20 Degrees of Separation

I recently had a ‘light bulb moment’, not about light bulbs, but about toilet paper. [Now there’s a sentence that I’m willing to bet has never been written before!]

Before I explain my idea, it is important to clearly state one major assumption that this blog post relies entirely upon. When toilet paper is used, two sheets are stripped from the roll and folded across the perforations in the centre. This has already ruled out any interest from one of my daughters who, after wondering why my toilet paper expenses were so high, I discovered wiped herself by taking about 12 sheets and wrapping it around her entire fist for fear of having to touch any of her own undesirable matter.

This is a diagram, to scale, of two sheets of toilet paper, the black line down the middle indicating where the perforations are.

Paper 0 degrees

When folded evenly, I can tell you that the length is 12.3 cm and the width is 10.2 cm, resulting in a total surface area of 125.46 square cm. It occurred to me that it is not necessary that the whole of that surface area needs to be a double-sheet thickness, but you still need that security of the bulk of the area being thicker. Can you tell I’m desperately trying not to be crude in my descriptions here?

Here is my idea. Instead of creating those perforations perpendicular to the paper edge, why not cut them at an angle? 20 degrees from that perpendicular seems perfect to me, as per the diagram below, again drawn to scale. The cutting process should alternate between a 90-degree perpendicular cut and a 20-degree cut (this is actually 70 degrees from the paper edge).

Paper 20 degrees

This is what the shape looks like then the paper is folded using the angled perforations.

Paper 20 degrees folded

The dark blue area is double thickness, and the pale blue sections represent single sheet thickness. If you hold the paper by the folded edge, then the single sheet parts are situated on the outer edges of the “action areas”. I have calculated using the formula for determining the area of a triangle (0.5 x base length x height) that this configuration of the end shape increases the surface area by 39 square centimetres. That’s a whopping 31.09% extra surface area, created by simply cutting perforations at a 20-degree angle!

Now, if you were Mr Andrex and you were presented with this idea, would you:

  1. Market it as a gimmick,
  2. Market it as an innovation that provides over 30% more efficiency,
  3. Cut the volume of paper used per sheet and therefore reduce raw material costs while still providing the same useful surface area?

I suspect I know the answer.


Ten-to-Two Feet

A few years ago my son said to me, “Dad, why are all my mates faster than me when we play 5-a-side football at the Sports Centre?” I told him it’s because he runs with ‘ten-to-two’ feet and it’s all to do with trigonometry.

“Oh well that explains it, so it’s not because my legs don’t go fast enough then?” he replied somewhat sarcastically.

I felt I ought to explain so I found a pen and a blank sheet of paper.

Suppose your foot is 30cm long. When you run perfectly straight with your feet pointing in exactly the same direction you are running you are propelling your stride to its maximum 30cm per step.

30cm Foot

Think about your feet and your running style, they point to 10 o’ clock (left foot) and 2 o’ clock (right foot).

10 to 2 feet

So when you spring forward you are not travelling in the same direction as your foot is pointing, therefore instead of maximising your 30cm foot length you are only pushing forward around 27cm per step and thereby losing 3cm in comparison to the perpendicular line of motion.

27cm Foot

Now suppose you are sprinting over 100m and for the sake of argument you take 70 strides in total. It means that as a direct consequence of your running style you are 70 x 3cm further behind where you could be, that’s over 2 whole metres that could be gained just by training your feet to point in a slightly different direction.

He looked at me and said, “Wow, that’s amazing, how do you know this stuff?”. I gently placed my pen on the paper, stretched my legs forward under the desk and leaned back in my chair, fingers interlocked around the back of my head with a feeling of smugness that a man can only experience a handful of times in his life. I thought about the ‘A’ level I achieved in further mathematics over 30 years ago and was grateful at last that all that hard work learning those trigonometric rules had reaped dividends.

After a minute or so, my son said, “Wait a minute, if I were to race my mates over 100m, I guarantee I’d lose by a lot more than 2m”. I replied, “Ah well, that’s because your legs don’t go fast enough!”.