The Helicone Numberscope: Mathematical Superpowers Hidden in a Simple Toy
Mathologer
0:00 Mathologer junior here, and look at all this cool stuff.
0:04 We have the cubes, the old calculator, and a maths gnome!
0:09 Look at that!
0:11 And over here, we have the barber pole—let’s turn it on.
0:17 Look at that!
0:19 And now, welcome to the Mathologer’s office.
0:25 Come in, if you dare!
0:32 Ah, thank you, Mathologer junior and Mathologer junioress.
0:36 I’ll take it from here.
0:39 So, welcome to my office and the 2024 Christmas Mathologer video!
0:44 Yes, this year, we are running with the Julian calendar,
0:49 like about one-tenth of the people celebrating Christmas each year.
0:53 According to the Julian calendar,
0:56 Christmas Day 2024 is on the 7th of January 2025.
1:03 Go figure!
1:04 Anyway, today’s video is about this really cool kinetic toy.
1:08 It’s called the Helicone.
1:09 What’s so cool about it?
1:11 Well, let me just show you.
1:12 So, this is the original version.
1:14 It’s a laser-cut version.
1:15 When I spin it, it turns into a pinecone—very cool.
1:21 Okay, here’s the second version.
1:24 It’s made from plastic, usually called a Lollipopter,
1:27 and you can see why—it really looks like a lollipopter.
1:31 Well, in this video, I’ll just be talking about the Helicone.
1:37 Alright, now, just like real plants that look like
1:40 this, the Helicone incorporates a lot of the numbers of nature:
1:44 the golden ratio, Fibonacci numbers, heaps and heaps of spirals.
1:48 It’s just a great model of nature!
1:52 Now, what even people who own this don’t know is that a multi-layer version—say,
1:58 like 2,000 layers—can act as a very
2:01 powerful microscope to probe the nature of numbers.
2:05 Just to whet your appetite in this respect,
2:07 I’ve prepared a slideshow consisting of some of the spectacular images
2:13 that you get when you zoom in on the square root of two.
2:26 Have a look.
2:29 Intrigued?
2:29 I sure hope so!
2:31 To be able to show you more,
2:33 I’ve programmed myself a virtual Helicone lab in Mathematica.
2:37 So, lots of maths and cool imagery
2:39 to look forward to—all the secrets of the Helicone,
2:43 the Helicone microscope, and the nature of numbers!
2:49 Enjoy:) This is a virtual Helicone with 31 layers using five different colours,
2:58 just like in the real thing.
3:00 With this slider I can recreate the full Helicone twist and untwist
3:04 actions that takes us between the two different states of this little marvel.
3:11 And with these three buttons I can switch between different views.
3:21 There is view no.
3:23 1, then there is the top view.
3:25 The five double spirals really show up nicely this way.
3:29 And here is the side view.
3:31 The cyclic repeat of the colours pops out nicely in this way.
3:36 Blue, turquoise, green, orange, red, back to blue,
3:39 and then repeat over and over.
3:41 Other controls allow me to modify parameters that cannot
3:44 be changed easily in the physical model: for example,
3:47 I can make a core appear to highlight the leaves
3:50 I can adjust the intensity of the colours like this.
3:54 I can change the number of colours.
3:57 Interesting, colouring the levels cyclically
4:00 with five colours ends up highlighting five
4:03 double spirals colouring with three colours
4:05 ends up highlighting three different double spirals.
4:08 We’ll explore this colour magic in detail in a minute:) But first,
4:12 let’s scroll through the layers from top to bottom.
4:16 As we scroll down it looks like something is spinning at a constant rate.
4:21 This is because consecutive layers are a constant special angle apart.
4:26 In biology this sort of angle is often referred to as a divergence angle.
4:31 To measure the divergence angle of our Helicone,
4:33 let’s switch back to the top view and go in reverse,
4:37 building the Helicone back up layer by layer.
4:40 Okay add the second layer.
4:43 The angle between these two layers is about 68.75 degrees and, as I said,
4:49 this angle is the same from layer to layer.
4:53 There the same angle.
4:55 And the same again.
