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J. M.'s missing motivation ♦
J. M.'s missing motivation ♦
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user64494
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How can weto exactly calculate the volume?

Fixed the question formation - missing auxiliary (or helping) verb - see e.g. <https://www.youtube.com/watch?v=t4yWEt0OSpg&t=1m49s> (see also <https://www.youtube.com/watch?v=kS5NfSzXfrI> (QUASM)) - alternatively, drop the question mark (title only).
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edit approved Feb 10, 2023 at 1:23
Peter Mortensen
edit approved Feb 10, 2023 at 1:23
Peter Mortensen
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How tocan we exactly calculate the volume?

reg = ImplicitRegion[(2 x/3 + y/3 + z/3)^2 + (2 y/3 + x/3 - z/3)^2 + (2 z/3 + x/3 - y/3)^2 <= 1 &&
(2 x/3 - y/3 - z/3)^2 + (2 y/3 - x/3 - z/3)^2 + (2 z/3 - x/3 - y/3)^2 <=  1 && 
(2 x/3 + y/3 - z/3)^2 + (2 y/3 + x/3 + z/3)^2 + (2 z/3 - x/3 + y/3)^2 <= 1 && 
(2 x/3 - y/3 + z/3)^2 + (2 y/3 - x/3 + z/3)^2 + (2 z/3 +  x/3 + y/3)^2 <= 1 && x^2 + y^2 <= 1, {x, y, z}];


RegionPlot3D[(2 x/3 + y/3 + z/3)^2 + (2 y/3 + x/3 - z/3)^2 + (2 z/3 + 
   x/3 - y/3)^2 <=   1 && (2 x/3 - y/3 - z/3)^2 + (2 y/3 - x/3 - z/3)^2 + (2 z/3 - x/3 - y/3)^2 <=  1 &&
 (2 x/3 + y/3 - z/3)^2 + (2 y/3 + x/3 + z/3)^2 + (2 z/3 - x/3 +
    y/3)^2 <=  1 && (2 x/3 - y/3 + z/3)^2 + (2 y/3 - x/3 + z/3)^2 + (2 z/3 + x/3 +
    y/3)^2 <= 1 && x^2 + y^2 <= 1, {x, -3/2, 3/2}, {y, -3/2, 3/2}, {z, -3/2, 3/2}, PlotPoints -> 50]

enter image description hereEnter image description here

It is very probablyprobable that the exact result equals 22/5. Just for sportive interest, how tocan I prove or disprove it with Mathematica? I don't find the answer here.

How to exactly calculate the volume?

reg = ImplicitRegion[(2 x/3 + y/3 + z/3)^2 + (2 y/3 + x/3 - z/3)^2 + (2 z/3 + x/3 - y/3)^2 <= 1 &&
(2 x/3 - y/3 - z/3)^2 + (2 y/3 - x/3 - z/3)^2 + (2 z/3 - x/3 - y/3)^2 <=  1 && 
(2 x/3 + y/3 - z/3)^2 + (2 y/3 + x/3 + z/3)^2 + (2 z/3 - x/3 + y/3)^2 <= 1 && 
(2 x/3 - y/3 + z/3)^2 + (2 y/3 - x/3 + z/3)^2 + (2 z/3 +  x/3 + y/3)^2 <= 1 && x^2 + y^2 <= 1, {x, y, z}];


RegionPlot3D[(2 x/3 + y/3 + z/3)^2 + (2 y/3 + x/3 - z/3)^2 + (2 z/3 + 
   x/3 - y/3)^2 <=   1 && (2 x/3 - y/3 - z/3)^2 + (2 y/3 - x/3 - z/3)^2 + (2 z/3 - x/3 - y/3)^2 <=  1 &&
 (2 x/3 + y/3 - z/3)^2 + (2 y/3 + x/3 + z/3)^2 + (2 z/3 - x/3 +
    y/3)^2 <=  1 && (2 x/3 - y/3 + z/3)^2 + (2 y/3 - x/3 + z/3)^2 + (2 z/3 + x/3 +
    y/3)^2 <= 1 && x^2 + y^2 <= 1, {x, -3/2, 3/2}, {y, -3/2, 3/2}, {z, -3/2, 3/2}, PlotPoints -> 50]

enter image description here

It is very probably that the exact result equals 22/5. Just for sportive interest, how to prove or disprove it with Mathematica? I don't find the answer here.

How can we exactly calculate the volume?

reg = ImplicitRegion[(2 x/3 + y/3 + z/3)^2 + (2 y/3 + x/3 - z/3)^2 + (2 z/3 + x/3 - y/3)^2 <= 1 &&
(2 x/3 - y/3 - z/3)^2 + (2 y/3 - x/3 - z/3)^2 + (2 z/3 - x/3 - y/3)^2 <=  1 &&
(2 x/3 + y/3 - z/3)^2 + (2 y/3 + x/3 + z/3)^2 + (2 z/3 - x/3 + y/3)^2 <= 1 &&
(2 x/3 - y/3 + z/3)^2 + (2 y/3 - x/3 + z/3)^2 + (2 z/3 +  x/3 + y/3)^2 <= 1 && x^2 + y^2 <= 1, {x, y, z}];


RegionPlot3D[(2 x/3 + y/3 + z/3)^2 + (2 y/3 + x/3 - z/3)^2 + (2 z/3 +
   x/3 - y/3)^2 <=   1 && (2 x/3 - y/3 - z/3)^2 + (2 y/3 - x/3 - z/3)^2 + (2 z/3 - x/3 - y/3)^2 <=  1 &&
 (2 x/3 + y/3 - z/3)^2 + (2 y/3 + x/3 + z/3)^2 + (2 z/3 - x/3 +
    y/3)^2 <=  1 && (2 x/3 - y/3 + z/3)^2 + (2 y/3 - x/3 + z/3)^2 + (2 z/3 + x/3 +
    y/3)^2 <= 1 && x^2 + y^2 <= 1, {x, -3/2, 3/2}, {y, -3/2, 3/2}, {z, -3/2, 3/2}, PlotPoints -> 50]

Enter image description here

It is very probable that the exact result equals 22/5. Just for sportive interest, how can I prove or disprove it with Mathematica? I don't find the answer here.

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user64494
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