A collection of lines drawn between points in a regular 13-gon - how to determine where the points sit...
$begingroup$
So I have 4 collections of lines drawn between points each making a path. The angles are measured. The problem I am attempting to solve is to determine whether or not each of the collections of points are a valid path between some subset of points on a regular 13-gon. So for instance, for the second shape are there 3 vertices in a regular 13 gon such that drawing a line between them gives a 60 degree angle (within 5 degrees of error, see context). I'm having a hard time figuring this out myself. I know the geometry involved but I'm just having a hard time applying it to this problem.
Some Context
The context of this problem is actually a clue given in the hunt for the secret zoo level of the video game Accounting Plus VR and this was my attempt to solve what these diagrams mean. So because I measured the angles with an actual protractor they have 5 degrees of precision. So if there are valid points with approximate angles that is also a valid answer. I hope this doesn't make the question unanswerable but this is what I have to work with geometry wise. This is effectively an actual "real" problem with imperfect diagrams, rather than an ideal theoretical problem coming out of a book.
And yes the images are the original images I obtained for reference blown up in size. So if one wants to remeasure the angles in case my protractor skill is flawed, they are welcome to.
geometry applications
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
$endgroup$
add a comment |
$begingroup$
So I have 4 collections of lines drawn between points each making a path. The angles are measured. The problem I am attempting to solve is to determine whether or not each of the collections of points are a valid path between some subset of points on a regular 13-gon. So for instance, for the second shape are there 3 vertices in a regular 13 gon such that drawing a line between them gives a 60 degree angle (within 5 degrees of error, see context). I'm having a hard time figuring this out myself. I know the geometry involved but I'm just having a hard time applying it to this problem.
Some Context
The context of this problem is actually a clue given in the hunt for the secret zoo level of the video game Accounting Plus VR and this was my attempt to solve what these diagrams mean. So because I measured the angles with an actual protractor they have 5 degrees of precision. So if there are valid points with approximate angles that is also a valid answer. I hope this doesn't make the question unanswerable but this is what I have to work with geometry wise. This is effectively an actual "real" problem with imperfect diagrams, rather than an ideal theoretical problem coming out of a book.
And yes the images are the original images I obtained for reference blown up in size. So if one wants to remeasure the angles in case my protractor skill is flawed, they are welcome to.
geometry applications
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
$endgroup$
$begingroup$
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845ldots^circ$.
$endgroup$
– Blue
Jan 13 at 7:28
$begingroup$
@Blue that definitely rules out 30 degree angles. Thanks for help. If you expand that into an answer I'd be likely to accept it.
$endgroup$
– The Great Duck
Jan 13 at 7:29
add a comment |
$begingroup$
So I have 4 collections of lines drawn between points each making a path. The angles are measured. The problem I am attempting to solve is to determine whether or not each of the collections of points are a valid path between some subset of points on a regular 13-gon. So for instance, for the second shape are there 3 vertices in a regular 13 gon such that drawing a line between them gives a 60 degree angle (within 5 degrees of error, see context). I'm having a hard time figuring this out myself. I know the geometry involved but I'm just having a hard time applying it to this problem.
Some Context
The context of this problem is actually a clue given in the hunt for the secret zoo level of the video game Accounting Plus VR and this was my attempt to solve what these diagrams mean. So because I measured the angles with an actual protractor they have 5 degrees of precision. So if there are valid points with approximate angles that is also a valid answer. I hope this doesn't make the question unanswerable but this is what I have to work with geometry wise. This is effectively an actual "real" problem with imperfect diagrams, rather than an ideal theoretical problem coming out of a book.
And yes the images are the original images I obtained for reference blown up in size. So if one wants to remeasure the angles in case my protractor skill is flawed, they are welcome to.
geometry applications
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
$endgroup$
So I have 4 collections of lines drawn between points each making a path. The angles are measured. The problem I am attempting to solve is to determine whether or not each of the collections of points are a valid path between some subset of points on a regular 13-gon. So for instance, for the second shape are there 3 vertices in a regular 13 gon such that drawing a line between them gives a 60 degree angle (within 5 degrees of error, see context). I'm having a hard time figuring this out myself. I know the geometry involved but I'm just having a hard time applying it to this problem.
Some Context
The context of this problem is actually a clue given in the hunt for the secret zoo level of the video game Accounting Plus VR and this was my attempt to solve what these diagrams mean. So because I measured the angles with an actual protractor they have 5 degrees of precision. So if there are valid points with approximate angles that is also a valid answer. I hope this doesn't make the question unanswerable but this is what I have to work with geometry wise. This is effectively an actual "real" problem with imperfect diagrams, rather than an ideal theoretical problem coming out of a book.
And yes the images are the original images I obtained for reference blown up in size. So if one wants to remeasure the angles in case my protractor skill is flawed, they are welcome to.
geometry applications
geometry applications
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
asked Jan 13 at 7:13
The Great DuckThe Great Duck
24432047
24432047
$begingroup$
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845ldots^circ$.
$endgroup$
– Blue
Jan 13 at 7:28
$begingroup$
@Blue that definitely rules out 30 degree angles. Thanks for help. If you expand that into an answer I'd be likely to accept it.
$endgroup$
– The Great Duck
Jan 13 at 7:29
add a comment |
$begingroup$
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845ldots^circ$.
$endgroup$
– Blue
Jan 13 at 7:28
$begingroup$
@Blue that definitely rules out 30 degree angles. Thanks for help. If you expand that into an answer I'd be likely to accept it.
