Allocated Dimensionality and Permutations

[vogon13]

<p>&nbsp;</p><p>Imagine a continuum of dimensions.</p><p>&nbsp;</p><p>Our world of 3 (not counting time) and our familiarity with a forth dimension, as embodied in the infamous tesseract.</p><p>&nbsp;</p><p>Now imagine, perhaps, a reality with 5 dimensions,</p><p>&nbsp;</p><p>-but-</p><p>&nbsp;</p><p>In the case of the this 5 dimensional reality, the dimensions correspond to our first and third, and the rest correspond to the ninth dimensions' forth, sixth, and seventh. </p><p>&nbsp;</p><p>From a 'many worlds' viewpoint, this would imply myriads more realities than anticipated.</p><p>&nbsp;</p><p>And I haven't even looked at fractional dimensionality yet . . .&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <div class="Discussion_UserSignature"> <p><font color="#ff0000"><strong>TPTB went to Dallas and all I got was Plucked !!</strong></font></p><p><font color="#339966"><strong>So many people, so few recipes !!</strong></font></p><p><font color="#0000ff"><strong>Let's clean up this stinkhole !!</strong></font> </p> </div>

 

[DrRocket]

<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>&nbsp;Imagine a continuum of dimensions.&nbsp;Our world of 3 (not counting time) and our familiarity with a forth dimension, as embodied in the infamous tesseract.&nbsp;Now imagine, perhaps, a reality with 5 dimensions,&nbsp;-but-&nbsp;In the case of the this 5 dimensional reality, the dimensions correspond to our first and third, and the rest correspond to the ninth dimensions' forth, sixth, and seventh. &nbsp;From a 'many worlds' viewpoint, this would imply myriads more realities than anticipated.&nbsp;And I haven't even looked at fractional dimensionality yet . . .&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br />Posted by vogon13</DIV></p><p>Forget abut fractals and fractional dimensions.&nbsp; They are basically the result of pathalogical topological spaces.</p><p>But there are a lot of applications of infinite-dimensional spaces.&nbsp; In fact infinite-dimensional spaces are quite common in mathematics and physics.&nbsp; Common enough that you need to be a bit careful about what you mean by dimension in those cases.</p><p>Finite-dimensional spaces originate with ordinary Euclidean vector spaces, and in that case the notion of dimension is fairly unambiguous.</p><p>In the infinite-dimensional case one needs to be a bit more careful.&nbsp; One of the most useful infinite-dimensional spaces is the separable Hilbert space that one encounters in quantum mechanics or in the theory Fourier series.&nbsp; The most natural notion of dimension is the cardinality of a complete orthonormal basis, which is the cardinality of the integers, i.e. such a basis is countable.&nbsp; That is what permits the representation of a function as an infinite series of sines and cosines. But if you demand a basis in which everything can be represented as a finite linear combination, then the necessary basis, called a Hamel basis is not countable.&nbsp; If yoo look at these spaces from a purely topological perspective then they are homeomorphic to a countable product of lines (this is not obvious) so you again are looking at what would be called a countably infinite dimension.</p><p>In any case, the basic message is that infinite-dimensional spaces are very commonly used and there is a rich theory already developed to describe them and their applications. You don't need to invent anything new. </p> <div class="Discussion_UserSignature"> </div>

 

[SHU]

Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>&nbsp;Imagine a continuum of dimensions.&nbsp;Our world of 3 (not counting time) and our familiarity with a forth dimension, as embodied in the infamous tesseract.&nbsp;Now imagine, perhaps, a reality with 5 dimensions,&nbsp;-but-&nbsp;In the case of the this 5 dimensional reality, the dimensions correspond to our first and third, and the rest correspond to the ninth dimensions' forth, sixth, and seventh. &nbsp;From a 'many worlds' viewpoint, this would imply myriads more realities than anticipated.&nbsp;And I haven't even looked at fractional dimensionality yet . . .&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br />Posted by vogon13</DIV><br /><br /><font size="2">Wow, I thought I was intentionally obtuse.</font>