension is one point, one dimension is two points, and two dimension is three points. We have to specify in this system that the added point is not in the same dimension as the points before it. The third point is non-co-linear with the first two points. That is, no single straight line can be drawn to connect all three points of the 3simplex.

Now we can make a table to show the relationships between the simplexes of zero, one, two, and three dimensions.

Code:
Dim.          Simplex     s0   s1   s2   s3   
zero             point      1     0     0    0
one                line      2     1     0    0
two           triangle     3     3     1    0
three   tetrahedron     4     6      4   1

Now we can see the Pascal triangle developing in this analysis of dimensionality. It seems reasonable to assume that we can continue the process started here to find the component parts of higher dimensional simplices. Below I have carried the table a few steps further.


Code:
Dim.          Simplex     s0   s1   s2     s3      s4     s5
zero             point      1     0     0      0       0       0
one                line      2     1     0      0       0       0
two           triangle     3     3     1      0       0       0
three   tetrahedron     4     6      4     1       0       0
four     hyper tet.       5    10    10     5       1       0
five                 ?        6    15    20    15      6       1
six                   ?       7     21    35    35     21      7

We now have three methods for finding the number of lower dimensional simplices in a higher dimensional simplex. We can use the Pascal triangle, or we can count using the N-choose-k method, or we can calculate using the formula for the N-choose-k method.

Can we extract any meaning from the numbers in the higher dimensionalities? And can we use the meaning to form some kind of visualization? We have seen how a one dimensional 1simplex (line) can be built up from two zero dimensional 0simplices. And we have seen how a two dimensional 2simplex can be built up from three 1simplices. We may easily be able to imagine the three dimensional tetrahedron built up from four 2simplices, joined by three 1simplice lines at four 0simplex apices.

Now how do we go on to visualize the four dimensional structure analogous to the tetrahedron? I have called it a hypertetrahedron here, for the simple reason that I do not know what else to call it. Probably we will have to find new words for these higher dimensional simplices. For now, I think I will use the convention of merely notating them as 5simplex and so on, even though that seems to me to lack a certain desireable elegance.

Now it is possible to represent a 3simplex (tetrahedron) in two dimensions by a number of methods. One can draw a fair perspective drawing, which looks like a three sided pyramid on a triangular base. Of course the base is only distinguished in the perspective, and really it is no different from the other sides. Also in the perspective, to create the illusion of depth, we do not draw all the lines the same length, but the ones that go down into the perspective or up out of it are drawn as shorter lines. That way we can arrange all the apices of the simplex in one two dimensional plane, and it seems quite natural for us to do so.

We should remember however that this method of perspective drawing is not really natural. Instead, it was invented in the renaisance by a scientist, DaVinchi, if i remember my history. He strung a frame with wires in squares, and fixing an observation point, he drew what he saw in each square, thus making a fair perspective. It seems simple to us now, but in the fifteenth century things were different. If you look at portraits of people from before the renaisance, you will see that they were drawn flat. Landscapes from that period look especially odd to us now, but at that time, it was thought natural to show distant objects as small, close objects as large, and no thought was given to the way lines between them looked. Small distant objects might be represented under the feet of a large close figure, so that today it looks to us as if the figure is floating in air above the tiny houses, churches, and people below.

We learned to draw and to view drawings in three dimensional perspective, and later we learned to make a series of such drawings, like a cartoon, to represent the passage of a fourth dimension, time. Now when we watch movies or television, we can see three dimensions on a two dimensional screen, and watch the flow of time as the two dimensional images flicker past us too fast to notice that they do not really, in themselves, one at a time, move at all. In fact, on television, if you understand how the image is formed, you can see that the data stream is really linear, a single line of information that is divided up across your television screen to give the illusion of a two dimensional image. So you see we can actually represent at least up to four dimensions even with a single line of data, a one dimensional object. We just have to know how to look at it.

Then the question, again, is how can we learn to see four dimensional objects? Is there some way to establish a frame and a viewpoint, like DaVinchi did, that will give us vision in 4d?

The viewpoint seems easy enough. A point is a point in any dimension up to four, as we have already seen, so we know what we mean when we choose one well enough. But what is a frame?

