kocka= Graphics3D[{Orange, Opacity[0.5], Polygon[{{{1,1,1},{1,1,-1}, {1,-1,-1}, {1,-1,1}}, {{1,1,1},{1,1,-1}, {-1,1,-1}, {-1,1,1}}, {{1,1,1},{1,-1,1}, {-1,-1,1}, {-1,1,1}}, {{-1,-1,-1},{-1,-1,1}, {-1,1,1}, {-1,1,-1}}, {{-1,-1,-1},{-1,-1,1}, {1,-1,1}, {1,-1,-1}}, {{-1,-1,-1},{-1,1,-1}, {1,1,-1}, {1,-1,-1}}}]}];

egyenes=ParametricPlot3D[{p,p,p}, {p,-1,3}];

sik=Plot3D[4-x-y, {x,-4,4}, {y,-4,4}, Mesh->None, PlotStyle->Directive[Gray, Opacity[0.5]], PlotRange->{{-4,4},{-4,4},{-4,4}}, BoxRatios->{1,1,1}, ClippingStyle->None];

Show[sik, egyenes, kocka,  Boxed->False,BoxRatios->{1, 1, 1},Axes-> False, SphericalRegion->True, Lighting->{{"Ambient", White}}]

hatszog=Graphics3D[{Orange, Opacity[0.8], Polygon[{{0,2,2}, {2/3,8/3,2/3}, {2,2,0}, {8/3,2/3,2/3}, {2,0,2}, {2/3,2/3,8/3}}]}];

Show[sik, egyenes, kocka, hatszog,  Boxed->False,BoxRatios->{1, 1, 1},Axes-> False, SphericalRegion->True, Lighting->{{"Ambient", White}}]

vonalak=Graphics3D[{Line[{{4/3,4/3,4/3},{0,2,2}}], Line[{{0,2,2},{-1,1,1}}], Line[{{1,1,1}, {-1/3,5/3,5/3}}]}];

Show[sik, vonalak,egyenes, kocka, hatszog,  Boxed->False,BoxRatios->{1, 1, 1},Axes-> False, SphericalRegion->True, Lighting->{{"Ambient", White}}]

kockalist={{1,1,1},{-1,1,1},{-1,1,-1},{1,1,-1},{1,-1,1},{-1,-1,1},{-1,-1,-1},{1,-1,-1}};

kocka2list=kockalist.RotationMatrix[Pi/4, {0,0,1}].RotationMatrix[ArcSin[1/Sqrt[3]], {1,1,0}];

kocka1= Graphics3D[{Orange, Opacity[0.8], Polygon[{{{1,1,1},{1,1,-1}, {1,-1,-1}, {1,-1,1}}, {{1,1,1},{1,1,-1}, {-1,1,-1}, {-1,1,1}}, {{1,1,1},{1,-1,1}, {-1,-1,1}, {-1,1,1}}, {{-1,-1,-1},{-1,-1,1}, {-1,1,1}, {-1,1,-1}}, {{-1,-1,-1},{-1,-1,1}, {1,-1,1}, {1,-1,-1}}, {{-1,-1,-1},{-1,1,-1}, {1,1,-1}, {1,-1,-1}}}]}];

kocka2=Graphics3D[{Purple, Opacity[0.5], Polygon[{{kocka2list[[1]],kocka2list[[2]], kocka2list[[3]], kocka2list[[4]]}, {kocka2list[[5]],kocka2list[[6]],kocka2list[[7]], kocka2list[[8]]}, {kocka2list[[1]],kocka2list[[4]], kocka2list[[8]],kocka2list[[5]]}, {kocka2list[[1]],kocka2list[[2]], kocka2list[[6]], kocka2list[[5]]}, {kocka2list[[2]],kocka2list[[3]], kocka2list[[7]], kocka2list[[6]]}, {kocka2list[[3]],kocka2list[[4]], kocka2list[[8]], kocka2list[[7]]}}]}];

Show[kocka1, kocka2, Boxed->False,Axes-> False, SphericalRegion->True, Lighting->{{"Ambient", White}}]

movelist[p_]=kocka2list+{{p,-p,p},{p,-p,p},{p,-p,p},{p,-p,p},{p,-p,p},{p,-p,p},{p,-p,p},{p,-p,p}};

movekocka[p_]=Graphics3D[{Purple, Opacity[0.5], Polygon[{{movelist[p][[1]],movelist[p][[2]], movelist[p][[3]], movelist[p][[4]]}, {movelist[p][[5]],movelist[p][[6]],movelist[p][[7]], movelist[p][[8]]}, {movelist[p][[1]],movelist[p][[4]], movelist[p][[8]],movelist[p][[5]]}, {movelist[p][[1]],movelist[p][[2]], movelist[p][[6]], movelist[p][[5]]}, {movelist[p][[2]],movelist[p][[3]], movelist[p][[7]], movelist[p][[6]]}, {movelist[p][[3]],movelist[p][[4]], movelist[p][[8]], movelist[p][[7]]}}]}];

Animate[Show[kocka1, movekocka[p], PlotRange->{{-3,3},{-3,3},{-3,3}}, Boxed-> False], {p,-3,3}]