Can the concepts of abstract algebra be visualized as in analysis?What do polynomials look like in the complex plane?Visual approach to abstract algebraEvery subgroup of cyclic group is cyclic,How do I visualize this fact graphically?Is there a plane algebraic curve with just 3-fold rotational symmetry, but without reflection symmetry?Symmetries of polynomialsElementary problems with group theoretic solutionsSubgroups of $S_4$ and homomorphismsA few questions about nonabelian cohomology of finite groups.Should I try to change the way Abstract Algebra is taught at my university? If so, how?What is the importance of examples in the study of group theory?Abstract algebra and the knight. (as in chess, not the knave that the box suggests) - How do I study the patterns?How does Dummit and Foote's abstract algebra text compare to others?Reference for guided exercises in Group TheoryThe order of $R^k in D_n$, the $n$th dihedral groupAdvice on how to overcome obstacles in studying Abstract Algebra
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Can the concepts of abstract algebra be visualized as in analysis?
What do polynomials look like in the complex plane?Visual approach to abstract algebraEvery subgroup of cyclic group is cyclic,How do I visualize this fact graphically?Is there a plane algebraic curve with just 3-fold rotational symmetry, but without reflection symmetry?Symmetries of polynomialsElementary problems with group theoretic solutionsSubgroups of $S_4$ and homomorphismsA few questions about nonabelian cohomology of finite groups.Should I try to change the way Abstract Algebra is taught at my university? If so, how?What is the importance of examples in the study of group theory?Abstract algebra and the knight. (as in chess, not the knave that the box suggests) - How do I study the patterns?How does Dummit and Foote's abstract algebra text compare to others?Reference for guided exercises in Group TheoryThe order of $R^k in D_n$, the $n$th dihedral groupAdvice on how to overcome obstacles in studying Abstract Algebra
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem.Is there any way of visualizing these things?Please suggest some good reference book/text also which discusses these things and also the motivation/idea behind different theorems and concepts.
real-analysis calculus abstract-algebra group-theory cyclic-groups
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
$endgroup$
|
show 4 more comments
$begingroup$
I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem.Is there any way of visualizing these things?Please suggest some good reference book/text also which discusses these things and also the motivation/idea behind different theorems and concepts.
real-analysis calculus abstract-algebra group-theory cyclic-groups
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
$endgroup$
3
$begingroup$
Group theory can be visualized as groups of symmetries.
$endgroup$
– Michael Burr
Jun 6 at 13:14
$begingroup$
See also math.stackexchange.com/q/1327016/589 and math.stackexchange.com/questions/607436/… and The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
$endgroup$
– lhf
Jun 6 at 13:38
3
$begingroup$
Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties.
$endgroup$
– André 3000
Jun 6 at 13:41
$begingroup$
I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups.
$endgroup$
– Mark Bennet
Jun 6 at 18:07
1
$begingroup$
There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not?
$endgroup$
– Anubhav Mukherjee
Jun 6 at 21:55
|
show 4 more comments
$begingroup$
I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem.Is there any way of visualizing these things?Please suggest some good reference book/text also which discusses these things and also the motivation/idea behind different theorems and concepts.
real-analysis calculus abstract-algebra group-theory cyclic-groups
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
$endgroup$
I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem.Is there any way of visualizing these things?Please suggest some good reference book/text also which discusses these things and also the motivation/idea behind different theorems and concepts.
real-analysis calculus abstract-algebra group-theory cyclic-groups
real-analysis calculus abstract-algebra group-theory cyclic-groups
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
asked Jun 6 at 13:11
Kishalay SarkarKishalay Sarkar
396 bronze badges
396 bronze badges
3
$begingroup$
Group theory can be visualized as groups of symmetries.
$endgroup$
– Michael Burr
Jun 6 at 13:14
$begingroup$
See also math.stackexchange.com/q/1327016/589 and math.stackexchange.com/questions/607436/… and The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
$endgroup$
– lhf
Jun 6 at 13:38
3
$begingroup$
Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties.
$endgroup$
– André 3000
Jun 6 at 13:41
$begingroup$
I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups.
$endgroup$
– Mark Bennet
Jun 6 at 18:07
1
$begingroup$
There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not?
$endgroup$
– Anubhav Mukherjee
Jun 6 at 21:55
|
show 4 more comments
3
$begingroup$
Group theory can be visualized as groups of symmetries.
$endgroup$
– Michael Burr
Jun 6 at 13:14
$begingroup$
See also math.stackexchange.com/q/1327016/589 and math.stackexchange.com/questions/607436/… and The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
$endgroup$
– lhf
Jun 6 at 13:38
3
$begingroup$
Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties.
$endgroup$
– André 3000
Jun 6 at 13:41
$begingroup$
I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups.
$endgroup$
– Mark Bennet
Jun 6 at 18:07
1
$begingroup$
There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not?