We might think of a picture frame, which is a sort of border to enclose and protect a picture. What is inside the frame is not the same thing, usually at least, as what is outside the frame. Inside may be a landscape painting, of mountains, trees, lakes and streams. Outside may be the wall of your living room. That would be a frame in two dimensions. Real frames have thickness, of course, but that is not part of the functionality we are discussing here. A picture of a picture frame is really as much a frame as the frame the picture of the picture frame is hung in. It may have the third dimension, but it doesn't really need it to be itself. Two dimensions are sufficient to a frame of this kind.

But a three dimensional frame needs its thickness as a part of its definition. The third dimension must also be limited. A frame in three dimensions is like a room, perhaps, or like a world, or like a universe. A frame in three dimensions is everything that fits inside some three dimesnional shape. It could be any shape that is convenient to our purpose. We might think of a train as a frame, as Einstein did in Special Relitivity, or as an elevator, as in Einstein's General Relitivity. But the simplest frame in three dimensions is probably the sphere, just as the circle is the simplest frame in two dimensions.

And what then is the four dimensional analog of the sphere?

Often when trying to visualize in four dimensions we think of something like M.C. Escher's strange castles, or Salvatore Dali's cross or his melting timepieces. But perhaps the simplest view of the fourth dimension is as a process, a blurring of the frame of reference, as in a cartoon the artist may draw a ball moving through the air as a sort of blurred series of images of itself, a sort of line that we recognise as the visual effect of an object pursueing a course through a limited amount of time and space: a four dimensional frame. We only see a part of the ball's movement in the picture, not its entire movement history. We only see the ball park, the player and the player's mit, not the entire universe. These regions of timespace are the frame of the drawing.

In any four dimensional visualization we will have to consider a point of view, and a frame enclosing three dimensions of space and one of time.

The tetrahedron fits inside a sphere. The sphere is the simplest representation of a frame in three space. If we want to represent a 3space frame in four space, we have to consider our point of view and for how long we are going to be watching. If we establish a point of view that is moving along with the 3frame sphere, for a limited amount of time, then the sphere does not seem to change at all. It looks just like a sphere in 3d, even though we are looking at it in 4d. But if the sphere is moving relitive to our point of view, then it takes on a blurr. If the movement is directly toward us or away from us, we will see the sphere getting larger or smaller. But if it is moving across our field of view, tangential to us at some distance, then it will appear like some kind of line. Imagine the line of a tracer bullet fired into the air. Your eye keeps the light for a while, and you do not just see a point of light where the bullet is, but you see a line. That is a four dimensional object you are looking at, from your point of view, and in the frame provided by the target range and the length of flight and burn of the tracer round.

A sphere moving in a straight line is probably the simplest image of a 4d object. But it does not reveal much structure, beyond that found in the first dimension. Really it is more. The sphere contains all the structures found in the lower dimensionalities, six lines, four apices, four surfaces. As it moves through time, all of those subspace structures must have continuity. So a line in a certain orientation will trace out a sheet within the line of motion of the sphere. We call this a world sheet. Of course, if the line is moving along its own length, it is still a line, a world line, I suppose.









Be well,

nc

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  • #2
this is just a date stamp. The first post in this thread has been updated.
 
  • #3
I will be posting drawings on the Society for the Investigation of Prescience board at Meetup.com under the presentation headed "Transformations." Shoshanna has posted some pretty pictures there, too. Please join us at:

http://paraphysics.meetup.com/1/

nc
 
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  • #4
I must apologise for my inability to find a way to post reasonable pictures from my presentation. I have been able to post fuzzy small reproductions on one site, and have tried to set up a url for the purpose, but continue to be foiled in my efforts. People put up such beautiful web pages, it still seems to me that i should be able to post some things too, but alas.

It seems my only option now is to make hard copy photos of my slides, then photograph them digitally, then try to post them as photographs. This seems unreasonable but it seems now to be the only unexplored option that doesn't cost a lot of megastringbigbucks, which I don't have. Anyway I tried.

nc
 
  • #5
One thing that bugs me about this is this:

The standard simplexes are not simply points. The zero-dimensional simplex is a point, but the one-dimensional standard simplex is the entire unit interval [0, 1]... not simply its endpoints. The two-dimensional standard simplex is the entire triangle with vertices at (0, 0), (1, 0), (0, 1). et cetera.

