## What is Rotational Symmetry?

A Rotation is a transformation of a shape around a central point.

Rotational Symmetry is when the rotated shape looks the same as in the original position.

In the diagram here, the point of rotation is where the lines meet at the centre. The shape that’s rotated is the line with a point.

Rotation symmetry leaves the object with the same shape but turned by a certain amount.

In the series of diagrams below, one line is highlighted red to demonstrate the shape being rotated. You can see that with each new position the shape remains the same as the original.

## What is the Order of Rotational Symmetry?

The **order** of rotational symmetry is the number of times the shape is the same as the original through 360 degrees. In all the diagrams we have looked at so far the order of rotation has been 5. The diagrams have all shown that the original can be turned 4 times to match the original. Making a total of 5 positions within 360 degrees which are all the same.

The diagrams so far have all been of a form of the pentagonal archetype. An archetype is a **shape that expresses the symmetry of that number**.

‘What is Rotational Symmetry?’ Factsheet and Rotational Symmetry Hunt sheet – are Free downloads available in the resource library

## What is the order of rotational symmetry of a regular pentagon?

A regular pentagon can be treated the same as the diagrams above. It can be rotated at its centre point 5 times (which includes the original). Each time showing a shape the same as the original. Therefore the order of rotational symmetry of a regular pentagon is 5.

In fact, any manifestation of the regular pentagonal symmetry will have an order of 5.

This 5 point star displays the same order of rotational symmetry as our other shapes so far. The 5 point star features in nature and when we see it we be able to see the rotational symmetry of the pentagon present in natural form.

## How many orders of rotational symmetry does a pentagon have?

This all depends on whether it is a **regular** pentagon or not.

## How does Rotational Symmetry work in nature?

So much of the symmetry we see in nature we take for granted we almost don’t even realise we are seeing it, that’s how used to it we are. But it is there and we can learn to see it anew.

What is important to remember here is that we are observing nature’s patterns and our understanding of these patterns is from a very maths centric point of view. Nature doesn’t actually rotate to produce these – at least not in the sense we are talking about here. What we see in nature is a manifestation of the phase appearance of a rotation.

In nature we see rotational symmetry mostly in the plant kingdom. We also see it in the animal kingdom in marine life.

Rotational symmetry in nature presents to us in two forms: Asymmetrical rotation and combined rotational and bilateral symmetries

## Rotational Symmetry in marine life

This starfish demonstrates to us rotational symmetry of order 5 combined with bilateral symmetry, the axis of which is down the centre of each of the starfish arms.

This starfish image we treated the same as the regular pentagonal shapes at the beginning of this post.

There are examples in flowers that demonstrate to us this combined rotational and bilateral symmetry. When the symmetry within each petal is also bilateral – each petal has a central axis and displays bilateral (reflective) symmetry. These are examples of the combined symmetries of Rotational and bilateral.

## Rotational Symmetry of a pentagon within the face of a flower

Just three examples of flowers that display rotational symmetry in the order of 5 and also bilateral symmetry, shown here from left to right are: primrose, hibiscus, vinca.

By examining the face of flowers with the rotational order of 5 we can come to understand the symmetry present within the 5 archetype. There is an experience of the infinite nature of the five archetype that we can relate to when we work with nature in this way. And understand our relationship to it. Spend calm moments examining and threading the patterns onto the flower in these Stitch Upon a Star kits.

There are loads of examples of both combined rotational and asymmetrical symmetry in the plant world. We can immerse ourselves in the simple beauty of rotational symmetry in the faces of so many flowers.

#### Don’t forget Circles.

When talking of rotations, you really can’t avoid mentioning circles being the form that the rotation takes. And circles in nature come in the form of spheres, slices, rotations and radials. Read more on a collection of circles in natural forms.

## Rotational symmetry in Nature: Asymmetrical.

Every flower form that has equally spaced petals or sepals around a centre displays rotational symmetry. This diagram has rotational symmetry of order five, it just isn’t combined with bilateral symmetry. Each shape that is rotated around the centre is asymmetrical in itself. This doesn’t alter the rotational symmetry and its order.

This cyclamen flower displays approx rotational symmetry around its centre. Unlike the primrose or vinca it doesn’t have bilateral symmetry in its petals and therefore the face of the flower doesn’t display the combined symmetries

Other petalled flowers give us opportunity to embrace other orders of rotational symmetries. We can examine nature and read the varying orders of rotational symmetry. Often all we need to do is count the petals.

Through The Smart Happy Project Lisa communicates to her followers a voice of natural connection in a fast paced world. Following in the footsteps of philosophers and geometers of all ages Lisa embarks on highlighting our journey as humans in a natural world governed by patterns we can see and understand.