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This story is a supplement to the feature "Using Causality to Solve the Puzzle of Quantum Spacetime" which was printed in the July 2008 issue of Scientific American.
A Whole New Dimension to Space
In everyday life the number of dimensions refers to the minimum number of measurements required to specify the position of an object, such as latitude, longitude and altitude. Implicit in this definition is that space is smooth and obeys the laws of classical physics.
But what if space is not so well behaved? What if its shape is determined by quantum processes in which everyday notions cannot be taken for granted? For these cases, physicists and mathematicians must develop more sophisticated notions of dimensionality. The number of dimensions need not even be an integer, as in the case of fractals—patterns that look the same on all scales.
Cantor Set : Take a line, chop out the middle third and repeat ad infinitum. The resulting fractal is larger than a solitary point but smaller than a continuous line. Its Hausdorff dimension [see next page] is 0.6309.
Sierpinski Gasket: A triangle from which ever smaller subtriangles have been cut, this figure is intermediate between a one-dimensional line and a 2-D surface. Its Hausdorff dimension is 1.5850.
Menger Sponge: A cube from which subcubes have been cut, this fractal is a surface that partially spans a volume. Its Hausdorff dimension is 2.7268, similar to that of the human brain.
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18 Comments
Add CommentHorrible. Instead of clearing the water, you've only muddied it. Thanks a lot.
Reply | Report Abuse | Link to thisHI;
Reply | Report Abuse | Link to thisAs far as I can tell, a dimension is just a variable (a basic variable but a variable nonetheless). The human mind cannot cope with drawings that try to show more than three spatial dimensions. But it can cope with a multitude of variables -- ordinary people do that everyday. Trying to deal with and live through human relationships is a good example of a multi-variable problem. Figuring out how my computer works is a multi-variable problem.
So why mathematicians, physicists and artists have struggled for 90 to 100 years to show extra dimensions graphically is beyond me. There are already available better multi-variable models than a Cartesian graph.
Regards Bill
Sir:
Reply | Report Abuse | Link to thisMathmaticians are attempting to show graphically what the mathmatically impaired of us can't grasp. I love reading pop science and even some hard science, yet, in college I literally hit the wall just past calculus, which, I might add, I barely passed and did not understand anyway.
My father was a gifted mathmatician who, despite his best efforts, could never get down to a level to explain math that to him was patently obvious. And, I had a great and gifted tutor, but to coin a phrase (here it comes), it was all greek to me. (Sorry)
Try as I might, I just did't get it then, and don't get it now, so, without graphic models to aid my understanding, I'm at a loss. Please be kind to us non-mathmatical types. We mean well.
Graphic models really do help. And, yes, I understand and can clearly picture that a black hole is a perfect sphere in space and not the whirlpool graphic used to explain one. And I know that gravity exists on all planes, not just the "rubber sheet" graphic of one plane with a heavy weight on it used to explain that concept.
To those of you trying to create graphics to explain more than four dimensions, keep at it. I'm positive it will confuse me no end, but your efforts are greatly appreciated.
I remain,
Jacomus d'Paganus-Fatuus
P.S. My IQ has repeatedly tested at 160 plus. However, words are my passion.
Haussdorff should not have conveyed his idea using the term dimension. Dimensions should be the number of variables required to span the space. This is an integer- only definition. The mathematicians are producing useful models but the term dimension is a misnomer.
Reply | Report Abuse | Link to thisexactly.. what is a dimension by the way????
Reply | Report Abuse | Link to thisThat answered nothing.
Reply | Report Abuse | Link to thisThe answer should have been authoritative. The article was rather hollow.
Reply | Report Abuse | Link to thisHello, Jacomus -
Reply | Report Abuse | Link to thisI would like to recommend this book to you:
[i]Where Mathematics Comes From -- the Theory of Embodied Mathematics[/i]
by Lakoff & Nunez
The text is about the cognitive science of mathematics, i.e., cognitive science applied to how we think about math, or more importantly, why we think mathematically at all.
It has been a tremendous aid to me, especially as I took on the task of teaching my child math from point one.
The first part of the book can be understood by anyone. The second part is an analysis of a particular idea meant to impress the reader that the theory also applies to higher mathematics.
