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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeAug 18th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs there a name for a kind of "geometry" that lies in between conformal and metric structure, consisting of a metric that is defined up to a *global* scaling factor (rather than a local one as in conformal geometry)? This seems to me to align roughly with our intuitive experience of the geometry of space: we can measure angles, and we can also measure distances and (crucially) compare a distance in one direction with a distance in another direction (which goes beyond conformal structure), but there is no canonical "unit length" (which a metric structure would supply) so we have to arbitrarily choose a system of units.

   Is there a name for a kind of “geometry” that lies in between conformal and metric structure, consisting of a metric that is defined up to a global scaling factor (rather than a local one as in conformal geometry)? This seems to me to align roughly with our intuitive experience of the geometry of space: we can measure angles, and we can also measure distances and (crucially) compare a distance in one direction with a distance in another direction (which goes beyond conformal structure), but there is no canonical “unit length” (which a metric structure would supply) so we have to arbitrarily choose a system of units.

2. • CommentRowNumber2.
   • CommentAuthorfastlane69
   • CommentTimeAug 18th 2016
   • (edited Aug 18th 2016)
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   Author: fastlane69
   Format: MarkdownItexA. S. Eddington's "Planoids" and "Uranoids" may be of use. He took a ensemble of particles and created a dimensionless measure of uncertainty relative to space-time, $\sigma$. He then uses $\sigma$ as the "unit length" which is now scale-free, non-local, and doesn't rely on the arbitrariness of the units used to define space-time with, as he believed that uncertainty was more fundamental than units in a sense.

   A. S. Eddington’s “Planoids” and “Uranoids” may be of use.

   He took a ensemble of particles and created a dimensionless measure of uncertainty relative to space-time, $\sigma$.

   He then uses $\sigma$ as the “unit length” which is now scale-free, non-local, and doesn’t rely on the arbitrariness of the units used to define space-time with, as he believed that uncertainty was more fundamental than units in a sense.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 19th 2016
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   Author: Urs
   Format: MarkdownItexAt least the two invariance properties have fairly standard names, one speaks of _[[scale invariance and conformal invariance]]_. A decent account of this is in * Yu Nakayama, _Scale invariance vs conformal invariance_ ([arXiv:1302.0884](http://arxiv.org/abs/1302.0884)) This might make one want to speak of "scale-invariant geometry". Google confirms that the term is in use, but not very widely so.

   At least the two invariance properties have fairly standard names, one speaks of scale invariance and conformal invariance.

   A decent account of this is in

   • Yu Nakayama, Scale invariance vs conformal invariance (arXiv:1302.0884)

   This might make one want to speak of “scale-invariant geometry”. Google confirms that the term is in use, but not very widely so.