Draw a triangle on a sphere with a base on the equator.

The internal angles are more than 180 deg.

Thats Elliptic for Ellipse , circle is special case rotate and you get sphere.

Galilean is like ordinary Eucledian space

Also cut a cone at different angles with a plane, to show cuts of the geometrys.

Perhaps give the equations that are satisfied.

Hyperbola as in xy=c

Juan Weisz

Dear Juan Weisz,

Thank you for your support and valuable contributions.

what a great question. some suggestions:

Spherical: Lenart Sphere. https://en.wikipedia.org/wiki/L%C3%A9n%C3%A1rt_sphere

Hyperbolic:"Henderson and Taimia, Experiencing Geometry: Euclidean and non-Euclidean with History. Pearson Prentice Hall, 2005 "have popularised “crocheted” representations of hyperbolic space.

Both spherical and hyperbolic spaces can be represented as projections in euclidean space using stereographic methods

There are several digital versions of non-euclidean spaces:

-Geogebra uses "paper and pencil" constructions of both spaces

-non-euclidean logo from me uses differential geometry with turtles!

Non-euclidean geometry is full of intersresting, confusing, thought provocking and entertaining activities. It should be part of every curriculum.

Thanks for the opportunity to get people thinking about these geometries.

--ian

Ian Stevenson

Dear Ian Stevenson,

Thank you for sharing these excellent resources and ideas. Non-Euclidean geometries offer a wealth of fascinating and thought-provoking tasks. Incorporating them into the curriculum would greatly enhance students' understanding of geometry. Your mention of GeoGebra and the non-Euclidean logo is particularly intriguing; I’ll definitely explore how these can be utilized in education. Thank you again for your valuable suggestions.

I believe, if you speak about Euclidean, Galilean and Minkowski geometries, the approach should be different than comparing Euclidean with elliptic and hyperbolic. You may start with physics and construct a geometry from it. A good example of such approach is described in book "Book A Simple Non-Euclidean Geometry and Its Physical Basis

Alexandru Popa

Dear Alexandru Popa,

Thank you for the suggestion! I appreciate the insight on approaching Euclidean, Galilean, and Minkowski geometries differently from a physics perspective. For further reference, I'll examine the book "A Simple Non-Euclidean Geometry and Its Physical Basis." Thanks again...

Minkoski has a mixed signature

-+++ instead of ++++ of E4.

Juan Weisz , involving signature does not help much in this concrete discussion, because Galilean geometry has no signature in classical meaning of this word.

Instead, we can define space specification {k1, k2,..., kn}, where n is space dimension and each km is a number -1, 0 or 1. Then for linear spaces the classical signature can be computed as: k2, k2k3,..., \prodm=2,n km. If all these numbers are 1 or -1, then each 1 becomes "+" and each -1 becomes "-". For non-linear spaces k1 indicates curvature: positive when k1 = 1 and negative when k1 = -1. For linear spaces k1 = 0.

Here are several examples of 4-dimensional spaces:

Euclidean has specification {0, 1, 1, 1} and its signature is ++++

Elliptic: {1, 1, 1, 1} | ++++ (positive curvature)

Hyperbolic: {-1, 1, 1, 1} | ++++ (negative curvature)

Galilean: {0, 0, 1, 1} | ????

Minkowski: {0, -1, 1, 1} | -+++ (linear)

De Sitter: {-1, -1, 1, 1} | -+++ (negative curvature)

Anti-De Sitter: {1, -1, 1, 1} | -+++ (positive curvature)

And so on.