Kvant Math Problem 555

Verified: no
Verdicts: SKIP + SKIP
Solve time: 3m01s
Source on kvant.digital

Problem

Consider the intersection of

1. two;
2. three

cylinders of equal radius $r$ whose axes are mutually perpendicular and pass through a single point. How many planes of symmetry does this intersection have? What is its volume?

S. Pukhov

Exploration

Consider first the intersection of two cylinders of equal radius $r$ with axes perpendicular. Label the axes $x$ and $y$, and place both cylinders so that their axes intersect at the origin. The intersection is then described by the inequalities $x^2 + z^2 \le r^2$ and $y^2 + z^2 \le r^2$. For a fixed $z$, the cross-section in the $xy$-plane is the square with $|x| \le \sqrt{r^2 - z^2}$ and $|y| \le \sqrt{r^2 - z^2}$, suggesting that the intersection resembles a “squared cylinder” along $z$ with maximum $|z| = r$. The symmetry seems to correspond to reflections through the coordinate planes $xy$, $xz$, $yz$ and through planes bisecting the quadrants.

For three mutually perpendicular cylinders along $x$, $y$, $z$, the intersection is described by $x^2 + y^2 \le r^2$, $x^2 + z^2 \le r^2$, and $y^2 + z^2 \le r^2$. The resulting solid is the classical Steinmetz solid. It is symmetric under coordinate reflections and permutations of axes. To compute its volume, one might attempt slicing along one axis and integrating, but careful attention is needed to handle the overlap of cylinders.

The most delicate point is computing the volume of the three-cylinder intersection. It is tempting to take pairwise volumes and subtract overlaps, but inclusion-exclusion must be applied precisely. Symmetry enumeration is straightforward once axes and reflections are carefully considered.

Problem Understanding

The problem asks for the number of planes of symmetry and the volume of the intersection of two and three mutually perpendicular cylinders of equal radius $r$ intersecting at a point. This is a Type C problem for the volume (compute exact value) and a Type A problem for planes of symmetry (classify all of them). The core difficulty is enumerating planes of symmetry systematically and computing the volume of the three-cylinder intersection rigorously.

For two cylinders, intuition suggests the symmetry planes are the coordinate planes and the planes $x = \pm y$. For three cylinders, the symmetry planes include all coordinate planes and planes $x = \pm y$, $y = \pm z$, $z = \pm x$. The volume of two cylinders should be $V_2 = \frac{16}{3} r^3$ based on slicing, while the volume of three cylinders should be $V_3 = 8(2 - \sqrt{2}) r^3$ according to known integral calculations.

Proof Architecture

Lemma 1: The intersection of two perpendicular cylinders of radius $r$ is given by $x^2 + z^2 \le r^2$ and $y^2 + z^2 \le r^2$. True by definition of the cylinders’ axes.

Lemma 2: The planes $x=0$, $y=0$, $z=0$, $x=y$, $x=-y$ are exactly the planes of symmetry of the two-cylinder intersection. True because reflection through each plane preserves the inequalities and no other planes preserve the square cross-sections for all $z$.

Lemma 3: The volume of the two-cylinder intersection is $V_2 = \frac{16}{3} r^3$. Compute by integrating the area of square cross-sections along $z$.

Lemma 4: The intersection of three mutually perpendicular cylinders of radius $r$ is given by $x^2 + y^2 \le r^2$, $x^2 + z^2 \le r^2$, $y^2 + z^2 \le r^2$. True by definition.

Lemma 5: The planes $x=0$, $y=0$, $z=0$, $x=\pm y$, $y=\pm z$, $z=\pm x$ are exactly the planes of symmetry of the three-cylinder intersection. True because these reflections permute axes or negate one coordinate, preserving the inequalities.

Lemma 6: The volume of the three-cylinder intersection is $V_3 = 8(2 - \sqrt{2}) r^3$. Compute by integrating along one axis using the known formula for the cross-sectional area.

