Question

Question: Which of the following has 3-fold rotational symmetry? Explain.

(a) An isosceles triangle

(b) An equilateral triangle

(c) A tetrahedron

(d) A cube

Short answer

(a) An isosceles triangle does not have a 3-fold rotational symmetry.

(b) An equilateral triangle has 3-fold rotational symmetry.

(c) A tetrahedron has 3-fold rotational symmetry.

(d) A cube has 3-fold rotational symmetry.

Step-by-step solution

Subpart (a) An isosceles triangle.

When rotating the object brings the same picture as visible from the front, then the object is said to have rotational symmetry. Many objects have more than one type of rotational symmetry.

An isosceles triangle has only two sides equal and can be rotated with C2 symmetry.

As it repeats itself only when rotated with ${180}^{\circ }$, therefore it does not have 3-fold rotational symmetry.

Subpart (b) An equilateral triangle.

Passing the C-2 or C-3 axis or rotating the equilateral triangle two times or three times brings the same picture.

As it repeats itself only when rotated with ${180}^{\circ }$or ${120}^{\circ }$, therefore, this triangle is said to have a three-fold rotational symmetry.

Subpart (c) A tetrahedron.

A tetrahedron has both 2-fold and 3-fold rotational symmetry.

The 2-fold rotational symmetry can be found three times and the3-fold rotational symmetry can be found four times.

The symmetry of 3-fold can be passed by making the base of one side of the tetrahedron and rotating it with ${120}^{\circ }$.

This rotation can be performed four times as shown below.

Subpart (d) A cube.

A cube carries, 2-fold symmetry (${180}^{\circ }$rotation),3-fold symmetry (${120}^{\circ }$rotation) and 4-fold symmetry (${90}^{\circ }$rotation).

The number of 3-fold symmetry is four as the axes pass through two opposite vertices four times as shown below.