It does not suffice to simply add a point to a simplex to get the next simplex... in fact, a disjoint collection of points is always zero-dimensional!

I don't think this is relevant to the rest of your post, though.
 
  • #6
Hurkyl said:
One thing that bugs me about this is this:

The standard simplexes are not simply points. The zero-dimensional simplex is a point, but the one-dimensional standard simplex is the entire unit interval [0, 1]... not simply its endpoints. The two-dimensional standard simplex is the entire triangle with vertices at (0, 0), (1, 0), (0, 1). et cetera.

It does not suffice to simply add a point to a simplex to get the next simplex... in fact, a disjoint collection of points is always zero-dimensional!

I don't think this is relevant to the rest of your post, though.

Hi Hurkyl, and thanks.

I guess I think about simplexes as the least possible structure required to define a minimal space. If it is a one space, then you have a line. If it is a two space, the least structure that can describe it entirely is a plane, and so on. You can get the least possible structure through an evolution like counting, by adding a point to the lower dimensional simplex, with the caveat that the added point cannot be contained in the lower dimensional space. Then it is just a matter of picking a point of view, a number of dimensions in which to project the structure, and the number of dimensions required to consider the process you wish to observe. I may look at a four dimensional process in a two dimensional picture from a point of view chosen to reveal all the relevant features of the process.

Be well,
nc
(186)
 
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  • #7
Each n+1 simplex is the join between the n-simplex and a point not in its n-dimensional subspace (also called in this instance the cone over the n-simplex). Thus to create the 1-simplex you take the two points that consitute the 0-simplex and join them to some (any) third point. This produces a sort of L shaped figure which is topologically the same as a unit interval, so the unit interval is taken as the standard 1-simplex. Pick a point not on the line and create the 2-simplex as the set of points spanned by all lines between the point and the interval. This makes a solid (i.e. filled in) triangle; all such triangles are topologically equivalent, and the triangle with vertices (0,0), (0,1), and (1,0) is chosen as the standard 2-simplex.

And so on.
 
  • #8
Basically this is good writing. I will try to say why later.
First, there are some little things, like spelling errors, to fix.
Whenever an Italian writes "ci" there is an unwritten H between the letters. CHinzano is spelled "cinzano", CHow is spelled "ciao".
Please fix the spelling of DaVinci.

----quote from Richard----
...Now how do we go on to visualize the four dimensional structure analogous to the tetrahedron? I have called it a hypertetrahedron here, for the simple reason that I do not know what else to call it. Probably we will have to find new words for these higher dimensional simplices. For now, I think I will use the convention of merely notating them as 5simplex and so on, even though that seems to me to lack a certain desireable elegance.

Now it is possible to represent a 3simplex (tetrahedron) in two dimensions by a number of methods. One can draw a fair perspective drawing, which looks like a three sided pyramid on a triangular base. Of course the base is only distinguished in the perspective, and really it is no different from the other sides. Also in the perspective, to create the illusion of depth, we do not draw all the lines the same length, but the ones that go down into the perspective or up out of it are drawn as shorter lines. That way we can arrange all the apices of the simplex in one two dimensional plane, and it seems quite natural for us to do so.

We should remember however that this method of perspective drawing is not really natural. Instead, it was invented in the renaisance by a scientist, DaVinchi, if i remember my history. He strung a frame with wires in squares, and fixing an observation point, he drew what he saw in each square, thus making a fair perspective. It seems simple to us now, but in the fifteenth century things were different. If you look at portraits of people from before the renaisance, you will see that they were drawn flat. Landscapes from that period look especially odd to us now, but at that time, it was thought natural to show distant objects as small, close objects as large, and no thought was given to the way lines between them looked. Small distant objects might be represented under the feet of a large close figure, so that today it looks to us as if the figure is floating in air above the tiny houses, churches, and people below.

We learned to draw and to view drawings in three dimensional perspective, and later we learned to make a series of such drawings, like a cartoon, to represent the passage of a fourth dimension, time. Now when we watch movies or television, we can see...
---end quote---

You call a 4D analog of a tetrahedron a "hypertetrahedron". I think this is an OK name. The conventional name mathematicians use is "4-simplex",
and you show that you are aware of that. So that is OK too.