4:58 This divergence angle is exactly half
5:02 the golden angle which is approximately 137.5 degrees.
5:06 And WHAT makes our divergence angle GOLDEN?
5:09 It is the fact that this division of the circle into two
5:15 parts is in the golden ratio phi, 1.618 dot dot dot.
5:20 As most of you will know, this ratio has many remarkable,
5:24 desirable and useful properties and pops up
5:27 in mathematics and nature all over the place.
5:30 In particular, the circle being divided
5:32 in the golden ratio means that this golden
5:34 ratio is present in two different ways in the picture in front of us.
5:38 On the one hand, it is equal to the green angle divided by the golden angle,
5:43 but on top of that it is also equal
5:46 to the full 360 degrees divided by the green angle.
5:50 Full divided by large is equal to large divided
5:54 by small is equal to 1.618 dot dot dot.
5:58 Great:) Also important for later,
6:01 the golden ratio is an irrational number and so cannot be written as a fraction.
6:06 In fact, in some sense it is the most
6:09 irrational of all irrational numbers and this turns
6:11 out to be one of the reasons why the golden ratio is so prevalent in nature.
6:16 We’ll also get to that in little while:) As well,
6:20 the golden angle and its complement are irrational angles.
6:23 Those angles up there are really just approximations… Anyway, remember,
6:27 half the golden angle is at the core of our Helicone.
6:34 Again, note that in this top down view
6:37 of the Helicone the five double spirals really jump out.
6:41 Okay, at the start of this video I
6:43 said that nature’s numbers are hiding in the Helicone.
6:45 Well, we’ve just seen the golden ratio.
6:47 Next are the Fibonacci numbers, nature’s favorite playlist:) 1,
6:51 2, 3, 5, 8, 13,… and so on.
6:53 I am sure you all know how the Fibonacci sequence is built,
6:57 but just in case you are late to this party,
7:00 the simple growth rule of the Fibonacci numbers
7:02 goes like this: Starting with 1 and 2,
7:04 two consecutive terms of the sequence always add up to the next term.
7:08 There 1+2 is 3 2+3 is 5 3+5 is 8 and so on.
7:14 So, where are the Fibonacci numbers hiding in the Helicone?
7:18 Well, we see one Fibonacci number, the number 5, right in front of us:
7:23 five colours highlight five double spirals in the Helicone.
7:25 And now it turns out that this is not a coincidence but part of a cool rule.
7:30 Cyclically colouring the layers with any small Fibonacci number
7:34 of colours highlights exactly that number of DIFFERENT double spirals.
7:37 Let me show you in a slightly larger Helicone with 60 layers.
7:42 There, just one colour.
7:44 And, featuring that constant angle between consecutive layers,
7:49 the whole Helicone can be considered to be just one big double spiral.
7:54 One colour, one double spiral.
7:56 2 colours, 2 double spirals, one red double spiral and one green double spiral.
8:02 Let me just highlight the pair of red spirals for you.
8:05 There that’s one of the two red spirals.
8:07 And the remaining red leaves combine into the second spiral.
8:11 Similarly, the green leaves form a second double spiral.
8:15 Using two colours, all the leaves really split into 2 double spirals.
8:21 Neat.
8:21 The next Fibonacci number is 3.
8:24 Again, three double spirals.
8:25 Next is 5.
8:26 5 double spirals.
8:28 8 and 13.
8:29 Bit harder to pick out those 13 double spirals.
8:33 Anyway, there’s one red spiral, one half of the red double spiral.
8:37 While we are at it, notice that the larger
8:40 the Fibonacci number the steeper the spiral.
8:43 Also the spirals alternate slanting to the left and right.
8:48 Have another look.
8:49 13, steep, slanting right.
8:51 Now 8.
8:52 A bit less steep, slanting left.
8:54 5 shallower, slanting right, and so on.
8:57 What happens when you use a number of colours that is not a Fibonacci number?
9:02 Well, let’s see.
9:03 Here is 6.
9:04 A mess.
9:05 Well, there is still a bit of structure there if you look closely,
9:09 but nothing as amazingly nice as with a Fibonacci number of colours.
9:14 There 7, another mess.
