Construction involving regular polygons inside a circle
Let's make a construction involving regular polygons:
► First, we begin with a equilateral triangle, with side $ell_3 = 1;$
► After, we draw a square on the middle point each side of the initial triangle, with side $ell_4 = frac{1}{2} = frac{ell}{2}.$
Now, the construction continues, taking one of these steps:
► If the regular polygon have an even number of sides $n$ with length $ell_n$, then we draw two regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the extreme segments.
► If the regular polygon have an odd number of sides $n$ with length $ell_n$, then we draw one regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the unique extreme segment in this case.
To clarify the explanation, we will obtain a figure like the one below:
I have two questions about this:
Q1. This figure is inside a circumference with center in the incenter of the initial equilateral triangle? In affirmative case, what is the radius $R$ of the circumference?
Q2. The sequence of the lengths I adopted in the construction is
$$ell_n = frac{1}{2^{n-3}}, quad forall n ge 3 $$
If I consider other sequence $ell_n$, when exists a circumference with center in the incenter of the initial equilateral triangle and radius $R$ in which the figure is inside?
sequences-and-series geometry euclidean-geometry circle polygons
asked Nov 20 at 3:14
674123173797 - 4
1407
add a comment |
Let's make a construction involving regular polygons:
► First, we begin with a equilateral triangle, with side $ell_3 = 1;$
► After, we draw a square on the middle point each side of the initial triangle, with side $ell_4 = frac{1}{2} = frac{ell}{2}.$
Now, the construction continues, taking one of these steps:
► If the regular polygon have an even number of sides $n$ with length $ell_n$, then we draw two regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the extreme segments.
► If the regular polygon have an odd number of sides $n$ with length $ell_n$, then we draw one regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the unique extreme segment in this case.
To clarify the explanation, we will obtain a figure like the one below:
I have two questions about this:
Q1. This figure is inside a circumference with center in the incenter of the initial equilateral triangle? In affirmative case, what is the radius $R$ of the circumference?
Q2. The sequence of the lengths I adopted in the construction is
$$ell_n = frac{1}{2^{n-3}}, quad forall n ge 3 $$
If I consider other sequence $ell_n$, when exists a circumference with center in the incenter of the initial equilateral triangle and radius $R$ in which the figure is inside?
sequences-and-series geometry euclidean-geometry circle polygons
asked Nov 20 at 3:14
674123173797 - 4
1407
Consider a path that starts at a corner of the triangle and meets the center of the opposite square, then the center of a connected pentagon, then a connected hexagon, etc, etc, etc. This path bends, so its length does not measure distance from the triangle's center; however, if the "limit" of the path's length is finite, then so is the "limit" of the distance from the center. Well, the path consists of a circumradius and an inradius of each $n$-gon; the formulas to calculate these from side-length are straightforward. The infinite sum, less so, but it's a place to start.
– Blue
Nov 20 at 10:38
add a comment |
Let's make a construction involving regular polygons:
► First, we begin with a equilateral triangle, with side $ell_3 = 1;$
► After, we draw a square on the middle point each side of the initial triangle, with side $ell_4 = frac{1}{2} = frac{ell}{2}.$
Now, the construction continues, taking one of these steps:
► If the regular polygon have an even number of sides $n$ with length $ell_n$, then we draw two regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the extreme segments.
► If the regular polygon have an odd number of sides $n$ with length $ell_n$, then we draw one regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the unique extreme segment in this case.
To clarify the explanation, we will obtain a figure like the one below:
I have two questions about this:
Q1. This figure is inside a circumference with center in the incenter of the initial equilateral triangle? In affirmative case, what is the radius $R$ of the circumference?
Q2. The sequence of the lengths I adopted in the construction is
$$ell_n = frac{1}{2^{n-3}}, quad forall n ge 3 $$
If I consider other sequence $ell_n$, when exists a circumference with center in the incenter of the initial equilateral triangle and radius $R$ in which the figure is inside?
sequences-and-series geometry euclidean-geometry circle polygons
asked Nov 20 at 3:14
674123173797 - 4
1407
Let's make a construction involving regular polygons:
► First, we begin with a equilateral triangle, with side $ell_3 = 1;$
► After, we draw a square on the middle point each side of the initial triangle, with side $ell_4 = frac{1}{2} = frac{ell}{2}.$
Now, the construction continues, taking one of these steps:
► If the regular polygon have an even number of sides $n$ with length $ell_n$, then we draw two regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the extreme segments.
