My OperaCalabi-Yau shapes
Thursday, December 1, 2005 6:59:44 AM
The imagery of a Calabi-Yau shape of one kind is an efflorenscence of a six-dimensional energetic pattern - an hyperspatial gewgaw.
Even as we live within the 3-D spatial dimensions, the hidden dimensions escape our attention because they are nestled in the domain of hyperreality. The equations of string theory actually determine more than just the number of spatial dimensions: ergo, they determine the kinds of shapes the extra dimensions can morph into.
String matrix picks out "a significantly more complicated class of six-dimensional shapes" known as Calabi-Yau shapes, or Calabi-Yau spaces. Calabi-Yau is named after two mathematicians - Eugenio Calabi and Shin-Tung Yau - who discovered them mathematically long before their relevance to string theory was 'realized'. It a particular Calabi-Yau shape constituted the extra 6-dimensions in string theory, on ultramicroscopic scales space would have these Calabi-Yau shapes tacked on to every point in the normal three-dimensions. So that in Superstring theorist Brian Greene's words, "You and I and everyone else would right now be surrounded by and filled with these little shapes". So, we literally navigate the nine dimensions everytime we move ourselves. "If these ideas are right, the ultrmicroscopic fabric of cosmos is embroidered with the richest of textures", suggests Brian Greene.
The beauty of general relativity is that the physics of gravity is controlled by the geometry of space.
A string that's constrained to vibrate only on the 2-dimensional surface will execute a variety of vibrational patterns but only those involving motion in the left/right and back/forth directions as permissible orientations within the 2-D matrix. At the third dimension, the motion in the up/down direction is accessible with all its variety consistent within the 3-D matrix. So, with every additional dimension, the string will vibrate in newer patterns. This is an important realization, underscores Brian Greene, because there is an equation in string theory that demands that the number of independent vibrational patterns meet a very precise constraint.
If the constraint is violated, the mathematics of string theory disintegrates and its equaitons are bereft of sense. Under successive dimensions from two till eight dimensions in the universe (also within the mathematical universe), the number of vibrational patterns continue to be smaller than necessity that the constraint is met. But with nine dimensions, the constraint on the number of vibrational patterns is satisfied perfectly. Thus, the string theory determines the number of space dimensions. But what we should not forget is that all the hypermathematical equations are only approximate at best.
Now the intimate tango of geometry of space and physics come into play. What bamboozles physicists is that fact that there were far too many massless string vibrational patterns and their properties did not necessarily match those of the known matter and force particles. So, mathematical accounting for extra dimensions alone was not enough; the
shapes of these extra dimenions are capable of morphing into matter. Hence, pops up the
Calabi-Yau shapes to meet this need.
Strings are so tiny resonances that they continue to vibrate in all nine space dimenions even if 'crumpled' into a Calabi-Yau shape.
Either shape or size change of the extra dimenions affect the precise properties of each possible vibrational pattern of a string. Since a string vibrational pattern determines its mass and charge, "the precise size and shape of the extra dimenions has a profound inpact on string vibrational patterns", affirms Brian Greene, "and hence on particle properties".
Calabi-Yau is just a probable/possible shape of a particular morphic resonance of an energy pattern, or a panoply of energetic patterns. This aggravates the mathematical choice for one Calabi-Yau shape or another; though each Calabi-Yau shape is valid as any other. Yet, Calabi-Yau shape yields string vibrational patterns that closely approximate the known particles.
Calabi-yau shapes have been found that give rise to string vibrational pattern in "agreement with the three families of fundamental particles and their masses (in multiples of the proton mass)." In the mid-1980s, Philip Candelas, Gary Horovitz, ANdrew Strominger and Edward Witten (who realized the relevance of Calabi-Yau shapes for string theory) discovered that each hole -defined precisely in mathematical sense- contained within a Calabi-Yau shape gives rise to a >family< of lowest-energy string vibration patterns. A Calabi-Yau shape with three holes would therefore provide an explanation for the repetitive structure of three families of elementary particles. Indeed, a unumber of such three holed Calabi-Yau shapes have been found. Among these preferred Calabi-Yau shapes are one that also give just the right number of messenger particles as well as just the right electric charges and nuclear force properties to match the particles.
But then, Physics/Mathematics has not gained an extraordinary understanding and means to calculate infinitesimal deviations of particles with Masses that deviate from the lowest=energy string vibrations - zero times the Planck mass - by less than one part in a million billion.
