Formula Region Projections 

A formula region representing a solid in .

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X \[ScriptCapitalR] = ImplicitRegion[ x^6 - 5 x^4 y z + 3 x^4 y^2 + 10 x^2 y^3 z + 3 x^2 y^4 - y^5 z + y^6 + z^6 <= 1, {x, y, z}];

Visualize the region.

In[2]:=

X RegionPlot3D[\[ScriptCapitalR], PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}, {-1.6, 1.6}}, PlotPoints -> 50]

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Find projections of ℛ on the coordinate planes.

In[3]:=

X {RegionPlot[Resolve[\!\( \*SubscriptBox[\(\[Exists]\), \(z\)]\({x, y, z} \[Element] \[ScriptCapitalR]\)\), Reals], {x, -1.5, 1.5}, {y, -1.5, 1.5}], RegionPlot[Resolve[\!\( \*SubscriptBox[\(\[Exists]\), \(y\)]\({x, y, z} \[Element] \[ScriptCapitalR]\)\), Reals], {x, -1.5, 1.5}, {z, -1.5, 1.5}], RegionPlot[Resolve[\!\( \*SubscriptBox[\(\[Exists]\), \(x\)]\({x, y, z} \[Element] \[ScriptCapitalR]\)\), Reals], {y, -1.5, 1.5}, {z, -1.5, 1.5}]}

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