Paradoxes also exist in human perceptions of geometric areas. When one makes conclusions based on visual perception instead of logical reasoning, figures that cannot logically exist may seem realistic while figures that can logically exist seem impossible. Geometric paradoxes exist because of the differences in people’s visual perceptions and the actual logical conclusion that can be drawn from geometric figures.
Part 2 - Geometry
Impossible figures are shapes that seem realistic in one’s visual perception, but logical reasoning will reveal they are impossible to create. Impossible Objects, one of the first articles surrounding these figures, was published by Lionel and Roger Penrose in 1958. This article introduces the Penrose Triangle and Penrose Staircase, both of which have become notable examples of impossible figures. As seen below, both figures appear realistic until one takes a closer look. Upon closer inspection, one realizes the following figures aren’t logically possible, because the spatial distribution doesn't add up. The Penrose Triangle appears to be warping in around itself, while the Penrose Staircase appears to continue infinitely without end. One can travel infinitely upwards or downwards on the infinite staircase without it coming to an end.
The figures can seem realistic on paper even though they cannot possibly exist in real life. This is because impossible figures use visual distortion to convince viewers of an impossible result. What one is seeing isn’t really accurate, since either the size or angle of the objects is distorted to make them seem possible.
Similarly, the Jigsaw Paradox uses visual distortion to create space where there isn’t any. The paradox claims that four figures composing a 13 by 5 triangle A can be rearranged into a 13 by 5 triangle B such that triangle B has an empty 1 by 1 square of space. As seen below, the paradox lies in the creation of additional space when the same four figures are rearranged.
The key to the Jigsaw Paradox is that neither A nor B are true triangles, because the hypotenuse of each is distorted. The hypotenuse of A and B are curved in such a way as to add to the area while still appearing straight to the human eye. This is done through the two triangle components of the figure; the smaller triangle has a ratio of 2.5 while the larger one has a slightly greater ratio of 2.67. When pieced together, the slope of the hypotenuse is not consistent throughout.
The four components also cannot create a true 13 by 5 triangle. When added together, the four figures total 32 units of area, while 32.5 units are needed in a 13 by 5 triangle. By taking advantage of the bent hypotenuse, triangle A is able to have an area of just 32 units while triangle B has an area of 33 units. Neither triangle has the 32.5 units of area as needed in a true 13 by 5 triangle.
Gabriel's Horn Paradox
Finally, Gabriel’s Horn Paradox contradicts the human perception of area and volume. The Torricelli Trumpet, also known as Gabriel’s Horn, is a figure with finite volume and infinite surface area. The horn is infinitely long but can be completely filled with a finite volume of paint. How can this be? One would assume that because the horn continues infinitely, an infinite volume of paint would be needed because there will always be at least a little opening for the paint to flow into.
As unbelievable as it may seem, it’s possible for a fixed volume to fill an infinite surface area. Gabriel’s Horn is the graph of y=1/x that starts at a height of one but gets infinitely close to the x-axis as the horn stretches out. Both the surface area and volume of the horn can be found by splitting the horn into discs of diminishing radii as the horn gets narrower. As seen in the illustration above, the radius of any given disc is equal to 1/x.
The surface area of Gabriel’s Horn is found with the equation for SA listed in the illustration above. As can be seen from the equation, the surface area of each disc is proportional to the radius, or 1/x. The total surface area of Gabriel’s horn is the sum of all surface areas of the infinite number of discs it’s split into. Total surface area can therefore be represented by the harmonic sequence of 1/x starting at x=1 and continuing to an infinite value of x. This sequence is divergent, meaning that the surface area gets infinitely larger as more discs are added.
The volume of Gabriel’s Horn is given by the equation for V in the illustration above. The volume is proportional to the square of the radius, or 1/x^2. Because the harmonic series of 1/x^2 is convergent, the horn’s volume eventually converges to a single finite value—in this case, the exact value of pi. This volume is finite, even though the surface area continues infinitely.
By analyzing geometric figures logically, one can clear up misconceptions in how these figures are viewed. One will be able to determine why figures can or cannot exist, without falling fool to visual manipulations and faulty logic. When dealing with geometric figures, one should rely on reasoning rather than visual perception to come to more accurate conclusions.
Stay tuned for part 3!
Author: Sophia Liu