In conventional math language, a line segment is a 1-simplex, a triangle is a 2-simplex (because it lives in 2D space), a tetrahedron is a 3-simplex (because it lives in 3D space) and what you call a hypertetrahedron is a conventionally termed a 4-simplex. You indicate this. Good.

To my taste, "4-simplex" is (from a purely literary point of view) weak.
I have thought about other names for the thing (even before you brought it up) but I have not been able to think of an alternative term that I really like and feel safe with.

You may as well say "hypertetrahedron" because it is more visually evocative by far than the usual "4-simplex". The word "4-simplex" handicaps the lay listener's understanding by making him think of 4 points (when he should think of 5). the word "hypertetrahedron" evokes the idea of jacking up the dimension of a tetrahedron---this is the right mental image.

the reason I am not entirely comfortable with "hypertetrahedron" is that (A) it has so many syllables it takes a long time to say and (B) I always worry a little when I depart from conventional mathspeak.

What you say in the above about lacking "desirable elegance" is a good point. But you misspelled "desirable" by injecting the terminal e. this is a type of spelling error I find very difficult to avoid myself.

the discussion of DaVinci and the wire frame seems to me very good.

It has the mark of a mystic who knows how to write.

there are good science writers and mediocre science writers
and there are good mystic writers and mediocre mystic writers

the job of a mystic writer is to see the whole world of mathematics in a simplex and Pascal's triangle
as in the wellknown lines of Blake ("to see a World in a grain of Sand
and a Heaven in a Wild Flower...")
so what you tend towards is to see dimensionality and the universe in the image of a hypertetrahedron. and it helps when you infuse it with
combinatorial stuff and number stuff like N-choose-k

it seems to me that you do a good job, at least at times it gets to where
visionary writing is supposed to get (involves a form of intelligence, not mere ecstasy)

mysticism and poetry are both dangerous lines to pursue (for the person doing it). i don't have to mention this, you are well aware of the risks

it is interesting what you suggest about human perception of dimension being able to develop (15th century acquistion of perspective, possible 20th century acquisition of cinematic rather than static imagery----BTW I am not sure about there being any progress since the 15th c but you toss out the idea and mention television and it is certainly a possible/evocative thought)
 
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  • #9
Sorry Marcus, my ignorance is an abyss, and names have a storage place very near to the edge. But the DaVinci mispelling will have to stand, because the edit feature on that post has expired.

And thank you, your criticism is most welcome. I know I should write these things in a word processer and spell check them before posting. It would merely be courteous to the reader. But I find I get a kind of stimulation from the sense of immediate contact the quick reply window gives me, I don't know why. More mysticism I suppose. I once was e-chatting with a psychic healer who wanted to do a healing on me, and told me to get up close to the screen. The idea made me laugh hysterically but I did it anyway, even though I don't believe in psychic healing. Would it make any difference it it were a plasma screen, or an LCD, or would an old fashioned cathode ray tube monitor work better?

You know, Marcus, you are right about the difficulties that come with mysticism and poetry, but my worst plague is that I am an anti-elitist. I have a great fondness for ordinary people who do not have advantage of intelligence, money, breeding, privilaged birth, and so on. I resent it when those who have special abilities use them for self-aggrandizement rather than for the benefit of the culture. But I am getting over it. I hardly ever write poetry any more, my mysticism is tempered by the desire to communicate what I have seen, and even the wealthy are only guilty of taking what they have and doing the best they can with it.

Be Well,

Richard
 
  • #10
Here is a poem i wrote on 2 January of this year. It is the first one to appear in a long series of notebooks. I won't go looking for the previous sib:

Final Orders
My final orders were to die in place.
Now all the others have gone, and
no more orders come.

I have killed them all who tried to kill me,
and I was not moved. Now what?
How long shall I stay on gaurd?
Until Death comes for me, I suppose.

I should like to have a talk with Him.
He was always so fleeting.
Yet when I looked in their eyes
I often thought I saw an answer,
as if they too had asked,
is this really all so stupid
and futile as it seems?
And do we sleep at last,
or waken into dreams?



