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Constructibility of the 17-gon
Clash Royale CLAN TAG#URR8PPP
up vote
6
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Comment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore.
Assume we are using this set of axioms $A$ for plane euclidean geometry and some sensible definition of the length $overlineab$ between two points $a$ and $b$. Then we can define the set $R$ to be a regular n-gon iff
- $R = x_j mid j in mathbbZ_n $ (has $n$ elements)
- $forall k in mathbbZ_n : ~overlinex_k-1x_k = overlinex_kx_k+1$ (is equilateral)
- $forall k in mathbbZ_n: angle ~x_k-1x_kx_k+1 = angle~ x_kx_k+1x_k+2 $ (is equiangular)
Now imagine someone simply presented you the following construction of a 17-gon, with an instruction of what he did. The construction yields 17 points of interest you collect in a set $R$.
Can you proof (or is there a known proof) by only using the Axioms of $A$, that $R$ is a regular 17-gon?
Comment: The linked construction is one by Herbert William Richmond which I found here, but my question would be the same for any other known construction which does the same job. The origins of the construction are of algebraic nature. Independantly of the origin, I want to know if the answer to my question is positiv, negative or not known.
geometry field-theory euclidean-geometry polygons
asked Jul 4 '16 at 17:15
Kettel
725418
 |Â
show 10 more comments
up vote
6
down vote
favorite
Comment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore.
Assume we are using this set of axioms $A$ for plane euclidean geometry and some sensible definition of the length $overlineab$ between two points $a$ and $b$. Then we can define the set $R$ to be a regular n-gon iff
- $R = x_j mid j in mathbbZ_n $ (has $n$ elements)
- $forall k in mathbbZ_n : ~overlinex_k-1x_k = overlinex_kx_k+1$ (is equilateral)
- $forall k in mathbbZ_n: angle ~x_k-1x_kx_k+1 = angle~ x_kx_k+1x_k+2 $ (is equiangular)
Now imagine someone simply presented you the following construction of a 17-gon, with an instruction of what he did. The construction yields 17 points of interest you collect in a set $R$.
Can you proof (or is there a known proof) by only using the Axioms of $A$, that $R$ is a regular 17-gon?
Comment: The linked construction is one by Herbert William Richmond which I found here, but my question would be the same for any other known construction which does the same job. The origins of the construction are of algebraic nature. Independantly of the origin, I want to know if the answer to my question is positiv, negative or not known.
geometry field-theory euclidean-geometry polygons
asked Jul 4 '16 at 17:15
Kettel
725418
1
But aren't the rules of algebra for construction derived to be compatible with and dependent on the rules of geometry? If something satisfies algebra mustn't it also satisfy geometry?
– fleablood
Jul 4 '16 at 17:36
2
Define geometric reasoning? I think the reason trisecting an angle was not shown to be viable until $1837$ is because 'geometric reasoning' can only go so far. en.wikipedia.org/wiki/Angle_trisection
– snulty
Jul 4 '16 at 17:37
7
this book shows how to do the 17-gon with an actual compass and straightedge. amazon.com/Mathographics-Dover-Recreational-Robert-Dixon/dp/… Given the construction, Euclid would have been able to fill in the details himself, and very happy about the whole thing.
– Will Jagy
Jul 4 '16 at 20:06
2
You do it by proving a theorem about geometric reasoning, to the effect that if some length or angle is constructible in the algebraic sense then one can write down a compass-and-straightedge construction of it in the geometric sense.
– Qiaochu Yuan
Jul 4 '16 at 20:25
1
@RobertFrost: Given a length $ell$, construct a circle of diameter $ell+1$, with this diameter divided at a point $P$ into segments of length $ell$ and length $1$. If the perpendicular at $P$ intersects the circle at $Q$, then $PQ$ has length $sqrtell$.
– Will Orrick
Aug 4 '16 at 11:40
 |Â
show 10 more comments
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Comment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore.
