@JonnElledge

A ball is 3D. A sphere is the surface of a ball and is only 2D. It has just 2 coördinates for every point, latitude and longitude (i.e. θ and φ without r).

I know that @standupmaths has definitely covered this particular one. I've seen Matt do it. This is Key Stage 3 National Curriculum #maths as taught in secondary school.

As I said, you've got a loudness of teeth grinding metric for how many mathematicians are in your readership who can further take that beyond KS3 to the 2 centuries of maths built from Karl Freidrich #Gauss and others that are some of the underpinnings of 20th century relativistic #physics.

I don't know whether you have a metric for the number of physicists in your readership. Physicists's teeth tend to start out frictionless, perfectly cuboid, inertial, and in vacuo. (-:

@oantolin @mapasmilhaud
#DifferentialGeometry #topology #curvature #cartography

@JonnElledge

[…Continued]

There are loads of resources on the WWW that do this far better justice, with more detail and less handwaving that I'm using here. There are also scads of pop science explanations by actual maths teachers. #Numberphile has #HannahFry discussing this with orange peel, for example. @standupmaths has probably covered it somewhere.

You'll know that you've reached the good stuff, the stuff that mathematicians and theoretical physicists deal in (knowing about intrinsic measure of curvature being necessary for a proper understanding of curved spacetime in General Relativity), when you come across the explanations that blithely throw in Christoffel symbols and Ricci tensors as if you know what the Hell they are on about. (-:

#maths #physics #Gauss #curvature

@JonnElledge

Bear in mind that I am not the mathematics teacher in my family. But I do know what makes mathematicians's teeth grind. (-:

The #maths involves Karl Freidrich #Gauss's Theorem Egregium that was published in 1827. The 3D-ness, the embedding (of what is actually still only a 2D manifold) within a higher dimensional space, does not matter, and isn't the reason that flat maps of the surface of Terra don't work.

The reason is that a plane has (constant) Gaussian #curvature zero and a sphere has positive Gaussian curvature. It's impossible to have an isometry between two spaces with different Gaussian curvatures. Something always has to give.

A torus is a 2D surface embedded in 3D space, like a sphere. But there are parts of a torus with zero Gaussian curvature, and those parts *can* be projected onto a plane without distortion. Likewise, a cone and a cylinder have constant zero Gaussian curvature and can also be projected onto a plane.

#physics
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@JonnElledge

You can measure how many mathematicians are in your readership by how loud the teeth-grinding noise is at that 3D-vs-2D explanation.

They'll be muttering darkly in corners about Gaussian curvature, you know. (-:

#maths #curvature