![]() ![]() Calling the definitional assertions into question merely creates heat without adding light. One way or the other we have to make definitional assertions and work from that standpoint. But if they had, they would have created a different set of 'basic' figures that made sense within that system. They could have used a different basis for geometry, I imagine, but they didn't, and this preference for straight-edged figures has persisted. They used straight lines and arcs of circles to make all their measurements and calculations, and so our geometry is (traditionally) based on straight-sided figures and evolutes of circles. Allowing curved sides effectively erases the distinctions between all the basic geometric shapes.Īncient mathematicians based their work on 'compass and straight-edge' geometry. Further, by artfully constructing a triangle with two straight sides and one curved side (with a right-angle curve in it), we can turn our triangle into a rectangle and vice-versa. In fact, we could take a circle, pick an arbitrary set of points on its circumference and call them vertices, and then claim our circle was in fact a triangle, or a quadrilateral, or a quintilateral. Keep in mind, that this isn’t just pure abstraction or imagination. For instance, allowing a triangle to have curved sides would mean that we could not distinguish between a triangle whose sides have a particular radius of curvature and a circle. Why are curved triangles, in Euclidean space, not considered triangles The only case of a curved triangle is a Reuleaux triangle, which is a highly symmetric one. Answer (1 of 10): Not only can a triangle have three acute angles (which is very common), you can also get triangles with angles so acute that their total sum is less than 180 degrees This happens in a ‘negative curved space’. So I think that my student’s answer is better than mine.It's with taking a moment to think about the ramifications of what this question asks. For one thing, what do you do when the two regions are equal in area? Also, that definition has the ugly property that you can move one vertex of a triangle a small amount and have the “inside” and “outside” flip in a discontinuous manner (if the triangle is close to half the sphere in size). But I have to say that that definition is pretty unnatural. You could, if you like, define a triangle to be the smaller of the two regions delimited by three line segments - that’s what I mean when I say that the difference between the two answers is a matter of definition. Any triangle could be opened up to form a straight angle, which has a measurement. The point is that, when you draw a closed curve on a sphere, there’s not necessarily a natural, principled distinction between the “inside” and the “outside” the way there is on an infinite plane. This image illustrates how the sum of the angles still add up to the straight angle from which it was formed. So the answer is that there are equilateral triangles with any area up to (but, I guess, not including), the surface area of the entire sphere. We can shrink the red triangle as small as we want and make the blue triangle as close as we want to the entire sphere. We naturally think of the red region as the triangle, but isn’t the blue region a triangle too? After all, it’s a region of the surface completely surrounded by three straight line segments. The largest triangle shown, ABC, will have angles that sum to 270 degrees. The triangle A'B'C' will have angles that sum to more than 180 degrees. However, as we explore larger areas, deviations will creep into our results. That would be so, near enough, for the triangle A''B''C'' below. Ultimately, the difference between his answer and mine is a matter of definition, so you can say that either is “right,” but I have to confess that I like his answer better than mine. Triangles will have angles that sum to 180 degrees. But a student in the class argued for a different answer. ![]() You take a single great circle (“straight line”), mark off three equally spaced point, and call those the vertices.Īt least, that’s what I intended the answer to be (and I’m pretty sure it’s what the textbook author intended too). The biggest one is one with three 180-degree angles, covering half the sphere. ![]() Here’s a little animation showing equilateral triangles of different sizes: It has the funny property that all of its angles are right angles. Frequently Asked Questions about Black Holes.Frequently Asked Questions About Black Holes. ![]()
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