I need to know how to find the contents of a sphone; however I have not been able to find an equation for it online. I noted that the equation for a cone is 1/3(h)(A base). So I thought that perhaps the formula for a sphone could be the volume of its base * h * 1/3 since a sphone is a continuous series of spheres terminating to a point similar to the continuous circles terminating to a point in a cone.
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3$\begingroup$ Can you be more precise what a sphone is? It seems to be unknown to Google. $\endgroup$– Ben McKayCommented Mar 15, 2020 at 13:57
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$\begingroup$ Also, please indicate more clearly why this is a research level mathematics problem, or move the question to math.stackexchange.com. $\endgroup$– Ben McKayCommented Mar 15, 2020 at 13:57
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$\begingroup$ Thanks Carlo! So... this was my first question, what qualifies as a research level mathematics problem? I didn't realize that there was a place called math.stackexchange.com. How do I move this there? $\endgroup$– OcéaneCommented Mar 15, 2020 at 14:08
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1 Answer
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The sphone is a 4-dimensional generalization of a cone (height $h$), where the base is a sphere (radius $r$). It is one of a collection of 4-dimensional objects, see this overview. The surface equation is $$|(x_1^2+x_2^2+x_3^2)^{1/2}+(r/h)x_4|+(x_1^2+x_2^2+x_3^2)^{1/2}=r,$$ the volume is $\frac{1}{3}\pi r^3 h$.
answered Mar 15, 2020 at 14:13
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$\begingroup$ Wait, that website says that the surcell volume is 4/3 * pi * r^3 * h $\endgroup$– OcéaneCommented Mar 15, 2020 at 23:16
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$\begingroup$ the 4D bulk volume is $\frac{1}{3}\pi r^3h$, the 3D surcell volume is $\frac{4}{3}\pi( r^3 +r^2\sqrt{r^2+h^2})$. $\endgroup$ Commented Mar 16, 2020 at 8:14