9:15 Alright.
9:16 Loads of spirals, golden ratio and golden angle,
9:19 Fibonacci numbers, tick, tick, tick, tick.
9:21 Helicones are definitely teeming with Nature’s numbers and phyllotactic magic.
9:26 Now, before I’ll show you my Helicone microscope,
9:29 let me introduce you to a super fun Christmas infused magic that I
9:34 discovered while playing with my Helicone
9:41 lab and decorating the Mathologer Christmas tree.
9:49 Okay, let’s reshape our Helicone into a Christmas tree.
9:52 Not bad, but we can do better.
9:55 Let’s use baubles instead of cylinder leaves.
9:58 Nice.
9:58 But now let’s also adjust the bauble sizes
10:01 a bit for a better fit around the tree.
10:04 Wow, super pretty, hm?
10:06 What else?
10:06 Well we are missing a crown.
10:08 How about a nice mathematical one, like a small stellated dodecahedron.
10:13 Or, how about a great stellated dodecahedron?
10:16 Also not bad.
10:17 But I’ve got more choices:) I programmed my Helicone lab
10:21 in Mathematica and so how about Wolfram’s spikey for a crown?
10:25 Last but not least a famous 3d Christmas star, the Moravian star,
10:30 or the Herrnhuter Stern as it’s known in Germany where I grew up.
10:35 Spent an hour creating this one from scratch:) Totally worth it:) Okay,
10:39 which one do you like best?
10:41 Small or great stellated dodecahedron?
10:43 Wolfram’s spikey?
10:43 Or Herrnhuter Stern?
10:44 Let me know in the comments:) My personal favourite
10:48 for this setup is this one:) The small stellated dodecahedron.
10:52 Okay, now here is a nice Christmas application of our Helicone magic:
10:57 a Fibonacci light show:) How did you like my Fibonacci Christmas tree?
11:17 The ONLY tree that comes pre-assembled with Fibonacci magic.
11:20 Baubles sold separately:) Spectacular, hm?
11:23 Challenge for the Christmas fanatics and programmers among you:
11:26 decorate a real Christmas tree this way and send me
11:30 a nice recording of the Fibonacci light show in action,
11:33 or create an online Helicone lab that mimics mine to share with others.
11:38 At the end of February, I’ll link to all submissions from this page
11:41 and select a winning entry from among all your submissions.
11:44 And I’ll send the winner a signed copy of one of my books:) Now before I forget,
11:50 here is a quick history lesson:
11:52 The Helicone was invented by John Edmark in 2008.
11:56 John teaches design at Stanford university
11:59 and is world-renowned for his spectacular kinetic artworks.
12:02 Check out the description of this video for more details and a link
12:07 to a presentation in which John talks about some of his masterpieces.
12:11 John told me that the original Helicone featuring 80 leaves
12:15 was a present for his professor on his professors’s 80th birthday.
12:20 Okay, but what about the number microscope that I promised you?
12:27 Coming right up:) At this point let me just fire
12:35 up my microscope and show it to you in action.
12:38 See whether you can fill in the missing details between what
12:41 I’ve shown you so far and what you are about to see.
12:44 I’ll also spell out those details later on.
12:48 Okay, The microscope has three main controls:
12:51 The number of circles or leaves which is currently 30,
12:54 but ranges all the way from 1 to 2000.
12:57 Then we have the number of spirals.
13:00 Currently we’ve got 5 spirals.
13:02 Hmm, interesting, not double spirals just single spirals.
13:06 Anyway, we can set the number of spirals to be
13:09 any number between 1 and the number of leaves.
13:12 Hmm, also interesting,
13:14 ANY number of spirals not just Fibonacci numbers of spirals.
13:19 And the third control is a divergence angle.
13:21 Now this divergence angle is not given as an angle
13:24 in degrees as before but as a number between 0 and 1.
13:29 We’ll make sense of this later.
13:31 To start with, the divergence angle is set to 0.618.
13:36 Note that 0.618… is the fractional part of the golden ratio.
13:40 Alright, let’s play.
13:41 First, let’s up the number of leaves a bit.
13:45 Let's go up in increments of 10.