Assume we are using this set of axioms $A$ for plane euclidean geometry and some sensible definition of the length $overlineab$ between two points $a$ and $b$. Then we can define the set $R$ to be a regular n-gon iff
- $R = x_j mid j in mathbbZ_n $ (has $n$ elements)
- $forall k in mathbbZ_n : ~overlinex_k-1x_k = overlinex_kx_k+1$ (is equilateral)
- $forall k in mathbbZ_n: angle ~x_k-1x_kx_k+1 = angle~ x_kx_k+1x_k+2 $ (is equiangular)
Now imagine someone simply presented you the following construction of a 17-gon, with an instruction of what he did. The construction yields 17 points of interest you collect in a set $R$.
Can you proof (or is there a known proof) by only using the Axioms of $A$, that $R$ is a regular 17-gon?
Comment: The linked construction is one by Herbert William Richmond which I found here, but my question would be the same for any other known construction which does the same job. The origins of the construction are of algebraic nature. Independantly of the origin, I want to know if the answer to my question is positiv, negative or not known.
geometry field-theory euclidean-geometry polygons
asked Jul 4 '16 at 17:15
Kettel
725418
Comment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore.
Assume we are using this set of axioms $A$ for plane euclidean geometry and some sensible definition of the length $overlineab$ between two points $a$ and $b$. Then we can define the set $R$ to be a regular n-gon iff
- $R = x_j mid j in mathbbZ_n $ (has $n$ elements)
- $forall k in mathbbZ_n : ~overlinex_k-1x_k = overlinex_kx_k+1$ (is equilateral)
- $forall k in mathbbZ_n: angle ~x_k-1x_kx_k+1 = angle~ x_kx_k+1x_k+2 $ (is equiangular)
Now imagine someone simply presented you the following construction of a 17-gon, with an instruction of what he did. The construction yields 17 points of interest you collect in a set $R$.
Can you proof (or is there a known proof) by only using the Axioms of $A$, that $R$ is a regular 17-gon?
Comment: The linked construction is one by Herbert William Richmond which I found here, but my question would be the same for any other known construction which does the same job. The origins of the construction are of algebraic nature. Independantly of the origin, I want to know if the answer to my question is positiv, negative or not known.
geometry field-theory euclidean-geometry polygons
asked Jul 4 '16 at 17:15
Kettel
725418
edited Jan 6 at 10:42
asked Jul 4 '16 at 17:15
Kettel
725418
asked Jul 4 '16 at 17:15
Kettel
725418
asked Jul 4 '16 at 17:15
Kettel
725418
725418
1
But aren't the rules of algebra for construction derived to be compatible with and dependent on the rules of geometry? If something satisfies algebra mustn't it also satisfy geometry?
– fleablood
Jul 4 '16 at 17:36
2
Define geometric reasoning? I think the reason trisecting an angle was not shown to be viable until $1837$ is because 'geometric reasoning' can only go so far. en.wikipedia.org/wiki/Angle_trisection
– snulty
Jul 4 '16 at 17:37
7
this book shows how to do the 17-gon with an actual compass and straightedge. amazon.com/Mathographics-Dover-Recreational-Robert-Dixon/dp/… Given the construction, Euclid would have been able to fill in the details himself, and very happy about the whole thing.
– Will Jagy
Jul 4 '16 at 20:06
2
You do it by proving a theorem about geometric reasoning, to the effect that if some length or angle is constructible in the algebraic sense then one can write down a compass-and-straightedge construction of it in the geometric sense.
– Qiaochu Yuan
Jul 4 '16 at 20:25
1
@RobertFrost: Given a length $ell$, construct a circle of diameter $ell+1$, with this diameter divided at a point $P$ into segments of length $ell$ and length $1$. If the perpendicular at $P$ intersects the circle at $Q$, then $PQ$ has length $sqrtell$.
– Will Orrick
Aug 4 '16 at 11:40
 |Â
show 10 more comments
1
But aren't the rules of algebra for construction derived to be compatible with and dependent on the rules of geometry? If something satisfies algebra mustn't it also satisfy geometry?
– fleablood
Jul 4 '16 at 17:36
2
Define geometric reasoning? I think the reason trisecting an angle was not shown to be viable until $1837$ is because 'geometric reasoning' can only go so far. en.wikipedia.org/wiki/Angle_trisection
– snulty
Jul 4 '16 at 17:37
7
this book shows how to do the 17-gon with an actual compass and straightedge. amazon.com/Mathographics-Dover-Recreational-Robert-Dixon/dp/… Given the construction, Euclid would have been able to fill in the details himself, and very happy about the whole thing.
