# Twosday's hidden analogue symmetry

22:22 is palindromic and ambigramic, but what about its analogue form?

## Intro

Yesterday, 20th Feb 2020 was supposedly Twosday (opens new window), both a palindromic (opens new window) and ambigramic (opens new window) date – at least on a calculator – and was celebrated (opens new window) by number nerds everywhere (myself included!).

However, those of us who get really geeky over times were hanging out for the double-whammy of 22:22 on 22/02/2022!

I got a Full House on #Ambigram Day yesterday!

— Dave Stewart (@dave_stewart) February 23, 2022

Note the battery; it took a day of meticulous planning and power management and an evening of telling people to go away and not bother me 😆#Twosday #TwosdayTuesday #22022022 pic.twitter.com/Pobe5HpLGs

22:22 has been my favourite time for a number of years now, because in addition to the digital symmetry, there is seemingly an *analogue* symmetry – the hands of the clock * look like they are in direct alignment*:

So the question is: does 22:22 – * and therefore Twosday* – have an analogue symmetry hidden inside its digital one!?

## Theory

### Conjecture

If you consider the hands of the clock:

- at 10 o’clock, the hour hand is
**⅓ before the 12** - at 20 past, the minute hand is
**⅓ before the 6**

But at 22:22 the proportions *cannot* be thirds because both hands will have travelled a bit further (by 22/60) so:

- the minute hand will be
**22/60 away from 12** - the hour hand will be
**22/60 between 10 and 11**

So the question is: with these additional small offsets, do the hands line up to 180°?

### Proof

The way to think about the problem is to imagine the minute hand driving the hour hand.

For every minute that passes:

- the minute hand moves by 1/60 of 360°
- the hour hand moves by 1/60 of (360° / 12) or 30°

Therefore at 22:22:

- the minute hand will be at 22/60 x 360 degrees
- the hour hand will be at (10/12 x 360 degrees) + (22/60 x 30 degrees)

If you subtract one from the other you get the difference in degrees between them… which we *hope* is 180°.

## Code

### Attempt 1: solving for 22:22

I like to solve things in code, so in JavaScript:

```
var f = 22/60
var t = 360
var m = f * t
var h = (10/12 * t) + (f * 30)
console.log(h - m)
```

The result is `179`

.

So clearly, this is very close, but not `180`

!

### Attempt 2: solving for 22:22:22

But what if we used 22 seconds as well? And how do we do that?

Well, 22 seconds as a fraction would be 22 / (60 x 60) or 22 / 3600, so the code is:

```
var f = (22/60) + (22/3600)
var t = 360
var m = f * t
var h = (10/12 * t) + (f * 30)
console.log(h - m)
```

And the result is: `176.98333333333335`

.

This feels very strange, that such a small adjustment would not only seemingly have such a large impact, but would move the value further away from the magical `180`

.

At this point I’m thinking my simple code experiments need some kind of geometric proof, so I can see the angles and start tweaking visually.

### Attempt 3: solving for 180°

Luckily, I found some clock code online (opens new window) that I could quickly edit and create something more interactive.

Tweaking the time value, it was surprisingly easy to work out what time would give 180°, and that time was `22:21:45`

as you can see in the interactive example below:

But here’s the thing…

If we *round off* the time to * just the minutes* (which was our original aim) we get

*and*

**22:22***!*

**180°**## Conclusion

If you’re happy to accept the rounding then * 22:22* on

*has*

**22/02/2022***multiple*embedded ambigrams and palindromes:

- the digital date:
**20/02/2022** - the digital time:
**22:22** - the analogue time:
**180°**

Is the 180° *strictly* pure? No, but by that token, is the date? The leading zero in the month component could have been omitted, but we wouldn’t be having the fun we are now. It’s all a fudge really, you just have to enjoy it for what it is.

Happy Twosday!