$endgroup$
– The Great Duck
Jan 13 at 7:29
$begingroup$
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845ldots^circ$.
$endgroup$
– Blue
Jan 13 at 7:28
$begingroup$
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845ldots^circ$.
$endgroup$
– Blue
Jan 13 at 7:28
$begingroup$
@Blue that definitely rules out 30 degree angles. Thanks for help. If you expand that into an answer I'd be likely to accept it.
$endgroup$
– The Great Duck
Jan 13 at 7:29
$begingroup$
@Blue that definitely rules out 30 degree angles. Thanks for help. If you expand that into an answer I'd be likely to accept it.
$endgroup$
– The Great Duck
Jan 13 at 7:29
add a comment |
1 Answer
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(Converting a comment to an answer, as requested.)
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$.
Therefore, for a $13$-gon, you'd expect the angles in your paths to be multiples of $13.845ldots^circ$.
Taking your measurements of $25^circ$, $30^circ$, $40^circ$, $45^circ$, $50^circ$, $60^circ$ as accurate, you'd seem to need an $n$-gon that allows for inscribed angles that are multiples of $5^circ$; in that case, $n=36$. This is close to $39$, which is a multiple of $13$. Note that $26$ is also a multiple of $13$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $6.932ldots^circ$.
In any event, you can use the $180^circ/n$ formula to check the viability of your guesses for $n$ and/or the accuracy of your angle measurements.
answered Jan 13 at 7:47
BlueBlue
49.1k870156
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1
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
add a comment |
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1 Answer
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$begingroup$
(Converting a comment to an answer, as requested.)
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$.
Therefore, for a $13$-gon, you'd expect the angles in your paths to be multiples of $13.845ldots^circ$.
Taking your measurements of $25^circ$, $30^circ$, $40^circ$, $45^circ$, $50^circ$, $60^circ$ as accurate, you'd seem to need an $n$-gon that allows for inscribed angles that are multiples of $5^circ$; in that case, $n=36$. This is close to $39$, which is a multiple of $13$. Note that $26$ is also a multiple of $13$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $6.932ldots^circ$.
In any event, you can use the $180^circ/n$ formula to check the viability of your guesses for $n$ and/or the accuracy of your angle measurements.
answered Jan 13 at 7:47
BlueBlue
49.1k870156
$endgroup$
1
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
add a comment |
$begingroup$
(Converting a comment to an answer, as requested.)
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$.
Therefore, for a $13$-gon, you'd expect the angles in your paths to be multiples of $13.845ldots^circ$.
Taking your measurements of $25^circ$, $30^circ$, $40^circ$, $45^circ$, $50^circ$, $60^circ$ as accurate, you'd seem to need an $n$-gon that allows for inscribed angles that are multiples of $5^circ$; in that case, $n=36$. This is close to $39$, which is a multiple of $13$. Note that $26$ is also a multiple of $13$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $6.932ldots^circ$.
In any event, you can use the $180^circ/n$ formula to check the viability of your guesses for $n$ and/or the accuracy of your angle measurements.
answered Jan 13 at 7:47
BlueBlue
49.1k870156
$endgroup$
1
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
add a comment |
$begingroup$
(Converting a comment to an answer, as requested.)
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$.
Therefore, for a $13$-gon, you'd expect the angles in your paths to be multiples of $13.845ldots^circ$.
Taking your measurements of $25^circ$, $30^circ$, $40^circ$, $45^circ$, $50^circ$, $60^circ$ as accurate, you'd seem to need an $n$-gon that allows for inscribed angles that are multiples of $5^circ$; in that case, $n=36$. This is close to $39$, which is a multiple of $13$. Note that $26$ is also a multiple of $13$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $6.932ldots^circ$.
In any event, you can use the $180^circ/n$ formula to check the viability of your guesses for $n$ and/or the accuracy of your angle measurements.
answered Jan 13 at 7:47
BlueBlue
49.1k870156
$endgroup$
(Converting a comment to an answer, as requested.)
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$.
Therefore, for a $13$-gon, you'd expect the angles in your paths to be multiples of $13.845ldots^circ$.
Taking your measurements of $25^circ$, $30^circ$, $40^circ$, $45^circ$, $50^circ$, $60^circ$ as accurate, you'd seem to need an $n$-gon that allows for inscribed angles that are multiples of $5^circ$; in that case, $n=36$. This is close to $39$, which is a multiple of $13$. Note that $26$ is also a multiple of $13$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $6.932ldots^circ$.
In any event, you can use the $180^circ/n$ formula to check the viability of your guesses for $n$ and/or the accuracy of your angle measurements.
answered Jan 13 at 7:47
BlueBlue
49.1k870156
answered Jan 13 at 7:47
BlueBlue
49.1k870156
answered Jan 13 at 7:47
BlueBlue
49.1k870156
answered Jan 13 at 7:47
BlueBlue
49.1k870156
49.1k870156
1
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
add a comment |
1
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
1
1
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
$begingroup$
Apparently the people I've been talking to said there are 12 main levels some having 3 versions. So perhaps $12*3=36$ does work if this is a map. Thanks.
$endgroup$
– The Great Duck
Jan 13 at 8:06
add a comment |
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$begingroup$
The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845ldots^circ$.
$endgroup$
– Blue
Jan 13 at 7:28
$begingroup$
@Blue that definitely rules out 30 degree angles. Thanks for help. If you expand that into an answer I'd be likely to accept it.
$endgroup$
– The Great Duck
Jan 13 at 7:29