– Will Jagy
Jul 4 '16 at 20:06
2
You do it by proving a theorem about geometric reasoning, to the effect that if some length or angle is constructible in the algebraic sense then one can write down a compass-and-straightedge construction of it in the geometric sense.
– Qiaochu Yuan
Jul 4 '16 at 20:25
1
@RobertFrost: Given a length $ell$, construct a circle of diameter $ell+1$, with this diameter divided at a point $P$ into segments of length $ell$ and length $1$. If the perpendicular at $P$ intersects the circle at $Q$, then $PQ$ has length $sqrtell$.
– Will Orrick
Aug 4 '16 at 11:40
1
1
But aren't the rules of algebra for construction derived to be compatible with and dependent on the rules of geometry? If something satisfies algebra mustn't it also satisfy geometry?
– fleablood
Jul 4 '16 at 17:36
But aren't the rules of algebra for construction derived to be compatible with and dependent on the rules of geometry? If something satisfies algebra mustn't it also satisfy geometry?
– fleablood
Jul 4 '16 at 17:36
2
2
Define geometric reasoning? I think the reason trisecting an angle was not shown to be viable until $1837$ is because 'geometric reasoning' can only go so far. en.wikipedia.org/wiki/Angle_trisection
– snulty
Jul 4 '16 at 17:37
Define geometric reasoning? I think the reason trisecting an angle was not shown to be viable until $1837$ is because 'geometric reasoning' can only go so far. en.wikipedia.org/wiki/Angle_trisection
– snulty
Jul 4 '16 at 17:37
7
7
this book shows how to do the 17-gon with an actual compass and straightedge. amazon.com/Mathographics-Dover-Recreational-Robert-Dixon/dp/… Given the construction, Euclid would have been able to fill in the details himself, and very happy about the whole thing.
– Will Jagy
Jul 4 '16 at 20:06
this book shows how to do the 17-gon with an actual compass and straightedge. amazon.com/Mathographics-Dover-Recreational-Robert-Dixon/dp/… Given the construction, Euclid would have been able to fill in the details himself, and very happy about the whole thing.
– Will Jagy
Jul 4 '16 at 20:06
2
2
You do it by proving a theorem about geometric reasoning, to the effect that if some length or angle is constructible in the algebraic sense then one can write down a compass-and-straightedge construction of it in the geometric sense.
– Qiaochu Yuan
Jul 4 '16 at 20:25
You do it by proving a theorem about geometric reasoning, to the effect that if some length or angle is constructible in the algebraic sense then one can write down a compass-and-straightedge construction of it in the geometric sense.
– Qiaochu Yuan
Jul 4 '16 at 20:25
1
1
@RobertFrost: Given a length $ell$, construct a circle of diameter $ell+1$, with this diameter divided at a point $P$ into segments of length $ell$ and length $1$. If the perpendicular at $P$ intersects the circle at $Q$, then $PQ$ has length $sqrtell$.
– Will Orrick
Aug 4 '16 at 11:40
@RobertFrost: Given a length $ell$, construct a circle of diameter $ell+1$, with this diameter divided at a point $P$ into segments of length $ell$ and length $1$. If the perpendicular at $P$ intersects the circle at $Q$, then $PQ$ has length $sqrtell$.
– Will Orrick
Aug 4 '16 at 11:40
 |Â
show 10 more comments
1 Answer
1
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up vote
0
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@Will Jagy is right. Euclid surely would have found the Gaussian method acceptable. And he would not have been afraid to use algebra.
Although the word "algebra" wasn't around for the ancient Greeks, they freely used methods that we would now recognize as algebra, or a combination of algebra with geometry. Even the construction of what we now call the "golden ratio", necessary to achieve a regular pentagon, requires using what we now call algebraic relationships between the sides of a right triangle having legs with the ratio 2:1. Euclid was unaware of the more advanced methods by which the construction of the regular heptadecagon was discovered, but had he seen them he would have recognized them as lying within the scope of his axioms and postulates.
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
@Will Jagy is right. Euclid surely would have found the Gaussian method acceptable. And he would not have been afraid to use algebra.