13:49 There 40, 50, That’s 180 leaves.
13:52 Right now we are still looking at 5 spirals.
13:58 Okay, let’s up the number of spirals by 1.
14:01 So 6 spirals is next.
14:03 Whoa, what just happened?
14:04 Don’t worry, let’s keep going.
14:06 7, 8.
14:07 Aha, Fibonacci, exactly 8 nice spirals.
14:10 Familiar territory;) Now 9 mess, but a nice mess:) 10 11, 12, 13 is next.
14:18 What do you expect to see?
14:21 Yep, 13 neat spirals:) 14, 15, 16.
14:24 Actually that does look pretty spirally, right?
14:27 Well, 16 is not a Fibonacci number.
14:30 However, 16 is 2 times 8 and 8 is a Fibonacci number.
14:35 There that was the picture for 8 again.
14:38 And back to 16.
14:39 So basically for 16, every one of those 8 spirals splits into two.
14:44 Neat.
14:44 Keep going, 17.
14:46 That’s a pretty insane one.
14:49 18, 19, 20.
14:50 Next one is the Fibonacci number 21 aaaand bingo.
14:56 26, two times the Fibonacci number 13.
14:59 34, Fibonacci.
15:00 Jump straight to the next Fibonacci number 55.
15:04 Now, let’s stay with those 55 spirals but up the number of leaves.
15:10 Now, count up to the next Fibonacci number.
15:14 Just take it in.
15:16 Another interesting one 63, 3 times the Fibonacci number 21.
15:20 And actually what you see here is three spirals each
15:23 for every one of those 21 spirals that we saw before.
15:30 68 that’s 2 times the Fibonacci number 34.
15:36 89 another Fibonacci number.
15:39 Okay, so currently the third control, the divergence angle is set to 0.618,
15:46 the fractional part of the golden ratio,
15:48 and in the first instance what the microscope allows us
15:51 to do is to visually generate the sequence of Fibonacci numbers.
15:55 Right, just increment the number of spirals and leaves and wait for some
15:59 of these spirals to actually look like real
16:01 spirally spirals to visually pinpoint the Fibonacci numbers,
16:05 without doing any calculations.
16:07 That’s pretty cool isn't it?
16:08 What if we change the divergence angle to another famous number between 0 and 1?
16:14 Like for example the fractional part of pi, 0.141592 dot dot dot?
16:19 Do we visually get a different sequence of integers?
16:22 What is it?
16:23 Is there a deeper meaning to all this?
16:26 Okay, let’s explore.
16:27 First dial back to smaller numbers of leaves and spirals.
16:32 91 leaves and three spirals.
16:34 Now switch the third control, the divergence angle,
16:37 to the fractional part of pi.
16:40 A starfish, cute:) No longer any clearly defined spirals for pi.
16:44 Let’s up the number of colours again, 4, 5, 6, 7, 7 spirals.
16:49 Definitely not Fibonacci.
16:50 But boy do those 7 spirals stand out:) Much more than any
16:56 of the spirals we’ve seen so far:) What has 7 to do with pi?
17:01 Well, what do you think of first when you hear 7 and pi?
17:06 Well, 22/7 of course that crazy good fractional,
17:10 small numbers approximation of pi.
17:12 Wait what?
17:13 Yes, this is not a coincidence:) Okay,
17:17 enough just mucking around with the microscope,
17:22 time to really explain what is going on here:) There,
17:31 that’s our 3-coloured Helicone.
17:33 There are actually plants that grow like this, with two opposing branches
17:38 at every level and consecutive levels placed half the golden angle apart.
17:42 However, what is much, much, much more common are plants with just one branch
17:47 per level and levels placed a full golden angle apart.
17:51 And here is a 1-branch Helicone that is built this way.
17:55 However, that’s not the end of it.
17:57 There are also plants that feature 3 branches at every level.
18:00 A corresponding Helicone would look like this.
18:03 Instead of single and double spirals we have triple spirals in this one.
18:07 Neat:) No reason to stop here of course, 4 branches per level is next,
18:12 but first what about the divergence angle of a trefoil Helicone?
