I hope it's not a silly question but what I wonder if we could make geometries with curves other than the circle (conics) like the lemniscate or folium.
That is not at all a silly question. First of all the geometry I am explaining here---hyperbolic geometry---has a direct generalization to the case of a more general conic. I will eventually explain this. However it is also possible to consider analogs say when one starts with a cubic curve. It turns out that we get a quite remarkable and new form of algebraic geometry! There is much more to be said about this... but I am still only learning...
Excellent as always. I believe the preferred pronunciation of Giovanni Ceva's last name is "Chay-vah," and that of his last name's adjectival form is "Chay-vee-an". Hardy.
In the past you've illustrated the connection (equivalence?) between hyperbolic geometry and projective relativistic geometry. Given your assertion that the ortho-axis is the most important line in hyperbolic triangle geometry, is there a meaningful interpretation of it in the relativistic setting, and in particular is there a way to conceptualise what it represents in terms of the physical world around us?
I hope it's not a silly question but what I wonder if we could make geometries with curves other than the circle (conics) like the lemniscate or folium.
rationalinteger 9 months ago
Hi rationalinteger
That is not at all a silly question. First of all the geometry I am explaining here---hyperbolic geometry---has a direct generalization to the case of a more general conic. I will eventually explain this. However it is also possible to consider analogs say when one starts with a cubic curve. It turns out that we get a quite remarkable and new form of algebraic geometry! There is much more to be said about this... but I am still only learning...
Excellent question.
njwildberger 9 months ago
Excellent as always. I believe the preferred pronunciation of Giovanni Ceva's last name is "Chay-vah," and that of his last name's adjectival form is "Chay-vee-an". Hardy.
peterhi503 9 months ago
Hi Hardy,
Thanks for that--will try to keep that in mind.
njwildberger 9 months ago
In the past you've illustrated the connection (equivalence?) between hyperbolic geometry and projective relativistic geometry. Given your assertion that the ortho-axis is the most important line in hyperbolic triangle geometry, is there a meaningful interpretation of it in the relativistic setting, and in particular is there a way to conceptualise what it represents in terms of the physical world around us?
simonsillitoe 9 months ago
Hi Simon
Those are both excellent questions, to which I currently do not have answers.
njwildberger 9 months ago