Question

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this image has rotational symmetry. what is the smallest number of degrees you need to rotate the image for it to look the same?

Explanation:

Step1: Analyze the soccer ball pattern

A soccer ball (a truncated icosahedron - like pattern) has a rotational symmetry related to its repeating units. The black and white pattern (pentagons and hexagons) has a rotational symmetry where the angle of rotation is determined by the number of repeating sectors. For a soccer - ball - like pattern with 5 - fold or 6 - fold symmetries, but more accurately, the smallest angle of rotation for a regular pattern like this (considering the symmetry of the pentagons and hexagons arrangement) is based on the fact that a full rotation is \(360^{\circ}\), and if we consider the symmetry around the center, the number of equivalent positions. For a pentagonal symmetry (since there are pentagons), the angle of rotation for rotational symmetry is \(\frac{360^{\circ}}{5}=72^{\circ}\)? Wait, no, actually, the soccer ball's pattern (truncated icosahedron) has both 5 - fold and 6 - fold symmetries. But the question is about the smallest number of degrees to rotate the image (the soccer ball) to look the same. Let's think about the basic rotational symmetry of the pattern. The black pentagons and white hexagons: if we look at the symmetry, the number of times the pattern repeats around the center. Let's consider the number of "slices" or the number of equivalent positions. For a regular pattern with \(n\) - fold symmetry, the rotation angle is \(\frac{360^{\circ}}{n}\). Now, looking at the soccer ball's image, the pattern has a 5 - fold symmetry? Wait, no, actually, the standard soccer ball (truncated icosahedron) has 12 pentagons and 20 hexagons. But when we look at the image, the key is to find the smallest angle such that rotating the image by that angle makes it look the same. Let's count the number of similar "units" around the center. If we look at the black and white pattern, the smallest angle of rotation for the image (the soccer ball) to coincide with itself is \(72^{\circ}\)? Wait, no, wait. Wait, the soccer ball's pattern: each pentagon is surrounded by hexagons. But maybe a better way: the image of the soccer ball, when rotated, the smallest angle to make it look the same. Let's think of the symmetry of the figure. The figure (soccer ball) has a rotational symmetry where the angle is \(72^{\circ}\)? Wait, no, let's check again. Wait, maybe I made a mistake. Let's consider the number of black pentagons? Wait, in the image, how many black pentagons are visible? Let's count: in the given image, there are 5 black pentagons (or parts of them) visible? Wait, no, the image shows a soccer ball with, let's see, the black regions: maybe 5? Wait, no, the standard soccer ball has 12 black pentagons. But in the given image, the pattern: let's assume that the rotational symmetry is based on the fact that the figure can be rotated by \(72^{\circ}\) (since \(360\div5 = 72\))? Wait, no, maybe it's \(60^{\circ}\)? Wait, no, let's think again. Wait, the correct answer for a soccer ball's rotational symmetry (the smallest angle) is actually \(72^{\circ}\)? Wait, no, I think I messed up. Wait, no, the standard rotational symmetry for a regular pentagon (which is part of the soccer ball) is \(72^{\circ}\) (since \(360\div5 = 72\)), but the soccer ball has both pentagonal and hexagonal symmetries. But the question is about the image of the soccer ball. Let's look at the image: the soccer ball in the picture has a pattern where the black and white regions repeat. Let's count the number of "spokes" or the number of times the pattern repeats. If we consider that the figure has a 5 - fold symmetry (because of the pentagons), then the rotation angle is \(360\div5 = 72\) degrees? Wait, no, maybe I am wrong. Wait, another approach: the smallest angle of rotation for a figure to have rotational symmetry is the smallest positive angle \(\theta\) such that rotating the figure by \(\theta\) about its center maps the figure onto itself. For the soccer ball image, the pattern is symmetric, and the smallest angle is \(72^{\circ}\)? Wait, no, let's check with a simple example. If you have a regular pentagon, the rotational symmetry angle is \(72^{\circ}\) (since \(360/5 = 72\)). The soccer ball has pentagons, so the rotational symmetry related to the pentagons is \(72^{\circ}\). But wait, maybe the answer is \(60^{\circ}\)? No, \(360/6 = 60\), but the soccer ball has hexagons too. Wait, the problem is about the image of the soccer ball. Let's look at the image again: the soccer ball in the picture has a pattern where the black regions are pentagons. Let's count the number of black pentagons visible: maybe 5? So the number of times the pattern repeats around the center is 5. So the rotation angle is \(360\div5 = 72\) degrees. Wait, but maybe I made a mistake. Wait, no, let's think of the rotational symmetry of the figure. If you rotate the soccer ball by \(72^{\circ}\) around its center, the pattern will match up again. Let's confirm: a full rotation is \(360^{\circ}\), and if there are 5 equivalent positions (due to the 5 - fold symmetry of the pentagons), then each rotation of \(72^{\circ}\) will map the figure onto itself.

Wait, but maybe the answer is \(60^{\circ}\)? No, because \(360\div6 = 60\), but the soccer ball's pattern is not 6 - fold in the way that would give \(60^{\circ}\) as the smallest. Wait, maybe I am overcomplicating. Let's recall that the soccer ball (truncated icosahedron) has rotational symmetries of order 2, 3, 5. The smallest non - zero rotation angle is \(360\div5 = 72^{\circ}\)? Wait, no, order of symmetry is the number of times it maps onto itself in a full rotation. So order 5 means rotation by \(72^{\circ}\) (since \(360/5 = 72\)) maps it onto itself. So the smallest angle is \(72^{\circ}\).

Step2: Conclusion

The smallest number of degrees to rotate the image (soccer ball) so that it looks the same is \(72^{\circ}\) (because of the 5 - fold rotational symmetry from the pentagonal pattern, where the angle of rotation is \(\frac{360^{\circ}}{5}=72^{\circ}\)).

Answer:

\(72\)