Mixing
Draw a Square Without a Compass, Only a Straightedge
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I remember seeing the following question in an old STEP question:
using only a straight-edge and a set of (unmarked) coordinate axes, construct a square.
I'm sure I knew how to do it when I was preparing for STEP, but I don't remember now. I also can't find the STEP question, otherwise there would likely be a solution for it on The Student Room. I just remember it being pretty cool!
geometry euclidean-geometry geometric-construction
asked 5 hours ago
Sam T
3,301828
add a comment |Â
up vote
2
down vote
favorite
I remember seeing the following question in an old STEP question:
using only a straight-edge and a set of (unmarked) coordinate axes, construct a square.
I'm sure I knew how to do it when I was preparing for STEP, but I don't remember now. I also can't find the STEP question, otherwise there would likely be a solution for it on The Student Room. I just remember it being pretty cool!
geometry euclidean-geometry geometric-construction
asked 5 hours ago
Sam T
3,301828
1
Are the coordinate axes marked or not?
– edm
4 hours ago
Sorry, unmarked, as otherwise it's trivial
– Sam T
3 hours ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I remember seeing the following question in an old STEP question:
using only a straight-edge and a set of (unmarked) coordinate axes, construct a square.
I'm sure I knew how to do it when I was preparing for STEP, but I don't remember now. I also can't find the STEP question, otherwise there would likely be a solution for it on The Student Room. I just remember it being pretty cool!
geometry euclidean-geometry geometric-construction
asked 5 hours ago
Sam T
3,301828
I remember seeing the following question in an old STEP question:
using only a straight-edge and a set of (unmarked) coordinate axes, construct a square.
I'm sure I knew how to do it when I was preparing for STEP, but I don't remember now. I also can't find the STEP question, otherwise there would likely be a solution for it on The Student Room. I just remember it being pretty cool!
geometry euclidean-geometry geometric-construction
geometry euclidean-geometry geometric-construction
asked 5 hours ago
Sam T
3,301828
asked 5 hours ago
Sam T
3,301828
edited 2 hours ago
asked 5 hours ago
Sam T
3,301828
asked 5 hours ago
Sam T
3,301828
asked 5 hours ago
Sam T
3,301828
3,301828
1
Are the coordinate axes marked or not?
– edm
4 hours ago
Sorry, unmarked, as otherwise it's trivial
– Sam T
3 hours ago
add a comment |Â
1
Are the coordinate axes marked or not?
– edm
4 hours ago
Sorry, unmarked, as otherwise it's trivial
– Sam T
3 hours ago
1
1
Are the coordinate axes marked or not?
– edm
4 hours ago
Are the coordinate axes marked or not?
– edm
4 hours ago
Sorry, unmarked, as otherwise it's trivial
– Sam T
3 hours ago
Sorry, unmarked, as otherwise it's trivial
– Sam T
3 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
Not that I've proved it, but I suspect this is impossible.
That's assuming that "using a straightedge" means exactly what it means in traditional Euclidean constructions.
Just to inject a minimal amount of content, I also have a conjecture how this is consistent with the fact that the OP says he saw a solution once. I conjecture that that solution cheated. For example, it's easy to construct a square if we're given an actual physical ruler and we're allowed to use the fact that the two edges are parallel.
And a vague notion of where a proof of impossibility might come from: The constructions we can make with a straightedge, starting with those two axes, are invariant under the transformation $(x,y)mapsto(2x,y)$.
answered 1 hour ago
David C. Ullrich
56.3k43788
add a comment |Â
up vote
1
down vote
How about joining $left(1,0right)$ to $left(0,1right)$ to $left(-1,0right)$ to $left(0,-1right)$ back to $left(1,0right)$ by straight lines?
answered 4 hours ago
Sam Streeter
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
 |Â
show 1 more comment
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Not that I've proved it, but I suspect this is impossible.
That's assuming that "using a straightedge" means exactly what it means in traditional Euclidean constructions.
Just to inject a minimal amount of content, I also have a conjecture how this is consistent with the fact that the OP says he saw a solution once. I conjecture that that solution cheated. For example, it's easy to construct a square if we're given an actual physical ruler and we're allowed to use the fact that the two edges are parallel.
And a vague notion of where a proof of impossibility might come from: The constructions we can make with a straightedge, starting with those two axes, are invariant under the transformation $(x,y)mapsto(2x,y)$.
answered 1 hour ago
David C. Ullrich
56.3k43788
add a comment |Â
up vote
3
down vote
Not that I've proved it, but I suspect this is impossible.
