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What should the value of pi be in taxicab geometry?
The value of pi in taxicab geometry technically does not exist as any taxicab shape would consist of right angles. A circle does not contain any right angles, therefore circles do not exist in taxicab geometry (so neither does the value pi).However, if you were to place a square around a circle with diameter 1, where each side is a tangent to the circle and then removed the corners of the square repeatedly without ever removing any or the area of the circle (see Figure 1), the value of 'pi' in that case would be 4 (as pi = diameter/circumference (or in this case perimeter)).Figure 1Here the problem arises, as perimeter and circumference are not exactly the same thing. Circumference only pertains to a circle (and circles do not exist in taxicab geometry) whereas perimeter pertains to any geometric shape (whether in taxicab geometry or Euclidean geometry).
What is the diameter of a 50mm area circle?
Area cannot be measured in mm. So an area of 50mm cannot exist and therefore, there can be no answer to this question.
A tessellation is an arrangement of repeating shapes that have what characteristics?
No spaces or overlaps exist between it's pieces.
Do parallel lines exist on a sphere?
No. The circular shape makes it impossible to have parallel lines just as you cannot have parallel lines in a circle that both reach the length of the diameter of the said circle.
What is the relationship between angle and distance?
Oh, dude, it's like this: the relationship between angle and distance is that they're kind of like distant cousins at a family reunion - they know each other exist, but they don't really hang out that much. Angles help us measure the direction or orientation of something, while distance tells us how far apart things are. So, they're related in the sense that they both play a role in understanding space, but they're not exactly best buddies.
