Why can't a (5,5,4) solid exist?

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In summary, the statement "and obviously there can be no solid where every vertex has a (5,5,4) arrangement" is an end of a proof by contradiction in geometry. This statement can be explained by understanding that a (5,5,4) arrangement implies that a pentagon is surrounded by alternating pentagons and squares, which is not possible since there would be two squares (or two pentagons) abutting each other. This can be further understood by looking at the explanation on the Wikipedia page for vertex configuration.
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GeometryIsHARD
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So i was reading my geometry textbook and it said "and obviously there can be no solid where every vertex has a (5,5,4) arrangement"; unfortunately, this is not an obvious fact to me. Can somebody explain to me what makes this so obvious? This statement was the end of a proof by contradiction so i feel I should really understand why it's true.
 
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This statement from the wikipedia article sums it up:

Specifically p.q.r implies that a p-gon is surrounded by alternating a q-gons and r-gons, so either p is even orq equals r.

If something is uniform 5.5.4 then focus on a pentagon. Each edge alternates being a pentagon or a square, but when you go all the way around the pentagon you will get two squares (or two pentagons) abutting each other.
 

1. Why can't a (5,5,4) solid exist?

A (5,5,4) solid, also known as a tetragonal trapezohedron, cannot exist because it violates the laws of geometry. In order for a solid to exist, it must have faces that are congruent and parallel to each other. However, in a (5,5,4) solid, the faces are not parallel and therefore cannot form a stable shape.

2. What are the properties of a (5,5,4) solid?

A (5,5,4) solid would have 14 faces, 24 edges, and 12 vertices. It would also have a tetragonal base with two trapezoidal faces on opposite sides.

3. Are there any real-life examples of a (5,5,4) solid?

No, there are no known real-life examples of a (5,5,4) solid. It is a purely theoretical shape that cannot exist in physical form.

4. Can a (5,5,4) solid be approximated in 3D modeling?

While a (5,5,4) solid cannot exist in reality, it can be approximated in 3D modeling software by using a combination of other shapes, such as tetrahedrons and pyramids.

5. How do scientists study the properties of shapes that cannot exist?

Scientists use mathematical equations, computer simulations, and theoretical models to study the properties of shapes that cannot exist in reality. They can also use these tools to predict the properties of potential future shapes that have not yet been discovered or created.

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