That's assuming that "using a straightedge" means exactly what it means in traditional Euclidean constructions.
Just to inject a minimal amount of content, I also have a conjecture how this is consistent with the fact that the OP says he saw a solution once. I conjecture that that solution cheated. For example, it's easy to construct a square if we're given an actual physical ruler and we're allowed to use the fact that the two edges are parallel.
And a vague notion of where a proof of impossibility might come from: The constructions we can make with a straightedge, starting with those two axes, are invariant under the transformation $(x,y)mapsto(2x,y)$.
answered 1 hour ago
David C. Ullrich
56.3k43788
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Not that I've proved it, but I suspect this is impossible.
That's assuming that "using a straightedge" means exactly what it means in traditional Euclidean constructions.
Just to inject a minimal amount of content, I also have a conjecture how this is consistent with the fact that the OP says he saw a solution once. I conjecture that that solution cheated. For example, it's easy to construct a square if we're given an actual physical ruler and we're allowed to use the fact that the two edges are parallel.
And a vague notion of where a proof of impossibility might come from: The constructions we can make with a straightedge, starting with those two axes, are invariant under the transformation $(x,y)mapsto(2x,y)$.
answered 1 hour ago
David C. Ullrich
56.3k43788
Not that I've proved it, but I suspect this is impossible.
That's assuming that "using a straightedge" means exactly what it means in traditional Euclidean constructions.
Just to inject a minimal amount of content, I also have a conjecture how this is consistent with the fact that the OP says he saw a solution once. I conjecture that that solution cheated. For example, it's easy to construct a square if we're given an actual physical ruler and we're allowed to use the fact that the two edges are parallel.
And a vague notion of where a proof of impossibility might come from: The constructions we can make with a straightedge, starting with those two axes, are invariant under the transformation $(x,y)mapsto(2x,y)$.
answered 1 hour ago
David C. Ullrich
56.3k43788
answered 1 hour ago
David C. Ullrich
56.3k43788
answered 1 hour ago
David C. Ullrich
56.3k43788
answered 1 hour ago
David C. Ullrich
56.3k43788
56.3k43788
add a comment |Â
add a comment |Â
up vote
1
down vote
How about joining $left(1,0right)$ to $left(0,1right)$ to $left(-1,0right)$ to $left(0,-1right)$ back to $left(1,0right)$ by straight lines?
answered 4 hours ago
Sam Streeter
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
 |Â
show 1 more comment
up vote
1
down vote
How about joining $left(1,0right)$ to $left(0,1right)$ to $left(-1,0right)$ to $left(0,-1right)$ back to $left(1,0right)$ by straight lines?
answered 4 hours ago
Sam Streeter
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
 |Â
show 1 more comment
up vote
1
down vote
up vote
1
down vote
How about joining $left(1,0right)$ to $left(0,1right)$ to $left(-1,0right)$ to $left(0,-1right)$ back to $left(1,0right)$ by straight lines?
answered 4 hours ago
Sam Streeter
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
How about joining $left(1,0right)$ to $left(0,1right)$ to $left(-1,0right)$ to $left(0,-1right)$ back to $left(1,0right)$ by straight lines?
answered 4 hours ago
Sam Streeter
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 4 hours ago
Sam Streeter
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 4 hours ago
Sam Streeter
313
answered 4 hours ago
Sam Streeter
313
313
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Sam Streeter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
 |Â
show 1 more comment
1
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
1
1
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Nice. I suspect it's impossible to construct a unit square, since that would require constructing $sqrt2$ without a compass.
– Ethan Bolker
4 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
Sorry, I should have said explicitly: unmarked coordinate axes. Basically just two (doubly-infinite) lines perpendicular to each other.
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
@EthanBolker is it not possible to construct $sqrt 2$ without a compass?
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
Note that we already have a set of perpendicular lines, so it's not like we're starting from nothing
– Sam T
2 hours ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
@samStreeter The OP has edited the question to say the axes are unmarked, so this nice solution isn't one.
– Ethan Bolker
1 hour ago
 |Â
show 1 more comment
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2940552%2fdraw-a-square-without-a-compass-only-a-straightedge%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Are the coordinate axes marked or not?
– edm
4 hours ago
Sorry, unmarked, as otherwise it's trivial
– Sam T
3 hours ago