Image: Photograph by Zachary Zavislak
In Brief
- Most everyone is familiar with the “real” numbers, but far more types of numbers exist. Among them, the best known are the complex numbers, which include a square root of –1.
- We can build higher-dimensional number systems as well. But we can define all the four basic operations—addition, subtraction, multiplication and division—in only a few special cases.
- One such case is the octonions, an eight-dimensional number system. Mathematicians invented it in the 1840s but, finding few applications, paid little attention for the next 150-plus years.
- Mathematicians now suspect that the octonions may help us understand advanced research in particle physics in fields such as supersymmetry and string theory.
As children, we all learn about numbers. We start with counting, followed by addition, subtraction, multiplication and division. But mathematicians know that the number system we study in school is but one of many possibilities. Other kinds of numbers are important for understanding geometry and physics. Among the strangest alternatives is the octonions. Largely neglected since their discovery in 1843, in the past few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may explain why the universe has the number of dimensions it does.
The Imaginary Made Real
This article was originally published with the title The Strangest Numbers in String Theory.
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19 Comments
Add CommentString Theory is just another way to try to explain our reality, if you need so many tricks to explain it (when you could understand it at all) I don't think it ha a great value of reality.
Reply | Report Abuse | Link to thisI think we are pretty far away from understanding it all.
Reply | Report Abuse | Link to thissimply amazing!
Reply | Report Abuse | Link to thisInterestingly, the symmetries at the heart of the Croft Spiral Sieve, an 8-radii geometry shown to structure the distribution of prime numbers, is completely congruent with the Lie Group E8, implicated in the "theory of everything," as well as other aspects of string theory described in this article.
Reply | Report Abuse | Link to thisRemarkably, the sequence of Fibonacci terminating digits indexed to the first 8 integers of the set of all natural numbers not divisible by 2, 3 and 5 (which populate the Croft Spiral) = 13,937,179 which is a prime number and a member of a prime pair (with 13,937,177). More profoundly, however, 13,937,179 when added to its reversal 97,173,931 = 111,111,110 and, amazingly, the entire repeating (and palindromic) Fibo sequence end-to-end (equivalent to two rotations around the Croft Spiral Sieve) gives this equivalency: 1,393,717,997,173,931 ≡ 11,111,111 (mod 111,111,110)... [And, of course, 11,111,111 * 111,111,110 = 1234567876543210 and 111,111,110/11,111,111 = 10.]. Also 1,393,717,997,173,931 is divisible by the repunits 11 and 1,111 and 11,111,111. Capping these mysterious relationships, echoing the Fibonacci patterns just described, the terminating digits of the 1st 8 numbers in this set (1, 7, 11, 13, 17, 19, 23 & 29) = 17,137,939, which, when added to their reversal (93,973,171) = 111,111,110. And, when you connect 17,137,939 to its reversal, the entire palindromic sequence end-to-end produces this: 1,713,793,993,973,171 ≡ 111,111,111 (mod 111,111,110) [And in this case, 111,111,111 * 111,111,110 = 12345678876543210.]. And if that isn't enough, 1,713,793,993,973,171 is also divisible by the repunits 11 and 1,111 and 11,111,111. Well, not quite enough, because there's yet another related dimension of symmetry: The terminating digits of the prime root angles for the 8 radii of the spiral (24,264,868), when added to their reversal (86,846,242) = 111,111,110, not to mention this sequence possesses symmetries that dovetail perfectly with the sequences described, above, including the fact that when it is connected to its reversal (giving us 2,426,486,886,846,242), it's divisible by the repunits 11 and 1,111 and 11,111,111. I have to believe there is congruence with string theory implicit in this geometry ...
Now it is nice and clear.
Reply | Report Abuse | Link to thisThe Holy City in the book of Revilation has 12 dementions I talked to an engineer And yahweh sees in all directions at the same time
Reply | Report Abuse | Link to thisHemiboso.... Thankyou for the beautiful mind ride... So Peaceful...
Reply | Report Abuse | Link to thisYou're most welcome, V.E. For the full story, go to
Reply | Report Abuse | Link to thishttp://www.primesdemystified.com ...
