What is an ambiguous object?
An ambiguous object is simply an object that has two different appearances. One appearance can be seen when viewed "head on" and the other appearance can be seen in conjunction by viewing its reflection in the mirror. This is depicted below. As you can see, the roof over the car appears differently when viewed head on compared to when viewing it in its reflection in the mirror behind the car.
(https://thekidshouldseethis.com/post/ambiguous-cylinder-illusion-by-kokichi-sugihara)
How does an ambiguous object work?
An ambiguous object works from the joining of two shapes or objects with mean curves that combine the curves and thus the shape of the objects and ultimately create a new shape entirely. For example, let us look at a series of squares that are reflected as circles (and vice versa) shown below.
(https://www.ndtv.com/offbeat/the-ambiguous-cylinder-illusion-is-blowing-peoples-mind-1840738)
The squares aren't really squares and the circles aren't really circles. Rather, the average curvature of the shapes or the averages of similarities between the two shapes are taken to create a new shape entirely that when viewed at different angles appears to be two different shapes. This illusion is largely due to the varience in height that is not apparent unless viewed from the side at a lower angle. The different heights between shapes, curves, and edges, allow certain edges to be seen when viewed at an angle where those edges are the only edges visable (usually due to being higher and overcasting the other shapes, curves, and edges) and from the opposite side the difference in heights of the edges form a different shape for a similar reason (usually due to being able to see the previously hidden shapes, curves, and edges that are lower and now in view from the other comppnents). This may seem complicated but its akin to having a cube with a different colored faces on each side. Each side you see from a different view will show you a different combination of colors. Our perception of something is limited by what we are able to see at that exact moment in time, in this case the angle we are looking at something. When we look at something from a different angle or view point we are able to see a different component or side of the object we are looking at. This ambiguous object project allows us to see both sides simeltaneously when the object is placed in front of a mirror. Again, these different shapes are created by averaging the two shapes together to make a single shape with variance in height that allows for both shapes to be seen from the uniform shape when viewed at different angles and sides. The illusion is believing that what you see are truly two compeltely seperate shapes and it blows your mind when you are able to see the "two different shapes" at the same time. However, once you know that the "two different shapes" are really just an amalgamation of two shapes the illusion fades and it becomes a simple concept of seeing two sides of the same coin. Or, seeing that a shape can be created by unconventional means and depending on your view you can see those means or not.
The curves I chose to analyze were the functions \[y=x^2\] and \[y=x^3\] shown below from (desmos.com/calculator).
(from my personal openscad renditions)
Here you can see by the yellow signs the two different functions, as well as how the blue shape represents the mean curvature of the two lines creating a new shape entirely.
(from my personal openscad renditions)
As you can see, the varying height and degree of visible curvature creates the illusion of two different shapes or in this case functions when in reality the two functions were combined into one shape that when viewed at different angles then appears to be two seperate entities.
I chose this design because it maintains the math component of understanding by using relatable functions esy to visualize in the construction of an ambiguous object. While not necesarry, I think it is easier to see how this ties into the mathematical realm by using math to explain math. Even though the functions are arbitrary, it is easier to see how the mean curvature is related to the combining of the two objects/functions when using mathematical functions as opposed to an elephant ear, a pineapple, or some other fun but possibly harder to grasp result of this concept.
An ambiguous object is simply an object that has two different appearances. One appearance can be seen when viewed "head on" and the other appearance can be seen in conjunction by viewing its reflection in the mirror. This is depicted below. As you can see, the roof over the car appears differently when viewed head on compared to when viewing it in its reflection in the mirror behind the car.
How does an ambiguous object work?
An ambiguous object works from the joining of two shapes or objects with mean curves that combine the curves and thus the shape of the objects and ultimately create a new shape entirely. For example, let us look at a series of squares that are reflected as circles (and vice versa) shown below.
The squares aren't really squares and the circles aren't really circles. Rather, the average curvature of the shapes or the averages of similarities between the two shapes are taken to create a new shape entirely that when viewed at different angles appears to be two different shapes. This illusion is largely due to the varience in height that is not apparent unless viewed from the side at a lower angle. The different heights between shapes, curves, and edges, allow certain edges to be seen when viewed at an angle where those edges are the only edges visable (usually due to being higher and overcasting the other shapes, curves, and edges) and from the opposite side the difference in heights of the edges form a different shape for a similar reason (usually due to being able to see the previously hidden shapes, curves, and edges that are lower and now in view from the other comppnents). This may seem complicated but its akin to having a cube with a different colored faces on each side. Each side you see from a different view will show you a different combination of colors. Our perception of something is limited by what we are able to see at that exact moment in time, in this case the angle we are looking at something. When we look at something from a different angle or view point we are able to see a different component or side of the object we are looking at. This ambiguous object project allows us to see both sides simeltaneously when the object is placed in front of a mirror. Again, these different shapes are created by averaging the two shapes together to make a single shape with variance in height that allows for both shapes to be seen from the uniform shape when viewed at different angles and sides. The illusion is believing that what you see are truly two compeltely seperate shapes and it blows your mind when you are able to see the "two different shapes" at the same time. However, once you know that the "two different shapes" are really just an amalgamation of two shapes the illusion fades and it becomes a simple concept of seeing two sides of the same coin. Or, seeing that a shape can be created by unconventional means and depending on your view you can see those means or not.
The curves I chose to analyze were the functions \[y=x^2\] and \[y=x^3\] shown below from (desmos.com/calculator).
Here you can see by the yellow signs the two different functions, as well as how the blue shape represents the mean curvature of the two lines creating a new shape entirely.
As you can see, the varying height and degree of visible curvature creates the illusion of two different shapes or in this case functions when in reality the two functions were combined into one shape that when viewed at different angles then appears to be two seperate entities.
I chose this design because it maintains the math component of understanding by using relatable functions esy to visualize in the construction of an ambiguous object. While not necesarry, I think it is easier to see how this ties into the mathematical realm by using math to explain math. Even though the functions are arbitrary, it is easier to see how the mean curvature is related to the combining of the two objects/functions when using mathematical functions as opposed to an elephant ear, a pineapple, or some other fun but possibly harder to grasp result of this concept.
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