► If the regular polygon have an odd number of sides $n$ with length $ell_n$, then we draw one regular polygons with $n + 1$ sides of length $ell_{n+1} = frac{ell_n}{2},$ from the middle point of the unique extreme segment in this case.
To clarify the explanation, we will obtain a figure like the one below:
I have two questions about this:
Q1. This figure is inside a circumference with center in the incenter of the initial equilateral triangle? In affirmative case, what is the radius $R$ of the circumference?
Q2. The sequence of the lengths I adopted in the construction is
$$ell_n = frac{1}{2^{n-3}}, quad forall n ge 3 $$
If I consider other sequence $ell_n$, when exists a circumference with center in the incenter of the initial equilateral triangle and radius $R$ in which the figure is inside?
sequences-and-series geometry euclidean-geometry circle polygons
sequences-and-series geometry euclidean-geometry circle polygons
asked Nov 20 at 3:14
674123173797 - 4
1407
asked Nov 20 at 3:14
674123173797 - 4
1407
edited Nov 20 at 3:27
asked Nov 20 at 3:14
674123173797 - 4
1407
asked Nov 20 at 3:14
674123173797 - 4
1407
asked Nov 20 at 3:14
674123173797 - 4
1407
1407
Consider a path that starts at a corner of the triangle and meets the center of the opposite square, then the center of a connected pentagon, then a connected hexagon, etc, etc, etc. This path bends, so its length does not measure distance from the triangle's center; however, if the "limit" of the path's length is finite, then so is the "limit" of the distance from the center. Well, the path consists of a circumradius and an inradius of each $n$-gon; the formulas to calculate these from side-length are straightforward. The infinite sum, less so, but it's a place to start.
– Blue
Nov 20 at 10:38
add a comment |
Consider a path that starts at a corner of the triangle and meets the center of the opposite square, then the center of a connected pentagon, then a connected hexagon, etc, etc, etc. This path bends, so its length does not measure distance from the triangle's center; however, if the "limit" of the path's length is finite, then so is the "limit" of the distance from the center. Well, the path consists of a circumradius and an inradius of each $n$-gon; the formulas to calculate these from side-length are straightforward. The infinite sum, less so, but it's a place to start.
– Blue
Nov 20 at 10:38
Consider a path that starts at a corner of the triangle and meets the center of the opposite square, then the center of a connected pentagon, then a connected hexagon, etc, etc, etc. This path bends, so its length does not measure distance from the triangle's center; however, if the "limit" of the path's length is finite, then so is the "limit" of the distance from the center. Well, the path consists of a circumradius and an inradius of each $n$-gon; the formulas to calculate these from side-length are straightforward. The infinite sum, less so, but it's a place to start.
– Blue
Nov 20 at 10:38
Consider a path that starts at a corner of the triangle and meets the center of the opposite square, then the center of a connected pentagon, then a connected hexagon, etc, etc, etc. This path bends, so its length does not measure distance from the triangle's center; however, if the "limit" of the path's length is finite, then so is the "limit" of the distance from the center. Well, the path consists of a circumradius and an inradius of each $n$-gon; the formulas to calculate these from side-length are straightforward. The infinite sum, less so, but it's a place to start.
– Blue
Nov 20 at 10:38
add a comment |
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Consider a path that starts at a corner of the triangle and meets the center of the opposite square, then the center of a connected pentagon, then a connected hexagon, etc, etc, etc. This path bends, so its length does not measure distance from the triangle's center; however, if the "limit" of the path's length is finite, then so is the "limit" of the distance from the center. Well, the path consists of a circumradius and an inradius of each $n$-gon; the formulas to calculate these from side-length are straightforward. The infinite sum, less so, but it's a place to start.
– Blue
Nov 20 at 10:38