Although the word "algebra" wasn't around for the ancient Greeks, they freely used methods that we would now recognize as algebra, or a combination of algebra with geometry. Even the construction of what we now call the "golden ratio", necessary to achieve a regular pentagon, requires using what we now call algebraic relationships between the sides of a right triangle having legs with the ratio 2:1. Euclid was unaware of the more advanced methods by which the construction of the regular heptadecagon was discovered, but had he seen them he would have recognized them as lying within the scope of his axioms and postulates.
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
add a comment |Â
up vote
0
down vote
@Will Jagy is right. Euclid surely would have found the Gaussian method acceptable. And he would not have been afraid to use algebra.
Although the word "algebra" wasn't around for the ancient Greeks, they freely used methods that we would now recognize as algebra, or a combination of algebra with geometry. Even the construction of what we now call the "golden ratio", necessary to achieve a regular pentagon, requires using what we now call algebraic relationships between the sides of a right triangle having legs with the ratio 2:1. Euclid was unaware of the more advanced methods by which the construction of the regular heptadecagon was discovered, but had he seen them he would have recognized them as lying within the scope of his axioms and postulates.
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
add a comment |Â
up vote
0
down vote
up vote
0
down vote
@Will Jagy is right. Euclid surely would have found the Gaussian method acceptable. And he would not have been afraid to use algebra.
Although the word "algebra" wasn't around for the ancient Greeks, they freely used methods that we would now recognize as algebra, or a combination of algebra with geometry. Even the construction of what we now call the "golden ratio", necessary to achieve a regular pentagon, requires using what we now call algebraic relationships between the sides of a right triangle having legs with the ratio 2:1. Euclid was unaware of the more advanced methods by which the construction of the regular heptadecagon was discovered, but had he seen them he would have recognized them as lying within the scope of his axioms and postulates.
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
@Will Jagy is right. Euclid surely would have found the Gaussian method acceptable. And he would not have been afraid to use algebra.
Although the word "algebra" wasn't around for the ancient Greeks, they freely used methods that we would now recognize as algebra, or a combination of algebra with geometry. Even the construction of what we now call the "golden ratio", necessary to achieve a regular pentagon, requires using what we now call algebraic relationships between the sides of a right triangle having legs with the ratio 2:1. Euclid was unaware of the more advanced methods by which the construction of the regular heptadecagon was discovered, but had he seen them he would have recognized them as lying within the scope of his axioms and postulates.
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
edited Jul 17 '16 at 10:57
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
answered Jul 17 '16 at 10:49
Oscar Lanzi
10k11632
10k11632
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
add a comment |Â
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
I agree that Euclid would be fine with the methodes that Gauss used, (in this case field theory) in the way that he would approve every step of reasoning. But Gauss used another set of axioms to prove it. My question is, if it is has also been proven using only the axioms of geometry.
– Kettel
Jul 17 '16 at 19:48
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
@ridac check my comments under the question itself. I am guessing now that in fact Erchinger may have solved this problem.
– Oscar Lanzi
Aug 3 '16 at 15:06
add a comment |Â
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1
But aren't the rules of algebra for construction derived to be compatible with and dependent on the rules of geometry? If something satisfies algebra mustn't it also satisfy geometry?
– fleablood
Jul 4 '16 at 17:36
2
Define geometric reasoning? I think the reason trisecting an angle was not shown to be viable until $1837$ is because 'geometric reasoning' can only go so far. en.wikipedia.org/wiki/Angle_trisection
– snulty
Jul 4 '16 at 17:37
7
this book shows how to do the 17-gon with an actual compass and straightedge. amazon.com/Mathographics-Dover-Recreational-Robert-Dixon/dp/… Given the construction, Euclid would have been able to fill in the details himself, and very happy about the whole thing.
– Will Jagy
Jul 4 '16 at 20:06
2
You do it by proving a theorem about geometric reasoning, to the effect that if some length or angle is constructible in the algebraic sense then one can write down a compass-and-straightedge construction of it in the geometric sense.
– Qiaochu Yuan
Jul 4 '16 at 20:25
1
@RobertFrost: Given a length $ell$, construct a circle of diameter $ell+1$, with this diameter divided at a point $P$ into segments of length $ell$ and length $1$. If the perpendicular at $P$ intersects the circle at $Q$, then $PQ$ has length $sqrtell$.
– Will Orrick
Aug 4 '16 at 11:40