18:16 Well, easy to guess.
18:18 It’s half the golden angle for two branches per level.
18:21 It’s a full golden angle for one branch per level.
18:24 And so it should be one third of a golden angle for the 3-branch Helicone,
18:26 one fourth of a golden angle for 4 branches, etc.
18:30 Right?
18:31 Challenge for the keen among you: 3d print one of those trefoil Helicones.
18:38 Has not been done yet.
18:40 There are largely identical microscopes
18:43 corresponding to the different Helicones,
18:45 but there are also some subtle differences.
18:47 Anyway, to start with, what I showed you is the simplest
18:51 microscope that corresponds to the 1-branch Helicone on the left.
18:54 So the default divergence angle for that one is the golden
18:58 angle and we have single spirals and not double spirals.
19:01 Okay, let’s make sense of this microscope.
19:05 At first glance it looks like the microscope just
19:09 shows a tidied up top view of a Helicone.
19:12 True but there are some important added features.
19:16 First, let’s label the leaves from the bottom up starting with 0.
19:20 There that’s the leaf at level 0.
19:23 level 1, 2, 3, and so on.
19:26 Here the lower the level the further away from the center the leaf is.
19:32 Let’s return to leaf 0.
19:34 What’s the angle between leaf 0 and leaf 1 again?
19:37 Well that’s supposed to be the golden angle.
19:41 Let’s check.
19:41 Well, yes, there is a golden angle down there.
19:44 There is another one between 1 and 2 between 2 and 3 and so on.
19:48 Business as usual:) But, remember,
19:50 unlike in our Helicone where the divergence angle is recorded in degrees,
19:56 in the microscope it’s recorded as 0.618.
19:59 which, remember, is the fractional part of the golden ratio… What we do here is,
20:06 instead of recording angles in degrees,
20:08 we represent them as parts of a full circle, a number between 0 and 1.
20:15 So, 0.618 indicates 0.618 of a full circle.
20:18 To convert this into degrees, you simply multiply it by 360.
20:22 In this case, 0.618 of a full circle
20:25 corresponds to 360 degrees times 0.618= 222.5 degrees.
20:28 which is… not the golden angle,
20:29 but almost:) Remember that’s the complement of the golden ratio.
20:29 Of course, for the purpose of recording
20:31 how far radially apart consecutive leaves are, both angles are just fine.
20:37 Anyway, let’s just run with this way of recording divergence angles.
20:51 Okay, now how are the spirals constructed?
20:59 In the microscope we don’t rely on spirals visually jumping out at us anymore.
21:05 Instead we use those numbers to connect the dots.
21:08 Let’s construct the 3 spirals corresponding to the number 3.
21:12 Start with 0 on the right and then keep adding 3.
21:16 Okay, so 0+3 is 3, plus 3 is 6, plus 3 is 9, then 12, 15, 18, 21, 24, 27.
21:25 First spiral finished.
21:26 For the second spiral start with 1 and again keep adding 3.
21:31 1+3 is 4, plus 3 is 7, 10, 13, and so on.
21:36 Third spiral starting with 2, keep adding 3,
21:39 done:) The same simple construction gives you n
21:43 spirals when you use the constant increment n.
21:46 Say n is 4.
21:47 0+4 is 4, plus 4 is 8, plus 4 is 12, and so on.
21:53 Doesn’t look like much of a spiral since it intersects itself.
21:57 Anyway, starting with 1, 2, and 3 gives three more such“spiral” abominations.
22:03 Now, at first glance these number-based spirals may appear to be totally
22:09 different from those purely colour-based ones
22:11 that we dealt with in the Helicone.
22:13 But that that’s actually not true.
22:16 On close inspection,
22:17 the only difference is that in the microscope all the leaves
22:21 of one colour are always connected into a curve by line segments.
22:25 Have a look.
22:26 If these are the levels coloured with four colours as before,
22:29 then our new number-based green curve really just connects the green leaves.
22:34 There, 0 plus 4 is 4, plus 4 is 8, and so on.
22:40 The blue curve connects the blue levels.
22:43 And so on.
