Multiple Inheritance
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A rectangle is a parallelogram. A rhombus is a parallelogram. A square is a rectangle. A square is a rhombus. In which geometries are all these still true, and why? Euclidean (obviously) non-Euclidean Riemannian etc. I'm thinking that some shapes may not exist in some geometries. Do any exist that still negate the statements made? P.S. - Geometry was my least favorite math.
There are II kinds of people in the world, those who understand binary and those who understand Roman numerals. Web - Blog - RSS - Math
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A rectangle is a parallelogram. A rhombus is a parallelogram. A square is a rectangle. A square is a rhombus. In which geometries are all these still true, and why? Euclidean (obviously) non-Euclidean Riemannian etc. I'm thinking that some shapes may not exist in some geometries. Do any exist that still negate the statements made? P.S. - Geometry was my least favorite math.
There are II kinds of people in the world, those who understand binary and those who understand Roman numerals. Web - Blog - RSS - Math
:-D
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles] -
:-D
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles] -
A rectangle is a parallelogram. A rhombus is a parallelogram. A square is a rectangle. A square is a rhombus. In which geometries are all these still true, and why? Euclidean (obviously) non-Euclidean Riemannian etc. I'm thinking that some shapes may not exist in some geometries. Do any exist that still negate the statements made? P.S. - Geometry was my least favorite math.
There are II kinds of people in the world, those who understand binary and those who understand Roman numerals. Web - Blog - RSS - Math
I try to guess that multiple inheritance applies also to non-Euclidean geometries if you generically define the rectangle having all angles equal and rhombus having all sides equal and, finally the square having both properties (i.e. you don't insist on 90 degrees angles). Probably I'm wrong since I'm really know nothing about non-Euclidean geaometry :-O . :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles] -
I try to guess that multiple inheritance applies also to non-Euclidean geometries if you generically define the rectangle having all angles equal and rhombus having all sides equal and, finally the square having both properties (i.e. you don't insist on 90 degrees angles). Probably I'm wrong since I'm really know nothing about non-Euclidean geaometry :-O . :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles]I don't think you can re-define the rectangle and the rhombus, it would be cheating... :-) The only difference between the Euclidian and non-Euclidian geometries is the 5th postulate, but the other ones stand. So you must have a definition of "right angle" in every geometry, which means you can define the rectangle in every geometry. Admittedly, the definition may become very obscure and complicated... and you probably would not be able to measure the right angle in degrees or radiants... So I think the multiple inheritance holds in every geometry. The square is defined to be something that is both a rectangle and a rhombus, and because you can define the rectangle and rhombus in every geometry, you can define the square in every geometry. [edit] I realised this may not be very clear... my point is that when you change geometry, the only thing you re-define is the basic concepts like line, lenght, and right angle, but the rectangle, rhombus and square have the same definitions, just expressed in the new interpretation of line, lenght, and angle.
-+ HHexo +-
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I don't think you can re-define the rectangle and the rhombus, it would be cheating... :-) The only difference between the Euclidian and non-Euclidian geometries is the 5th postulate, but the other ones stand. So you must have a definition of "right angle" in every geometry, which means you can define the rectangle in every geometry. Admittedly, the definition may become very obscure and complicated... and you probably would not be able to measure the right angle in degrees or radiants... So I think the multiple inheritance holds in every geometry. The square is defined to be something that is both a rectangle and a rhombus, and because you can define the rectangle and rhombus in every geometry, you can define the square in every geometry. [edit] I realised this may not be very clear... my point is that when you change geometry, the only thing you re-define is the basic concepts like line, lenght, and right angle, but the rectangle, rhombus and square have the same definitions, just expressed in the new interpretation of line, lenght, and angle.
-+ HHexo +-
Nope, I'm not cheating, a rectangle may really be defined, in Euclidean geometry as a quadrilateral having all of the angles equal, the fact that then they are right angles is then a mere consequence ;P .
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles] -
Nope, I'm not cheating, a rectangle may really be defined, in Euclidean geometry as a quadrilateral having all of the angles equal, the fact that then they are right angles is then a mere consequence ;P .
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles]CPallini wrote:
mere consequence
Visually it is mere, but trying to prove that may get you stuck on the 5th Postulate, which made me wonder if my original statements always hold true.
There are II kinds of people in the world, those who understand binary and those who understand Roman numerals. Web - Blog - RSS - Math
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CPallini wrote:
mere consequence
Visually it is mere, but trying to prove that may get you stuck on the 5th Postulate, which made me wonder if my original statements always hold true.
There are II kinds of people in the world, those who understand binary and those who understand Roman numerals. Web - Blog - RSS - Math
IMHO the point is not if you need or don't need the fifth postulate. If you define a rectangle as the quadrialteral having all its angles equal then yes, you may need the fifth postualte to demonstrate that the angle are in fact right ones, but it doesn't matter: in the context of Euclidean Geometry the fifth postulate is fine (correct me if I'm wrong, since I'm not an expert about), but with the above definition (the all agles equals one), the proposed inheritance tree is valid also in not-Euclidean geometries (Maybe I'm wrong again). :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
[my articles] -
A rectangle is a parallelogram. A rhombus is a parallelogram. A square is a rectangle. A square is a rhombus. In which geometries are all these still true, and why? Euclidean (obviously) non-Euclidean Riemannian etc. I'm thinking that some shapes may not exist in some geometries. Do any exist that still negate the statements made? P.S. - Geometry was my least favorite math.
There are II kinds of people in the world, those who understand binary and those who understand Roman numerals. Web - Blog - RSS - Math
But a rhombus is a diamond... leading to diamond-shaped Multiple Inheritance which is naughty. :cool: