Higher dimensional spheres cause contradictions?

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According to this Numberfile video, if you tightly pack hyper-dimensional Spheres into a hyper-dimensional box and then find the radius of the largest sphere that could possibly fit in the remaining space, the resulting sphere would somehow exceed the confines of the box that contained all of the spheres. (where the number of dimensions are greater or equal to 10)

Isn't a logical contradiction generally considered a disproof of something?

Wouldn't this disprove the generalised formula being used to find the radius of the resulting sphere on n dimensions?

Is it possible that mathematicians simply do not understand extra dimensional geometry and its inherent rules?

logic axioms dimension-theory

share|cite|improve this question

edited 21 mins ago

asked 2 hours ago

Lorry Laurence mcLarry

4641511

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sorry, where is the logical contradiction here?
  – Rahul
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  There is no contradiction. It's simply true that a sphere of appropriate size to touch the other spheres that just touch the box from the inside will have some points outside the box in a high enough dimension.
  – Mark S.
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is a consequence of the pythagorean theorem extended to higher dimensions. For a 100-dimensional hypercube, a diagonal is ten times the length of any edge; but the hyperspheres you're packing in (excluding this weird center one) stay the same radius in every dimension if your cube's edge length stays the same. So it shouldn't be hard to imagine that the distance between a sphere in one "corner" gets farther from the one in the opposite corner of the cube as the dimension goes up.
  – Malice Vidrine
  34 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I guess they explained it poorly towards the end, talking about "spiky spheres"...
  – Lorry Laurence mcLarry
  19 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  That is a bit of a weird description. If anything my intuition has always been that it's cubes that get "spiky".
  – Malice Vidrine
  10 mins ago
  

  
  
  
  

add a comment | 

up vote
3
down vote

favorite

According to this Numberfile video, if you tightly pack hyper-dimensional Spheres into a hyper-dimensional box and then find the radius of the largest sphere that could possibly fit in the remaining space, the resulting sphere would somehow exceed the confines of the box that contained all of the spheres. (where the number of dimensions are greater or equal to 10)

Isn't a logical contradiction generally considered a disproof of something?

Wouldn't this disprove the generalised formula being used to find the radius of the resulting sphere on n dimensions?

Is it possible that mathematicians simply do not understand extra dimensional geometry and its inherent rules?

logic axioms dimension-theory

share|cite|improve this question

edited 21 mins ago

asked 2 hours ago

Lorry Laurence mcLarry

4641511

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sorry, where is the logical contradiction here?
  – Rahul
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  There is no contradiction. It's simply true that a sphere of appropriate size to touch the other spheres that just touch the box from the inside will have some points outside the box in a high enough dimension.
  – Mark S.
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is a consequence of the pythagorean theorem extended to higher dimensions. For a 100-dimensional hypercube, a diagonal is ten times the length of any edge; but the hyperspheres you're packing in (excluding this weird center one) stay the same radius in every dimension if your cube's edge length stays the same. So it shouldn't be hard to imagine that the distance between a sphere in one "corner" gets farther from the one in the opposite corner of the cube as the dimension goes up.
  – Malice Vidrine
  34 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I guess they explained it poorly towards the end, talking about "spiky spheres"...
  – Lorry Laurence mcLarry
  19 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  That is a bit of a weird description. If anything my intuition has always been that it's cubes that get "spiky".
  – Malice Vidrine
  10 mins ago
  

  
  
  
  

add a comment | 

up vote
3
down vote

favorite

up vote
3
down vote

favorite

According to this Numberfile video, if you tightly pack hyper-dimensional Spheres into a hyper-dimensional box and then find the radius of the largest sphere that could possibly fit in the remaining space, the resulting sphere would somehow exceed the confines of the box that contained all of the spheres. (where the number of dimensions are greater or equal to 10)

Isn't a logical contradiction generally considered a disproof of something?

Wouldn't this disprove the generalised formula being used to find the radius of the resulting sphere on n dimensions?

Is it possible that mathematicians simply do not understand extra dimensional geometry and its inherent rules?

logic axioms dimension-theory

share|cite|improve this question

edited 21 mins ago

asked 2 hours ago

Lorry Laurence mcLarry

4641511

According to this Numberfile video, if you tightly pack hyper-dimensional Spheres into a hyper-dimensional box and then find the radius of the largest sphere that could possibly fit in the remaining space, the resulting sphere would somehow exceed the confines of the box that contained all of the spheres. (where the number of dimensions are greater or equal to 10)

Isn't a logical contradiction generally considered a disproof of something?

Wouldn't this disprove the generalised formula being used to find the radius of the resulting sphere on n dimensions?

Is it possible that mathematicians simply do not understand extra dimensional geometry and its inherent rules?

logic axioms dimension-theory

logic axioms dimension-theory

share|cite|improve this question

edited 21 mins ago

asked 2 hours ago

Lorry Laurence mcLarry

4641511

share|cite|improve this question

edited 21 mins ago

asked 2 hours ago

Lorry Laurence mcLarry

4641511

share|cite|improve this question

share|cite|improve this question

edited 21 mins ago

edited 21 mins ago

edited 21 mins ago

asked 2 hours ago

Lorry Laurence mcLarry

4641511

asked 2 hours ago

Lorry Laurence mcLarry

4641511

asked 2 hours ago

Lorry Laurence mcLarry

4641511

4641511

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sorry, where is the logical contradiction here?
  – Rahul
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  There is no contradiction. It's simply true that a sphere of appropriate size to touch the other spheres that just touch the box from the inside will have some points outside the box in a high enough dimension.
  – Mark S.
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is a consequence of the pythagorean theorem extended to higher dimensions. For a 100-dimensional hypercube, a diagonal is ten times the length of any edge; but the hyperspheres you're packing in (excluding this weird center one) stay the same radius in every dimension if your cube's edge length stays the same. So it shouldn't be hard to imagine that the distance between a sphere in one "corner" gets farther from the one in the opposite corner of the cube as the dimension goes up.
  – Malice Vidrine
  34 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I guess they explained it poorly towards the end, talking about "spiky spheres"...
  – Lorry Laurence mcLarry
  19 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  That is a bit of a weird description. If anything my intuition has always been that it's cubes that get "spiky".
  – Malice Vidrine
  10 mins ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Sorry, where is the logical contradiction here?
  – Rahul
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  There is no contradiction. It's simply true that a sphere of appropriate size to touch the other spheres that just touch the box from the inside will have some points outside the box in a high enough dimension.
  – Mark S.
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is a consequence of the pythagorean theorem extended to higher dimensions. For a 100-dimensional hypercube, a diagonal is ten times the length of any edge; but the hyperspheres you're packing in (excluding this weird center one) stay the same radius in every dimension if your cube's edge length stays the same. So it shouldn't be hard to imagine that the distance between a sphere in one "corner" gets farther from the one in the opposite corner of the cube as the dimension goes up.
  – Malice Vidrine
  34 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I guess they explained it poorly towards the end, talking about "spiky spheres"...
  – Lorry Laurence mcLarry
  19 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  That is a bit of a weird description. If anything my intuition has always been that it's cubes that get "spiky".
  – Malice Vidrine
  10 mins ago
  

  
  
  
  

1

1

Sorry, where is the logical contradiction here?
– Rahul
2 hours ago

Sorry, where is the logical contradiction here?
– Rahul
2 hours ago

There is no contradiction. It's simply true that a sphere of appropriate size to touch the other spheres that just touch the box from the inside will have some points outside the box in a high enough dimension.
– Mark S.
1 hour ago

There is no contradiction. It's simply true that a sphere of appropriate size to touch the other spheres that just touch the box from the inside will have some points outside the box in a high enough dimension.
– Mark S.
1 hour ago

This is a consequence of the pythagorean theorem extended to higher dimensions. For a 100-dimensional hypercube, a diagonal is ten times the length of any edge; but the hyperspheres you're packing in (excluding this weird center one) stay the same radius in every dimension if your cube's edge length stays the same. So it shouldn't be hard to imagine that the distance between a sphere in one "corner" gets farther from the one in the opposite corner of the cube as the dimension goes up.
– Malice Vidrine
34 mins ago

This is a consequence of the pythagorean theorem extended to higher dimensions. For a 100-dimensional hypercube, a diagonal is ten times the length of any edge; but the hyperspheres you're packing in (excluding this weird center one) stay the same radius in every dimension if your cube's edge length stays the same. So it shouldn't be hard to imagine that the distance between a sphere in one "corner" gets farther from the one in the opposite corner of the cube as the dimension goes up.
– Malice Vidrine
34 mins ago

I guess they explained it poorly towards the end, talking about "spiky spheres"...
– Lorry Laurence mcLarry
19 mins ago

I guess they explained it poorly towards the end, talking about "spiky spheres"...
– Lorry Laurence mcLarry
19 mins ago

That is a bit of a weird description. If anything my intuition has always been that it's cubes that get "spiky".
– Malice Vidrine
10 mins ago

That is a bit of a weird description. If anything my intuition has always been that it's cubes that get "spiky".
– Malice Vidrine
10 mins ago

add a comment | 

2 Answers
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4
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No. It just means that (hyper)cubic lattice sphere packing (where the centers of the spheres are placed in a cubic grid, say spheres with radius $frac12$ centered at each point with integer cartesian coordinates) is very inefficient in higher dimensions, and the room between the spheres become large enough to fit even larger spheres.

Counterintuitive? Yes, but mostly because we are relatively low-dimensional beings with limited imagination. Paradox or contradiction? No.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Arthur

102k797178

add a comment | 

up vote
3
down vote

This is not a logical contradiction, only a counterintuitive result.

share|cite|improve this answer

answered 1 hour ago

Santana Afton

2,2041426

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote

No. It just means that (hyper)cubic lattice sphere packing (where the centers of the spheres are placed in a cubic grid, say spheres with radius $frac12$ centered at each point with integer cartesian coordinates) is very inefficient in higher dimensions, and the room between the spheres become large enough to fit even larger spheres.

Counterintuitive? Yes, but mostly because we are relatively low-dimensional beings with limited imagination. Paradox or contradiction? No.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Arthur

102k797178

add a comment | 

up vote
4
down vote

No. It just means that (hyper)cubic lattice sphere packing (where the centers of the spheres are placed in a cubic grid, say spheres with radius $frac12$ centered at each point with integer cartesian coordinates) is very inefficient in higher dimensions, and the room between the spheres become large enough to fit even larger spheres.

Counterintuitive? Yes, but mostly because we are relatively low-dimensional beings with limited imagination. Paradox or contradiction? No.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Arthur

102k797178

add a comment | 

up vote
4
down vote

up vote
4
down vote

No. It just means that (hyper)cubic lattice sphere packing (where the centers of the spheres are placed in a cubic grid, say spheres with radius $frac12$ centered at each point with integer cartesian coordinates) is very inefficient in higher dimensions, and the room between the spheres become large enough to fit even larger spheres.

Counterintuitive? Yes, but mostly because we are relatively low-dimensional beings with limited imagination. Paradox or contradiction? No.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Arthur

102k797178

No. It just means that (hyper)cubic lattice sphere packing (where the centers of the spheres are placed in a cubic grid, say spheres with radius $frac12$ centered at each point with integer cartesian coordinates) is very inefficient in higher dimensions, and the room between the spheres become large enough to fit even larger spheres.

Counterintuitive? Yes, but mostly because we are relatively low-dimensional beings with limited imagination. Paradox or contradiction? No.

share|cite|improve this answer

edited 1 hour ago

answered 2 hours ago

Arthur

102k797178

share|cite|improve this answer

share|cite|improve this answer

edited 1 hour ago

edited 1 hour ago

edited 1 hour ago

answered 2 hours ago

Arthur

102k797178

answered 2 hours ago

Arthur

102k797178

answered 2 hours ago

Arthur

102k797178

102k797178

add a comment | 

add a comment | 

up vote
3
down vote

This is not a logical contradiction, only a counterintuitive result.

share|cite|improve this answer

answered 1 hour ago

Santana Afton

2,2041426

add a comment | 

up vote
3
down vote

This is not a logical contradiction, only a counterintuitive result.

share|cite|improve this answer

answered 1 hour ago

Santana Afton

2,2041426

add a comment | 

up vote
3
down vote

up vote
3
down vote

This is not a logical contradiction, only a counterintuitive result.

share|cite|improve this answer

answered 1 hour ago

Santana Afton

2,2041426

This is not a logical contradiction, only a counterintuitive result.

share|cite|improve this answer

answered 1 hour ago

Santana Afton

2,2041426

share|cite|improve this answer

share|cite|improve this answer

answered 1 hour ago

Santana Afton

2,2041426

answered 1 hour ago

Santana Afton

2,2041426

answered 1 hour ago

Santana Afton

2,2041426

2,2041426

add a comment | 

add a comment | 

 

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