Keep in mind that after 5000 years of math, no one person understands or practices more than a small fraction of all the mathematics that have been invented, although to those who understand less it appears that the mathematician must know it all.
If you ask, you will find that mathematicians specialize in one or two particular areas.
There have been many people who having failed arithmetic, algebra or basic calculus then concluded that they were just not mathematically gifted, or were just plain stupid. But it is usually not the student who is at fault. Some folks cannot teach, and some texts are more difficult than others even when the texts cover exactly the same topics. Further, time and energy are finite resources for all students, and if one or the other is lacking, little learning will take place.
I once had a calculus teacher inform me that it has been shown that most trigonometry students forget all the trig the learned a month after their class ends. That certainly is not because most trig students are stupid.
Anther interesting book you might consider, just for your own personal understanding of the subject, is [i]Calculus for Dummies[/i], by Mark Ryan. This may seem a bit simple for you but I found it especially useful as a source of ideas on how to introduce calculus to my child. There are no problems to work in this text; it simply explains what calculus is and how it works.
--
Edited by Bradley at 06/28/2008 2:15 AM
Perhaps what the author tries to convey is the complexity of the concept of Dimension, there isnt any straight forward answer.
Reply | Report Abuse | Link to thisTo a layperson, a dimension is often understood as a magnitude that serves to define the location of an element within a given set, as of a point on a line, an object in a space, or an event in space-time. These are physically tangible.
A person is able to visualize and comprehend a 4-D space-time continuum, but unlikely (if not impossible) to do so for any dimension beyond that.
Multidimensional notion (5-D and above) is mathematically sound, unfortunately, it can be quite meaningless to creatures long confined in space and time. Mathematics deals with axioms and algorithms, it is self contained. Thus it can work on any number of dimensions, integer or fractal. This is not the case for physics. Maybe multidimensional mathematical physics could only be considered as a branch of math, not physics.
What actually is a Dimension anyway, physically speaking? mathematically speaking?
(Tan Boon Tee)
we have to beyond emperical understanding to define a dimention.yet we can only comprehend it but not understand it in detail.
Reply | Report Abuse | Link to thisI would like to add to Bradley's (excellent) comments about maths- it is an underrated subject because many people are put off by school experiences,to this end I have added many webpages about maths onto the net (http://members.fortunecity.com/templarseries/index0.html) I have many sites - but too many to sort into any kind of order.
Reply | Report Abuse | Link to thisI have found maths to be an excellently entertaning subject from fractals to the Riemann hypothesis and I too am an advocate that maths is not for the gifted but part of everyone's cultural heritage and to be left out is probably the biggest mortal sin that could happen to anyone before they die.
If there IS a God (which I do not believe),even he would probably thank believers for taking the time out to understand how he constructed the whole shebang. The fact that Carl Sagan used the rouse of the PRIME numbers in contact as a communications medium indicates the basic level that one assumes all creatures to be versed,in order to be assumed intelligent. One notes with respect to the article on MIND and consciousness that even snails can avail themselves of a spiral even if they do not count it consciously.
http://members.fortunecity.com/templarseries/padovan.html
One wonders whether Bradley is a the David Bradley mentioned in that article?? What are the chances - only maths can tell you!
Please see
Reply | Report Abuse | Link to thishttp://members.fortunecity.com/templarseries/dimens.html
and
http://members.fortunecity.com/templarseries/tourist4a.html
I note Jacomus that you require numbers in order to measure your IQ - it might be quite reasonable to say "One hundred and sixty" but numbers give you the brevity - for the sake of Obscure Hinges - the tourist4a link explains DIMENSIONS and the dimens link explains where modern physics links with dimensions.
Reply | Report Abuse | Link to thisIf one can grasp the idea of a shadow being a 2 dimensional projection of a 3 dimensional object then one has to keep extending the analogy upwards - with respect to Hausdorff etc - the dimensions become fractional or "fractional dimensional" - thus FRACTAL - that is there actually measured dimension lives between the normal integer ones - tricky to get your head round - as are those above 3 - as it is outside of experience - I am sure there is plenty of web material to aid the avid mind these days and a lot of it does not need a hi IQ - I don't know what mine is - but having defeated a couple of mensa tests (http://members.fortunecity.com/templarseries/mensa2.html) I would imagine it is capable of handling Hausdorrf.