22:45 Anyway, the number-based spirals used in the microscope are
22:48 definitely the way to go when it comes to further
22:51 highlighting the overall distribution and structure of the leaves
22:54 of one colour for large numbers of leaves and spirals.
22:58 Alright, let’s stick with four spirals and up the number of leaves a bit.
23:04 Alright, now 5 spirals.
23:06 5 is Fibonacci.
23:07 6, 7, 8 is Fibonacci.
23:09 Okay, this is what we get for the divergence
23:13 angle of 0.618… In the previous chapter,
23:17 where we just played around with the microscope,
23:20 we now switched to the fractional part of pi for the divergence angle.
23:25 But before we try to make sense of irrational
23:29 divergence angles like this fractional part of pi,
23:32 let’s first make sense of the much simpler rational divergence angles,
23:42 divergence angles that are fractions.
23:48 What’s the angle associated with 1/4th?
23:51 Easy, right?
23:52 A quarter of 360 degrees, that’s 90 degrees.
23:55 Can you picture what the distribution of leaves will look like for 90 degrees?
23:59 Super simple.
24:00 Makes sense, right?
24:01 From leaf to leaf we rotate by 90 degrees.
24:05 There, there, and, in this way,
24:08 the leaves arrange themselves into four straight arms at 90 degrees.
24:12 If the number is 1/7, we get 7 straight arms.
24:16 Actually, it’s clear that if the number is a fraction,
24:20 a/b then the leaves will all end up on b equally spaced straight arms.
24:26 But what’s also important to note is
24:29 that these straight arms may only become visible
24:31 in the microscope once the number of leaves
24:34 is much larger than the denominator b.
24:37 For example, here is the picture for 1/23 and 41 leaves the number of leaves
24:44 is only about twice the denominator 23 and so the 23 arms are not visible yet.
24:51 However, when we crank up the number
24:54 of leaves to 200 the arms become clearly visible.
24:58 And with 2200 leaves the 23 arms is all we see.
25:02 Also, if you perturb our number by a tiny little bit,
25:07 for example, add one 1/1000th then the straight arms get twisted slightly.
25:13 If you minus the tiny bit instead of adding it,
25:19 the arms twist in the opposite direction.
25:24 Now 1/23- 1/1000 is equal to 977/23000 And this means that if
25:32 we up the number of leaves to say… ten times the denominator,
25:36 we’ll end up seeing 23000 equally spaced straight arms.
25:40 All clear?
25:41 Good:) So that gives us a pretty good feel for what the overall leave
25:47 pattern will look like if we are
25:49 dealing with a divergence angle that’s a fraction.
25:51 And how does our number-based game of connecting
25:54 the leaves into spirals play out for these rational angles?
26:00 Well, basically all the nice clearly visible spirally spirals run along the arms
26:09 and correspond to numbers that are factors or multiples of the number of arms.
26:15 On the other hand, all the spiral abominations jump
26:18 from arm to arm and form spiderwebs between the arms.
26:22 These spiderwebs also make for nice Christmas decorations:) Have a look.
26:26 In this example 2 spirals winding around the 23 arms look like this.
26:34 3 spirals, 4, and so on.
26:38 That’s 23 spirals running along the 23 arms.
26:43 46, double 23, would also run along those 23 arms, and so on.
26:49 Everything else, Christmas decorations:) Nice!
26:51 So, what happens for rational angles in the microscope, that is,
26:55 angles that can be expressed as fractions is really easy to make sense of.
27:01 Don’t worry if you did not get all of this 100%.
27:04 As long as you got the gist of it all,
27:06 you’ll be fine in the rest of the video:) Anyway,
27:09 I think it’s also clear that if the number we are dealing with is
27:13 an irrational number like the fractional parts of the golden ratio or pi,
27:17 then it’s impossible for the leaves to ever
27:21 settle into perfectly straight arms emanating from the center.
27:24 And with these remarks about rational divergence angles,
27:27 we are now ready to make sense
27:29 of the irrational fractional part of pi 0.141592...
27:30 What’s the angle in degrees that corresponds to this fractional part of pi?
27:35 Easy, 360 degrees times 0.1415 dot dot dot, that’s about 51 degrees.