Well it isn't really much of a poem and the kick line at the end was stolen from Shakespeare. Hamlet me thinks. And I have never been a soldier who was ordered to kill anyone. My infantry training was never put to the test. Actually I was thinking of two things: a training exercise where my first sergeant had me hunker down in the snow to take on two advanceing platoons. It was just a game, played with lasers. I took out the PL, the first squad leader, and the radio man from the platoon to my right, then rolled over to get sights on the platoon on the left, when my buzzer went off. I never heard the shot that got me. Actually I think when I rolled over I probably hit my helmet sensors with the laser box, effectively shooting myself in the head. Oh well. Hazards of war.

The other source of the poem was an old Laurel and Hardy movie. It starts out with a scene from the trenches in The Great War, and Laurel is ordered by his tough sergeant to stay and gaurd the trench while the rest of the company charges off into no man's land, never to return. The war ends and years pass and Laurel, in his enviable simplicity, stands gaurd until discovered by some passing goatherders. The film jumps to a veteran's home and Hardy discovers his old boon companion and it goes on from there, surely one of the funniest things ever to come out of The War To End All Wars.

Now that I look at it, it is really hardly a poem at all, just a stolen rhyme.

Yet getting past Death is probably one of the strongest motivators in the quest to higher dimensional reality. G_d is the arborist for the tree of life, which has its roots in the preCambrian and it's latest branches hereabouts and its lofty flowering tops who knows wherewhen? All our existence, our three dimensions of space and one of time, are in those branches, and G_d, if you like, picks the lofty fruit and fertilizes the root to relief, as G-d wills, we have nothing to say about it. Yet there is some comfort to us in the idea that the tree stands forever, eternal, outside of the time we know, and G_d is free of it, can come and go through our little time as is pleasing. I don't know why that is comforting, it just is. I guess it means that we, too, have some part in eternity, however small, supporting the higher branches, perhaps, or trying to bear fruit.

If we could learn to see ourselves through G_d's eyes, death as we know it, the end of the branch, would have no meaning. Just another limit. I weigh two hundred and fifty pounds, and I have lived fifty five years. I have no wish to weigh three hundred pounds, nor should I desire to live three hundred years. Why is that?

Sorry Marcus. I'll leave this post for a day or two and then remove it. You should not have disturbed my mystic poetry. I thought I was trying to understand mathematics.

Thanks,
and be well,

nc
 
  • #11
nightcleaner said:
... I'll leave this post for a day or two and then remove it. You should not have disturbed my mystic poetry. I thought I was trying to understand mathematics.
...

If I should not have done so, then it was I who made the error. I will
watch my step in the future :smile:
Since your main aim is to try to understand mathematics, in this thread,
I will respond to hits in that department by preference.

Interesting story though, about the laser/buzzer combat in Basic.
Must have been around 1970 or 1971 and they were training you to go to Viet Nam, but then things changed and you didnt have to go.

back to mathematics?
 
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  • #12
Ok.
I have noticed a couple things in the past few days of musing. One is that a convergent series in addition must have vaules more or less evenly distributed around zero, while a convergent series in multiplication must have values more or less evenly distributed around unity. This is one of those observations that seem to me to be hanging out there in isolation, waiting for their true connectivity to be revealed. That seems to me to be a form of prescience, not to elevate my abilities, but just as an observation of how these things seem to work.

Then, I was thinking about the math formulas I have seen which start out with a summation operator followed by a product operator and then the algebraic formula which the operators act upon. How is this read? The sum of all products of (x)? And if the operators were exchanged so that the product sign came before the sum sign, would that be read: the product of all sums of (x)? And if I have intuited this correctly, would the sum of all products be different from the product of all sums? And if they are different, is this an example of an algebra that does not commute, such that (A,B) does not equal (B,A)?

The ground here is uncertain and I tremble, but perhaps if I think of these things a while it will become more clear, or else the ground will collapse under me and I will find a new, lower, more stable level to tread upon.

Btw, VN was a good guess, but it was a later war I was training for, in a decade I remember fondly for its relative peace. My unit was told to mobilize for deployment, but the war ended quickly, unexpectedly, and after that we were told to stand down, then deactivated. I was never sent overseas. My grandfather had an almost identical experience in The Great War. His unit was loaded on transport ships, ready to put to sea, but sat idle in port, all locked down, for three days. No news. Then one morning the captain came through the companionway and told them they were going home. The war was over.