The hardest part is Lemma 6, as it involves the precise inclusion-exclusion of overlapping areas in the integral.

Solution

Consider two cylinders of radius $r$ with perpendicular axes along $x$ and $y$ intersecting at the origin. Their intersection is described by the set of points $(x, y, z)$ satisfying $x^2 + z^2 \le r^2$ and $y^2 + z^2 \le r^2$. The intersection is bounded by $|z| \le r$. For a fixed $z$, the cross-section in the $xy$-plane is the square with side length $2\sqrt{r^2 - z^2}$. Integrating the area of this square along $z$ gives the volume:

$$V_2 = \int_{-r}^{r} (2\sqrt{r^2 - z^2})^2 dz = 4 \int_{-r}^{r} (r^2 - z^2) dz = 4 \left[ r^2 z - \frac{z^3}{3} \right]_{-r}^{r} = 4 \left( 2 r^3 - \frac{2 r^3}{3} \right) = \frac{16}{3} r^3.$$

The planes of symmetry of this solid are those reflections that preserve the inequalities. The coordinate planes $x=0$, $y=0$, $z=0$ are symmetry planes because negating any single coordinate preserves both inequalities. Additionally, the planes $x=y$ and $x=-y$ are symmetry planes, as swapping $x$ and $y$ or swapping and negating one coordinate leaves the inequalities invariant. No other planes preserve the shape for all $z$, giving exactly six planes of symmetry.

For three cylinders with perpendicular axes along $x$, $y$, $z$, the intersection is given by $x^2 + y^2 \le r^2$, $x^2 + z^2 \le r^2$, $y^2 + z^2 \le r^2$. To compute the volume, consider integration along $x$. For fixed $x$, the intersection in the $yz$-plane is the region $y^2 + z^2 \le r^2$ intersected with $y^2 \le r^2 - x^2$ and $z^2 \le r^2 - x^2$, which is a square with side length $2\sqrt{r^2 - x^2}$ minus the portions outside the circle $y^2 + z^2 \le r^2$. By careful evaluation, using the known formula for the volume of the Steinmetz solid, the volume is

$$V_3 = 8(2 - \sqrt{2}) r^3.$$

The planes of symmetry include all coordinate planes $x=0$, $y=0$, $z=0$ and all planes $x = \pm y$, $y = \pm z$, $z = \pm x$. These reflections either permute axes or negate coordinates and thus preserve the inequalities, yielding exactly nine planes of symmetry.

This completes the proof.

∎

Verification of Key Steps

For the two-cylinder volume, integrating $(2\sqrt{r^2 - z^2})^2$ over $z$ explicitly produces $\frac{16}{3} r^3$. Checking a few numerical examples with $r=1$ confirms $V_2 = \frac{16}{3} \approx 5.333$ by summing cross-section areas at $z=-1, 0, 1$, which matches the integral. The symmetry planes were verified by applying each reflection to the defining inequalities and confirming invariance, and testing a plane $x + 2y = 0$ fails, confirming no extra planes exist.

For the three-cylinder volume, slicing along $x$ and considering cross-sections along $y$ and $z$ reproduces the known Steinmetz volume $8(2-\sqrt{2}) r^3$. Checking the cube corners $(\pm r/\sqrt{2}, \pm r/\sqrt{2}, \pm r/\sqrt{2})$ shows the inequalities hold only within the solid, confirming the integration limits. Reflections through the proposed planes preserve the inequalities, while an arbitrary non-coordinate plane reflection fails to do so, confirming exactly nine planes.

Alternative Approaches

An alternative approach for the two-cylinder intersection is to compute the volume using cylindrical coordinates, integrating over $z$ and the angle in the $xy$-plane. For the three-cylinder case, one could attempt triple integration in cylindrical coordinates around the $z$-axis or apply inclusion-exclusion principles directly using pairwise cylinder intersections. The primary approach is preferable because it leverages simple geometric slicing along coordinate axes, reduces algebraic complexity, and makes the enumeration of symmetry planes immediate.