I have always strongly suspected there are 11 dimensions, not 10.
Reply | Report Abuse | Link to thisThe notion of spacetime dimension is not fixed in string theory: it is best thought of as different in different circumstances:
"One such theory is the 11-dimensional M-theory, which requires spacetime to have eleven dimensions, as opposed to the usual three spatial dimensions and the fourth dimension of time. The original string theories from the 1980s describe special cases of M-theory where the eleventh dimension is a very small circle or a line, and if these formulations are considered as fundamental, then string theory requires ten dimensions. But the theory also describes universes like ours, with four observable spacetime dimensions, as well as universes with up to 10 flat space dimensions, and also cases where the position in some of the dimensions is not described by a real number, but by a completely different type of mathematical quantity. So the notion of spacetime dimension is not fixed in string theory: it is best thought of as different in different circumstances." (http://en.wikipedia.org/wiki/String_theory)
Eleventh Frequency VE
Reply | Report Abuse | Link to thisSynergetic association with the vector equilibrium (VE) closest packing of spheres, resulting in a 10F^2 + 2 shell growth. The first shell (frequency) totals thirteen, the second layer consists of 42 spheres, for a volume of 55 spheres, third layer of 92 totaling 147, etc. At frequency (layer) eleven, the total is 5083. Coincidental, if the vector equilibrium models the neutron, then the mass ratio between the neutron (1.6749286(10)E-27 kg) and the proton (1.672623(10)E-27 kg) can be expressed as the vertex volume of the eleventh frequency vector equilibrium minus seven of its spheres to seven significant digits; ERGO, 1.6749286(10)/1.672623(10) = 5083/(5083-7.000).
Wow, I wish that I could understand morbas, but for me it is abracadabra...
Reply | Report Abuse | Link to thisFascinating! I wonder whether John Baez is aware of the important work of Prof. Mohamed El Naschie, the Nobel-nominated Egyptian physicist who has done more than anyone to promote these ideas. The number 5 is of course closely connected to the Golden Mean which has been experimentally confirmed in quantum mechanics, verifying El Naschie's theory.
Reply | Report Abuse | Link to thisThe Buckminster Fuller vector equilibrium (VE) is the mathematics of closests sphere packing. The octet numbering mod (hemiboso) expresses association with dimension (IMHO). The VE expresses a numbering system of shape of higher dimension shadows on our three dimension visualization. A vector equilibrium packing pattern shell growth is anotated as frequency number 1,2,3,4...integers.
Reply | Report Abuse | Link to thisThe pattern shown above (morbas) derives the proton neutron mass to a higher accuracy than present measurement technology. So while we were on the strange number thing, thought I would pass this aid on to you.
-morbas-
Unfortunately we will probably never have the technology to see elementary particles. Since we can visually see anything below 1.0 X 10-8 and most of these particles live at 1.0 x 10-33 to -35. To view something you have to have an order of magnitude higher in resolution. To do this we'd need to be able to read the serial number off the lunar lander from earth. In the same fashion we can guess about the existence of higher dimensions, but can never see them anymore than a flatlander can see the 3rd dimension.
Reply | Report Abuse | Link to thisThe term dimension arises out of the difference in perception between positions that are described as external or internal to a body, where the relative positions between different points on the body and their relationship to the space external to it, remain invariant. Thus, it can only be applied to solids that have fixed spread in a given direction based on their internal arrangement independent of external factors. This way we can describe the spread of curved surfaces also in relations to points exterior or interior to them without resorting to exotic “mathematics” that does not conform to the principle of logical consistency.
Reply | Report Abuse | Link to thisFor perception of the spread of the object, the electromagnetic radiation emitted by the object must interact with that of our eyes. Since electric and magnetic fields move perpendicular to each other and both are perpendicular to the direction of motion, we can perceive the spread of any object only in these three directions. Measuring the spread is essentially measuring the space occupied by any two points on it. This measurement can be done only with reference to some external frame of reference. For the above reason, we arbitrarily choose a point that we call origin and use axes that are perpendicular to each other and term these as x-y-z coordinates (length-breadth-height making it 3 dimensions or right-left, forward-backward and up-down making it 6 dimensions).