My good friend's nephew and his buddy are draft age now. They never learned anything about WWII in high school. They don't even know who the main combatants were, or why they were fighting. And I read that one of Diana's sons attended a costume party in a Nazi uniform. Has England also neglected her history? Le Pen in France is allowed to say without contradiction that the Nazi occupation was not brutal. I fear for our next generation, as it has been observed that a people who forget their history are doomed to repeat it.

Be well, Marcus. Ciao.

nc
 
  • #13
nightcleaner said:
Ok.
I have noticed a couple things in the past few days of musing. One is that a convergent series in addition must have vaules more or less evenly distributed around zero, while a convergent series in multiplication must have values more or less evenly distributed around unity. This is one of those observations that seem to me to be hanging out there in isolation, waiting for their true connectivity to be revealed. That seems to me to be a form of prescience, not to elevate my abilities, but just as an observation of how these things seem to work.

This is absolutely right. The terms of a convergent infinite sum have to go to zero, and the factors of a convergent infinite product have to go to 1.

These are necessary but not sufficient condiditons. The elements going to their limit (0 or 1) does not make the operation converge by itself. They have to go to the limit fast enough. There is a famous series of reciprocals of the integers:
[tex]\sum_{i=2}^{\infty} 1/i[/tex]
where the terms do go to zero, but the series doesn't converge. How fast is fast enough? Mathematicians have proved theorems to provide convergence tests. Some of the tests for sums are discussed in second or third semester Calculus. Tests for products are mostly discussed in complex variables courses where infinite products loom very large.

Then, I was thinking about the math formulas I have seen which start out with a summation operator followed by a product operator and then the algebraic formula which the operators act upon. How is this read? The sum of all products of (x)? And if the operators were exchanged so that the product sign came before the sum sign, would that be read: the product of all sums of (x)? And if I have intuited this correctly, would the sum of all products be different from the product of all sums? And if they are different, is this an example of an algebra that does not commute, such that (A,B) does not equal (B,A)?

The best way to start to see these is to think how they work in finite cases.

(a*b) + (c*d) versus (a+b)*(c+d)

I think you can see these are not equal because the second one has cross product terms and the first one hasn't. This is not the same as noncommutativity (the differrence doesn't depend on the order of a, b, c, and d), but this reasoning has an important role in quantum entanglement math (using state vectors instead of numbers).

The ground here is uncertain and I tremble, but perhaps if I think of these things a while it will become more clear, or else the ground will collapse under me and I will find a new, lower, more stable level to tread upon.

Famous old mathematics advice: Go on and faith will come to you.
 
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  • #14
Do you do any astronomy? There is a near perfect square of stars close to and about the same apparent size as Cassiopea. Do you know it?
 
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  • #15
Could you mean the Great Square of Pegasus? The four stars form the horse's body.
 
  • #16
yes, yes, that's it. A kind of delta. Delta sigma or sigma delta, i don't know if it commutes. But it makes a pretty picture.

Code:
          *         A       *
                        *
    *           *         *
                         *
           *                  *

Andromeda is located at about A. Very clear and cold tonight, should be good viewing around midnight. Those Greeks, you know, spent a lot of time looking at the night sky.

Be well,

nc
 
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  • #17
All of these constellations, Cassiopea, Cepheus, Pegasus, Andromeda, Perseus, and even Draco, are taken from the myth of Perseus and Andromeda.
 
  • #18
Draco

selfAdjoint said:
All of these constellations, Cassiopea, Cepheus, Pegasus, Andromeda, Perseus, and even Draco, are taken from the myth of Perseus and Andromeda.

Good day!

Is anyone doing serious scientific research on Draco as it has a very important connection to a tradition I am interested in.

Suzanne Elizabeth Seitz
 
  • #19
Hi Suzanne

I don't know anything about Draco. I just mentioned Cassiopea and The Great Square because they look like Greek letters to me, and I am trying now to get the hang of Greek letters in physical formulae. Sometimes when studying a book I find it is good to look up and away at something distant. It sort of rests the eyes or something, gives the mind an assimilation break, a chance to catch up. Andromeda is about as far away and as large as anything I ever hope to be able to see with my own eyes. Gives me a nice rest from all those squiggley little letters.