I with Bradley - there is plenty of exciting and interesting things to be had for those who think that they have been put off maths for some reason. The fact that the authors have attempted to convey fractional and multi dimensional objects graphically aids the mind.
http://members.fortunecity.com/templarseries/mathex8.html
Will the information that was in all of the discussions forum before the recent changes to the Scientific American web site be made available in any way for those who would like to access that data? There was much useful information in those discussions. It would be a shame to have it all lost. If it won't be made available I would like to hear from anyone that made a copy of the discussions before they were erased. I have copies of my postings, but would like to get the whole forum or whatever parts that anyone has available to have it all in perspective for future reference.
Reply | Report Abuse | Link to thisDr. F. Tengelin was quicker than me or more brave than me. I was just thinking of saying the same thing. Let me make his statement more precise. Prof. Steve Weinberg who developed the electro weak theory and shared the Nobel prize with two others is the author of the most authoritative book on quantum field theory. In volume 3 of his book The Quantum Theory of Fields published by Cambridge University Press in 2000 he states on page 192 that the inverse super symmetric unification coupling of all fundamental gauge forces is 17.5. This value is given by his equation 28.2.19. Finding this result scared me quite a bit because I used Prof. El Naschies result which comes to 24.28. This is a large discrepancy. I repeated the calculation again and again but I always found 24.28 and never 17.5. To make things worse Prof. El Naschie noticed immediately that 17.5 must be a miscalculation and said that the exact integer value must be 26. That means 17.5 must be wrong and 24.28 is only an approximation to the exact value which is 26. He said it is obvious that 26 must be correct. He directed me to his paper in Chaos, Solitons & Fractals 35, p. 862 (2008) entitled Non-perturbative super symmetric quantum gravity coupling. I am desperate to know who is right and who is wrong? This result will not affect either the career of a Nobel laureate or the career of a well established professor but it could be devastating for me. I would be extremely grateful to anyone who could help me decide who is right, Prof. Weinberg or Prof. El Naschie. Please send me your answers as quick as possible to the address below.
Reply | Report Abuse | Link to thisAyman Elokaby
Dept. of Physics
University of Alexandria
Egypt
Dr. F. Tengelin was quicker than me or more brave than me. I was just thinking of saying the same thing. Let me make his statement more precise. Prof. Steve Weinberg who developed the electro weak theory and shared the Nobel prize with two others is the author of the most authoritative book on quantum field theory. In volume 3 of his book The Quantum Theory of Fields published by Cambridge University Press in 2000 he states on page 192 that the inverse super symmetric unification coupling of all fundamental gauge forces is 17.5. This value is given by his equation 28.2.19. Finding this result scared me quite a bit because I used Prof. El Naschies result which comes to 24.28. This is a large discrepancy. I repeated the calculation again and again but I always found 24.28 and never 17.5. To make things worse Prof. El Naschie noticed immediately that 17.5 must be a miscalculation and said that the exact integer value must be 26. That means 17.5 must be wrong and 24.28 is only an approximation to the exact value which is 26. He said it is obvious that 26 must be correct. He directed me to his paper in Chaos, Solitons & Fractals 35, p. 862 (2008) entitled Non-perturbative super symmetric quantum gravity coupling. I am desperate to know who is right and who is wrong? This result will not affect either the career of a Nobel laureate or the career of a well established professor but it could be devastating for me. I would be extremely grateful to anyone who could help me decide who is right, Prof. Weinberg or Prof. El Naschie. Please send me your answers as quick as possible to the address below.
Reply | Report Abuse | Link to thisAyman Elokaby
Dept. of Physics
University of Alexandria
Egypt
i don't under stand the image from the 4th dimetion it looks to me just like a 3 dimentional figure exept it changes shapes.... i gess we cant piture the 4 dimention with our eyes but don't publish stuff that doesn't makes sense.
Reply | Report Abuse | Link to thisMy thoughts are that the search for multiple dimensions is a way to attempt to account for the phenomena within QED which is far better explained by assuming that this physical universe is the only component of the 'whole' which runs to linear time. Non-linear potential and probability acting on each-other brings about quite a satisfactory counter explanation as an alternative model.
Reply | Report Abuse | Link to thisCheers,.. Craig
http://www.quantumtemporaldynamics.com