What's the difference between Euclidean & Cartesian space?

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SUMMARY

The discussion clarifies that Euclidean space and Cartesian space are often conflated terms, with Euclidean space adhering to Euclid's axioms and allowing for higher dimensions, while Cartesian space typically refers to two or three dimensions with mutually perpendicular axes. The term "Cartesian coordinates" is correctly associated with Euclidean spaces, which are flat and devoid of curvature, unlike curved surfaces such as the moon. The distinction lies in the dimensionality and the context in which these terms are used.

PREREQUISITES
  • Understanding of Euclidean geometry
  • Familiarity with Cartesian coordinates
  • Knowledge of dimensionality in mathematical spaces
  • Basic concepts of geometric axioms
NEXT STEPS
  • Research the properties of Euclidean spaces in higher dimensions
  • Explore the implications of curvature in non-Euclidean geometries
  • Study the application of Cartesian coordinates in various mathematical contexts
  • Learn about the historical development of Euclidean geometry and its axioms
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Mathematicians, physics students, educators, and anyone interested in the foundational concepts of geometry and spatial analysis.

swampwiz
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What's the difference between Euclidean & Cartesian space?
 
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swampwiz said:
What's the difference between Euclidean & Cartesian space?
One exists, the other doesn't.
One doesn't speak of Cartesian spaces. What you mean is probably Euclidean spaces and thus there is no difference, but only because you invented a term. One speaks of Cartesian coordinates in Euclidean spaces, which means the coordinate directions are pairwise perpendicular. Euclidean space mean, there is no curvature. E.g. the surface of the moon is curved and so no Euclidean space. The screen on which I read this now is flat, and thus Euclidean.
 
swampwiz said:
What's the difference between Euclidean & Cartesian space?
I've never heard the term "Cartesian space," but if I search for it on the web, I find some hits. More often I see "Cartesian coordinates."

From one of the definitions I saw, a Cartesian space is one of either two or three dimensions, in which the axes are mutually perpendicular.

A Euclidean space also has mutually perpendicular axes, but can represent spaces of higher than three dimensions.
 
Most likely authors are conflating the terms of Cartesian space to mean Cartesian coordinates in a Euclidean space.
 
jedishrfu said:
Most likely authors are conflating the terms of Cartesian space to mean Cartesian coordinates in a Euclidean space.
Better than what had happened to me here on PF. I innocently abbreviated orthonormal system ...
 
A Euclidean space is geometric space satisfying Euclid's axioms. A Cartesian space is the set of all ordered pairs of real numbers e.g. a Euclidean space with rectangular coordinates.
 

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