$endgroup$
– Anubhav Mukherjee
Jun 6 at 21:55
3
3
$begingroup$
Group theory can be visualized as groups of symmetries.
$endgroup$
– Michael Burr
Jun 6 at 13:14
$begingroup$
Group theory can be visualized as groups of symmetries.
$endgroup$
– Michael Burr
Jun 6 at 13:14
$begingroup$
See also math.stackexchange.com/q/1327016/589 and math.stackexchange.com/questions/607436/… and The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
$endgroup$
– lhf
Jun 6 at 13:38
$begingroup$
See also math.stackexchange.com/q/1327016/589 and math.stackexchange.com/questions/607436/… and The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
$endgroup$
– lhf
Jun 6 at 13:38
3
3
$begingroup$
Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties.
$endgroup$
– André 3000
Jun 6 at 13:41
$begingroup$
Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties.
$endgroup$
– André 3000
Jun 6 at 13:41
$begingroup$
I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups.
$endgroup$
– Mark Bennet
Jun 6 at 18:07
$begingroup$
I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups.
$endgroup$
– Mark Bennet
Jun 6 at 18:07
1
1
$begingroup$
There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not?
$endgroup$
– Anubhav Mukherjee
Jun 6 at 21:55
$begingroup$
There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not?
$endgroup$
– Anubhav Mukherjee
Jun 6 at 21:55
|
show 4 more comments
2 Answers
2
active
oldest
votes
$begingroup$
One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
$endgroup$
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
1
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
|
show 1 more comment
$begingroup$
To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
$endgroup$
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
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votes
active
oldest
votes
$begingroup$
One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
$endgroup$
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
1
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
|
show 1 more comment
$begingroup$
One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
$endgroup$
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
1
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
|
show 1 more comment
$begingroup$
One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
$endgroup$
One nice visual approach to basic results on cyclic groups is using periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Some of these have concrete implementations in toys like Spirograph - which can be employed to motivate these more abstract algebraic ideas at very elementary levels.
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
answered Jun 6 at 13:27
Bill DubuqueBill Dubuque
218k30 gold badges209 silver badges669 bronze badges
218k30 gold badges209 silver badges669 bronze badges
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
1
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
|
show 1 more comment
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
1
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
Do the colors in the images have any meaning or are they there to make them pretty?
$endgroup$
– B.Swan
Jun 6 at 16:35
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
So how do you use these pictures to visualize the concepts?
$endgroup$
– Don Thousand
Jun 6 at 18:00
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
Tell me something more on the above pictures as Don Thousand said?
$endgroup$
– Kishalay Sarkar
Jun 6 at 18:15
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
$begingroup$
@KishalaySarkar Sure, I'll elbaorate in a bit.
$endgroup$
– Bill Dubuque
Jun 6 at 18:45
1
1
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
$begingroup$
If the OP wants plane curves with cyclic symmetry groups then there are simpler versions like this with a 3-fold rotational symmetry. Complex power mappings lead to simple equations. Here is a 5-fold symmetry. I didn't bother to kill the dihedral symmetry of the latter curve. Sorry about that. Can't offer curves related to toys or pieces of art either :-(
$endgroup$
– Jyrki Lahtonen
Jun 6 at 20:03
|
show 1 more comment
$begingroup$
To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
$endgroup$
add a comment |
$begingroup$
To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
$endgroup$
add a comment |
$begingroup$
To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
$endgroup$
To add a bit of history here - the famous German mathematician Felix Klein in 1872 launched the so-called Erlangen Program, proposing a method of characterizing geometries based on group theory and projective geometry. This was a novel and visionary idea a the time of his writings. It preceded later developments that showed how the symbiosis of (Complex) Analysis, Geometry and Algebra led to new insights and theories.
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
answered Jun 6 at 16:26
Nicky HeksterNicky Hekster
29.7k6 gold badges35 silver badges57 bronze badges
29.7k6 gold badges35 silver badges57 bronze badges
add a comment |
add a comment |
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Group theory can be visualized as groups of symmetries.
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– Michael Burr
Jun 6 at 13:14
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See also math.stackexchange.com/q/1327016/589 and math.stackexchange.com/questions/607436/… and The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
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– lhf
Jun 6 at 13:38
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Cayley graphs provide a way to visualize groups. There is a textbook called Visual Group Theory that gives visual interpretations for many group theoretic properties.
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– André 3000
Jun 6 at 13:41
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I seem to recall a book by Fraleigh (First Course on Abstract Algebra) which had some visuals to go with quotient groups/cosets. Also simply the vertices of a regular $n-gon$ are a starting point for visualising subgroups of the finite cyclic groups.
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– Mark Bennet
Jun 6 at 18:07
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There is a reason it called "abstract algebra" but there is ofcourse there are some ways to visualize it (if not all). But I am not sure if you have that enough background yet or not?
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– Anubhav Mukherjee
Jun 6 at 21:55