These are not absolute terms, but are related to the order of placement of the object in the coordinate system of the field in which the object is placed. Thus, they remain invariant under mutual transformation. If we rotate the object so that x-axis changes to y-axis or z-axis, there is no effect on the structure (spread) of the object, i.e. the relative positions between different points on the body and their relationship to the space external to it. Based on the positive and negative directions (spreading out from or contracting towards) the origin, these describe six unique positions (x,0,0), (-x,0,0), (0,y,0), (0,-y,0), (0,0,z), (0,0,-z), that remain invariant under mutual transformation. Besides these, there are four more unique positions, namely (x, y), (-x, y), (-x, -y) and (x, -y) where x = y for any value of x and y, which also remain invariant under mutual transformation. These are the ten dimensions and not the so-called mathematical structures. These are described in detail in our book. Since time does not fit in this description, it is not a dimension.
basudeba
The term dimension arises out of the difference in perception between positions that are described as external or internal to a body, where the relative positions between different points on the body and their relationship to the space external to it, remain invariant. Thus, it can only be applied to solids that have fixed spread in a given direction based on their internal arrangement independent of external factors. This way we can describe the spread of curved surfaces also in relations to points exterior or interior to them without resorting to exotic “mathematics” that does not conform to the principle of logical consistency.
Reply | Report Abuse | Link to thisFor perception of the spread of the object, the electromagnetic radiation emitted by the object must interact with that of our eyes. Since electric and magnetic fields move perpendicular to each other and both are perpendicular to the direction of motion, we can perceive the spread of any object only in these three directions. Measuring the spread is essentially measuring the space occupied by any two points on it. This measurement can be done only with reference to some external frame of reference. For the above reason, we arbitrarily choose a point that we call origin and use axes that are perpendicular to each other and term these as x-y-z coordinates (length-breadth-height making it 3 dimensions or right-left, forward-backward and up-down making it 6 dimensions).
These are not absolute terms, but are related to the order of placement of the object in the coordinate system of the field in which the object is placed. Thus, they remain invariant under mutual transformation. If we rotate the object so that x-axis changes to y-axis or z-axis, there is no effect on the structure (spread) of the object, i.e. the relative positions between different points on the body and their relationship to the space external to it. Based on the positive and negative directions (spreading out from or contracting towards) the origin, these describe six unique positions (x,0,0), (-x,0,0), (0,y,0), (0,-y,0), (0,0,z), (0,0,-z), that remain invariant under mutual transformation. Besides these, there are four more unique positions, namely (x, y), (-x, y), (-x, -y) and (x, -y) where x = y for any value of x and y, which also remain invariant under mutual transformation. These are the ten dimensions and not the so-called mathematical structures. These are described in detail in our book. Since time does not fit in this description, it is not a dimension.
basudeba
M-THEORY
Reply | Report Abuse | Link to this-- James Ph. Kotsybar
Physicists are theorists so extreme
that they can get lost in speculation
and even lose track, or so it would seem,
of designated abbreviation.
They’ll say string theory has many parts
that are likely aspects of just one thing
they like to call “M,” but, with all their smarts,
when asked what it means, they start stammering.
“What does M stand for?” -- a simple query
that receives an obfuscating reply:
“Membrane, Mother or Master Theory.”
They really don’t know, and they don’ know why
nor grasp that they’ve lost their unified voice,
thus need to respond with multiple-choice.
I think string theory makes simple sense. Points don't exist in reality, so why "point particles". Our physical universe requires a minimum of physical extension. How about a one-dimensional entity of shortest possible distance ---i.e. Planck length? And that's a string. In effect, nothing physically conceivable could be more basic, so it's extremely likely that a superstring is the basic entity of the physical universe. As to how the rest of the theory plays out, that's anyone's guess.
Reply | Report Abuse | Link to thisHay guyz i thinks its all poopy. thats right the universe is made of poopy. i'm right and your wrong. now ttap into its powa!
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