Somewhere down the line, the strings and Higgs ocean may conjugally mate into a reciprocal version of complementarity; but the climax is not the Climax with a capital C.
eoi.
Even as we live within the 3-D spatial dimensions, the hidden dimensions escape our attention because they are nestled in the domain of hyperreality. The equations of string theory actually determine more than just the number of spatial dimensions: ergo, they determine the kinds of shapes the extra dimensions can morph into.
String matrix picks out "a significantly more complicated class of six-dimensional shapes" known as Calabi-Yau shapes, or Calabi-Yau spaces. Calabi-Yau is named after two mathematicians - Eugenio Calabi and Shin-Tung Yau - who discovered them mathematically long before their relevance to string theory was 'realized'. It a particular Calabi-Yau shape constituted the extra 6-dimensions in string theory, on ultramicroscopic scales space would have these Calabi-Yau shapes tacked on to every point in the normal three-dimensions. So that in Superstring theorist Brian Greene's words, "You and I and everyone else would right now be surrounded by and filled with these little shapes". So, we literally navigate the nine dimensions everytime we move ourselves. "If these ideas are right, the ultrmicroscopic fabric of cosmos is embroidered with the richest of textures", suggests Brian Greene.
The beauty of general relativity is that the physics of gravity is controlled by the geometry of space.
A string that's constrained to vibrate only on the 2-dimensional surface will execute a variety of vibrational patterns but only those involving motion in the left/right and back/forth directions as permissible orientations within the 2-D matrix. At the third dimension, the motion in the up/down direction is accessible with all its variety consistent within the 3-D matrix. So, with every additional dimension, the string will vibrate in newer patterns. This is an important realization, underscores Brian Greene, because there is an equation in string theory that demands that the number of independent vibrational patterns meet a very precise constraint.
If the constraint is violated, the mathematics of string theory disintegrates and its equaitons are bereft of sense. Under successive dimensions from two till eight dimensions in the universe (also within the mathematical universe), the number of vibrational patterns continue to be smaller than necessity that the constraint is met. But with nine dimensions, the constraint on the number of vibrational patterns is satisfied perfectly. Thus, the string theory determines the number of space dimensions. But what we should not forget is that all the hypermathematical equations are only approximate at best.
Now the intimate tango of geometry of space and physics come into play. What bamboozles physicists is that fact that there were far too many massless string vibrational patterns and their properties did not necessarily match those of the known matter and force particles. So, mathematical accounting for extra dimensions alone was not enough; the
shapes of these extra dimenions are capable of morphing into matter. Hence, pops up the
Calabi-Yau shapes to meet this need.
Strings are so tiny resonances that they continue to vibrate in all nine space dimenions even if 'crumpled' into a Calabi-Yau shape.
Either shape or size change of the extra dimenions affect the precise properties of each possible vibrational pattern of a string. Since a string vibrational pattern determines its mass and charge, "the precise size and shape of the extra dimenions has a profound inpact on string vibrational patterns", affirms Brian Greene, "and hence on particle properties".
Calabi-Yau is just a probable/possible shape of a particular morphic resonance of an energy pattern, or a panoply of energetic patterns. This aggravates the mathematical choice for one Calabi-Yau shape or another; though each Calabi-Yau shape is valid as any other. Yet, Calabi-Yau shape yields string vibrational patterns that closely approximate the known particles.
Calabi-yau shapes have been found that give rise to string vibrational pattern in "agreement with the three families of fundamental particles and their masses (in multiples of the proton mass)." In the mid-1980s, Philip Candelas, Gary Horovitz, ANdrew Strominger and Edward Witten (who realized the relevance of Calabi-Yau shapes for string theory) discovered that each hole -defined precisely in mathematical sense- contained within a Calabi-Yau shape gives rise to a >family< of lowest-energy string vibration patterns. A Calabi-Yau shape with three holes would therefore provide an explanation for the repetitive structure of three families of elementary particles. Indeed, a unumber of such three holed Calabi-Yau shapes have been found. Among these preferred Calabi-Yau shapes are one that also give just the right number of messenger particles as well as just the right electric charges and nuclear force properties to match the particles.
But then, Physics/Mathematics has not gained an extraordinary understanding and means to calculate infinitesimal deviations of particles with Masses that deviate from the lowest=energy string vibrations - zero times the Planck mass - by less than one part in a million billion.
Somewhere down the line, the strings and Higgs ocean may conjugally mate into a reciprocal version of complementarity; but the climax is not the Climax with a capital C.
eoi.
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