# Curiosity about infinity in geometry(Not homework just curious)

• caters
In summary, using area and volume formulas for different shapes, it is possible to get infinitely large shapes from an infinitely long line segment. This includes shapes such as rectangles, triangles, squares, rhombi, trapezoids, pentagons, hexagons, and other regular polygons. For volume, shapes such as spheres, triangular prisms, other prisms, dodecahedrons, octahedrons, and icosahedrons can also become infinitely large. Toruses with holes and cones are also included in this possibility. However, it is uncertain if the formulas would still be applicable in these infinite cases.
caters
Okay, let's say you have an infinitely long line segment.

Using area and volume formulas for different shapes, would you get an infinitely large shape that the formula was for in the first place?
Area:
For rectangles, would you get an infinitely large rectangle with A = l * w?
What about triangles with 1/2 * b * h?
What about squares with l^2
what about rombi with A = b * h?
what about trapezoids with A = ((b1 + b2)*h)/2?
what about pentagons with A = 5/2 * l * a(apothem)?
hexagons with A = (3√3 s2)/ 2?
what about other regular polygons with Area = (a(apothem) x p(perimeter))/2?
what about ellipses with π * vertical radius * horizontal radius?
What about circles with π * r^2
Volume:
What about spheres with V = ⁴⁄₃πr³?
What about triangular prisms with 1/2 x b x h x l?
what about other prisms with V = area of base * l (yes that includes the cylinder at the infinite end)?
what about dodecahedrons with (15+7×√5)/4 × (Edge Length)^3
what about octahedrons with (√2)/3 × (Edge Length)^3
what about Icosahedrons with 5×(3+√5)/12 × (Edge Length)^3
What about toruses that have holes with 2 × π^2 × R(radius of hole) × r^2(radius of circular cross section)?
What about cones with π × r^2 × (h/3)?

What do you think?

I think that you would get an infinitely large shape from an infinitely long line segment but I honestly don't know.

## 1. What is infinity in geometry, and how is it defined?

Infinity in geometry is the concept of a value or quantity that is unbounded, limitless, and has no definite or fixed value. It is often symbolized as ∞. In geometry, infinity is defined as a point or line that extends infinitely in all directions. It is also used to describe the size, shape, or scale of geometric figures that have no boundaries or endpoints.

## 2. How does infinity relate to the concept of a fractal?

A fractal is a geometric figure that exhibits self-similarity at different scales. This means that as you zoom in on a fractal, you will see the same patterns repeated at smaller and smaller scales. Infinity and fractals are closely related because they both involve endless repetition and self-similarity. In fact, some fractals have an infinite perimeter or surface area, making them examples of infinity in geometry.

## 3. Can infinity be visualized in geometry?

While infinity cannot be fully grasped or comprehended by the human mind, it can be represented visually in geometry. For example, a line with no endpoints can be used to represent infinity in one dimension, and a plane with no boundaries can represent infinity in two dimensions. In three dimensions, a shape such as a Möbius strip or a Klein bottle can be used to visualize infinity.

## 4. How is infinity used in practical applications of geometry?

Infinity is used in practical applications of geometry, such as computer graphics and animation, to create smooth and continuous curves and shapes. It is also used in mathematical modeling and simulations to represent infinite or unbounded phenomena, such as the expansion of the universe or the behavior of complex systems.

## 5. What are some common misconceptions about infinity in geometry?

One common misconception is that infinity is a number. In reality, infinity is a concept that represents something that is unbounded and has no fixed value. Another misconception is that infinity is always the largest possible number, when in fact, there are different levels or types of infinity in mathematics and geometry. Additionally, some people may think that infinity exists in the physical world, when in fact it is an abstract concept used in mathematical and scientific contexts.

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