Did you see Marcus in his thread on calculations with natural units gloating about the warmth of the sun on the peach trees in his garden? Hah! Tonight is the coldest nite yet this year, thirty three below zero by my thermometer. My poor truck will barely start, and the wheel bearing grease is so thick it feels like I am driving with the brakes on for the first few hundred yards. After that it loosens up ok. The shocks are so tight and the tires so hard that every little bump in the road feels like an explosion.

However, I did notice that there was still light in the sky last night at almost six PM, a real noticible difference from last month when it was dark by four thirty. So the first signs of the return of spring are not hidden from me either, even if they don't include peach blossums. Marcus! Where the heck are you, anyway, and are you taking any refugees?

I have been reading about Fermi-Dirac statistics and Bose-Einstein statistics in my book full of little Greek letters, among other squiggles. Princeton Guide to Advanced Physics. Well I can tell you it makes me feel pretty elementary. Anyway as I was saying before getting starstruck, it turns out that Fermi statistics look just like the N-choose-k math of Pascal's triangle, which Marcus and selfAdjoint showed me is related to dimensional analysis. Each line in the Pascal triangle reads out the number of simplexes in the dimension. Well this is mysterious enough, but then the Bose statistics are tantalizingly similar.

The Fermi stats ask how many ways are there to choose k objects out of a group of n objects. The Bose stats ask how many ways are there to choose g objects such that they sum up to n objects. I don't even know what that means. Been thinking on it working and sleeping for three days and still havn't a clue.


Going to go Wiki and see what I can find.

Be well,

nc
 
  • #20
Well I found the page on Bose-Einstein statistics, but could make nothing of it. It sent me on to the page on Maxwell-Boltzmann statistics, and I can't make them out either. Here is the formula as copied from Wiki (I just had to translate into Latex)

The Maxwell-Boltzmann distribution can be expressed as:

[tex]
\frac{N_i}{N} = \frac{\exp\left(-E_i/kT \right) } { \sum_{j}^{} {\exp\left(-E_j/kT\right)} }
\qquad\qquad (1([/tex]

where [tex]N_i[/tex] is the number of molecules at equilibrium temperature ''T'', in energy level ''i'' which has energy [tex]E_i[/tex], ''N'' is the total number of molecules in the system and ''k'' is the [[Boltzmann constant]]

Now I am confronted with a question which led me to the Physics Forums site in the first place, two years ago. What does that exp in the formula mean? Something about exponent, I think. But what? And, more important right now, how do I find out what it means? What branch of math is it? The Princeton guide assumes I already know, as do most of the advanced texts, but the calc texts don't mention it. What the heck is that anyway?

Still crazy after all these years...

nc
 
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  • #21
exp is just another notation for raising e to the power of the expression in the parentheses.

[tex] exp(something) \equiv e^{something} [/tex]

Powers of e are a constant feature of physical math. It used to be hard to set regular exponents in type, so they developed the exp notation to save the printers trouble. It could be printed on one line and didn't require a special font.
 
  • #22
Ok that seems reasonable. So how should I interpret when I see exp with something as a superscript after it? For example in Fermi Dirac statistics (p336 of Tribble if you have it) I see the formula

[tex]\frac{g_i-n_i}{n_i}=exp^{+\beta(\epsilon_i-\mu)}[/tex]
 
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  • #23
I haven't the slightest notion how to interpret that. Anybody know?
 
  • #24
[tex]\frac{g_i-n_i}{n_i}=exp^{+\beta(\epsilon_i-\mu)}[/tex]


I have checked the notation and what I have shown above is a fair copy, with the one exception that the [tex]n_i[/tex] is really shown in Tribble with a bar over it, a notation I have not mastered in Latex. The exp is followed with a superscript, shown here just as it is in Tribble.

Now I am looking at Pauling's general Chemistry, p. 333, which has about a page on "The Distribution Laws for Bosons and Fermions", and gives the Fermi-Dirac distribution law without derivation as:

[tex]N_i=\frac{1}{C'' exp (\frac{E_i}{kT})+N}[/tex]

where C'' is a constant with a value such that [tex]\Sigma N_i =N[/tex]and [tex]N[/tex] is the number of particles.

I don't see any obveous relationship. I could suppose the n_i could be the same as the N_i.

Pageing through Tribble, I find he uses the [tex]exp^(x)[/tex] notation frequently in the statistical physics chapter, and it is also found in his treatment of the Fourier transform, and once toward the end of his discussion of the HUP. He also uses the exp(x) notation, but not in the same formula.

I will provisionally assume that the two forms are equivalent and the superscript form is an idiosynchracy.

My other two texts on quantum, Quantum Theory by David Bohm and Mathematics of Classical and Quantum Physics by Frederick W. Byron, Jr, and Robert ZW. Fuller, do not treat the statistics of Bose and Fermi gasses.

I suppose I could push on by trying to match Tribble's forms of Fourier with that found in Bohm and in Byron, but I have exhausted my frustration with varient notations, and it will be more profitable for me to study in other areas. G_d knows there is no end to my lack of understanding.

I did find on side trips to this excursion that Bose and Fermi stats do not only apply to fundamental particles, but are also useful in describing the behavior of liquid helium gas, which make me wonder if spin state has anything to do with the stats. Helium molocules are not likely to have much spin in the super-cold liquid phase. Of course, spin at the fundamental level is not likely to be directly comparable to spin on the molecular level. Still, I had been hopeful to learn something about the nature of quantum spin from the Bose and Fermi stats, since Bosons have integral spin and Fermions have half-integral spin.

So writing

[tex]\frac{g_i-n_i}{n_i}=exp(+\beta(\epsilon_i-\mu))[/tex]

and

[tex]n_i=\frac{1}{C'' exp (\frac{E_i}{kT})+N}[/tex]

and neglect the C'' since we are only interested in units of dimensional analysis here and not calculating numbers, and subbing for n defined by the second equation into the first equation, we would have

[tex]g_i exp(\frac{E_i}{kT})+N =exp(+\beta(\epsilon_i-\mu))[/tex]

"number of ways to choose g objects such that the sum is N objects"

well let's assume the objects to be summed are whole objects so the g's are integers which add to N. N and g are counting numbers from one to less than infinity. And g is less than or equal to N. Also, the number of ways is a real positive integer less than infinity, so, the exp(x) should be a real positive integer less than infinity.


I see some errors above, will return soon to reconsider the formula



nc
 
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  • #25
The ni in Tribble and the Ni in the distribution law are not the same. There is no common meaning for n, or N. Frequently they denote some integer that comes in, but it would be a differently defined integer in each case. Read the discussion in your Chemistry book around the distribution law. Even though it doesn't derive the formula it should define the variables that come into it.
 
  • #26
ok. Pauling is pretty good with giving the meanings of variables, but Tribble doesn't seem to bother. I guess Tribble was writing a reference book for people who already know what he is talking about.

That's the bad thing about trying to learn on my own...without a proper guide, I spend a lot of time exploring in tunnels that don't go anywhere.

nc
 

Related to Attempt to visualize higher dimensions

What are higher dimensions?

Higher dimensions refer to mathematical spaces beyond the standard three dimensions of length, width, and height. These additional dimensions are not visible to the human eye, but can be conceptualized and studied through mathematical models and visualizations.

Why is it difficult to visualize higher dimensions?

It is difficult to visualize higher dimensions because our brains are only able to process information in three dimensions. We are used to living and perceiving the world in three-dimensional space, so it is challenging to imagine anything beyond that. Additionally, our visualizations are limited by the tools and techniques we use to represent higher dimensions.

How can scientists attempt to visualize higher dimensions?

Scientists use various techniques and tools to attempt to visualize higher dimensions. One common method is through analogies and projections. For example, a 4D cube (known as a tesseract) can be represented as a 3D cube within a 3D cube. Other methods include using mathematical equations, computer simulations, and virtual reality.

What are some real-world applications of visualizing higher dimensions?

Visualizing higher dimensions has many practical applications in fields such as physics, computer science, and engineering. For example, understanding higher dimensions can help in developing theories about the structure of the universe, designing complex computer algorithms, and creating more efficient and accurate models for engineering projects.

Are there any limitations to visualizing higher dimensions?

Yes, there are limitations to visualizing higher dimensions. As mentioned earlier, our brains are limited in their ability to perceive beyond three dimensions. Additionally, the mathematical models and visualizations used to represent higher dimensions may not always be accurate or complete. It is also important to consider that our understanding of higher dimensions is still evolving and may change as we continue to study and explore this concept.

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