27:43 51 degrees between leaves.
27:44 What’s the angle in degrees that corresponds to this fractional part of pi?
27:47 Easy, 360 degrees times 0.1415 dot dot dot, that’s about 51 degrees.
27:55 51 degrees between leaves.
27:57 Interesting but does not give much insight.
27:59 Much more important is what we saw
28:02 in the microscope for small numbers of leaves.
28:06 Remember?
28:06 7 clearly defined spirals.
28:08 So that’s pretty much the same picture as for 1/7th,
28:12 with a tiny bit of a perturbation thrown in.
28:15 And, of course that’s true.
28:17 That fractional part of pi turns out to be 1/7th minus
28:22 a tiny little bit,minus 0.00126 dot dot dot, to be precise.
28:26 Add 3 on both sides of the equal sign and you get pi on one side and 3
28:32 and 1/7th on the other side and of course
28:37 3 plus 1/7th that just our familiar 22/7:) Okay,
28:41 so what turns out to be the case is that those special integers detected
28:47 by our number microscope correspond to really
28:49 good fraction approximations of the divergence angle,
28:52 as well as of all numbers whose fractional part IS that divergence angle.
28:58 Let’s zoom in on the fractional part of pi by adding more leaves.
29:03 Unlike in the case of the fractional part of the golden ratio,
29:07 you really have to add a lot of leaves before you see new spirals popping out.
29:12 Here we’ve upped things to about 2000 leaves and what
29:15 we see highlighted here is still our 7 spirals from before.
29:19 This is one of them.
29:21 Winding around and around.
29:23 Let’s see what we get here in terms of the individual leaves.
29:30 Okay, pretty obvious what numbers will give nicely visible spirals.
29:37 Add 113, add 113 again and again.
29:41 106 would also work or 120.
29:44 Okay, let’s check.
29:45 Here is what you get for 106.
29:48 And this is what you get for 113.
29:53 And here is 120.
29:55 Among these three numbers 106, 113 and 120 which one’s the best?
30:02 Well 113 is the best in the sense that it produces the straightest
30:06 spirals and therefore should correspond
30:08 to a crazy good fraction approximation of pi.
30:11 To figure out what approximation we are dealing with here is also very easy.
30:16 Just multiply pi by 113.
30:18 As you can see we get something very close to 355.
30:22 And this means that 355 divided by 113
30:26 should be the approximation we are looking for.
30:30 And just to check, here are the first few digits
30:34 of this fraction that coincide with those of pi highlighted in red.
30:38 Pretty amazing:) We have a coincidence in the first seven digits of pi.
30:45 Really very very good.
30:47 In fact after 3 and 22/7,
30:50 355/113 is the most famous rational approximation of pi.
30:55 This amazing approximation was first discovered by the Chinese mathematician Z?
30:59 Ch?ngzh?
31:00 in the 5th century.
31:01 Here is a very nice way of remembering it.
31:05 11 33 55 the first three odd numbers all doubled up.
31:09 Split in half and you get
31:11 the denominator and numerator of our fantastic approximation.
31:13 Commit this to memory.
31:15 On your deathbed this will be part of the quiz that decides whether you’ll go
31:21 to mathematical heaven or hell:) Now 106
31:24 is another number that’s visible in the microscope.
31:27 However, here the spirals are nowhere near as straight and, consequently,
31:31 we expect the approximation corresponding to it to be
31:34 not as good as the one corresponding to 113.
31:38 Let’s check.
31:39 And so the approximation corresponding to 106 is 333/106 a 5 digits coincidence,
31:45 definitely very good.
31:47 However, nowhere near the 7 digit coincidence that we get with 355/113.
31:54 What about 120, our third special number?
31:58 Only 4 digits.
31:59 Anyway, 113 is the standout.
32:02 But, of course, since pi is irrational even those 113 spirals are not perfectly
32:09 straight and eventually will twist around the circle
32:12 many times to reveal even straighter spirals.
32:15 Here the next magical number after 113
32:21 is 33215 and the corresponding fraction is 104348/33215.
32:29 10 digits of coincidence, insane:) But, of course,
32:34 to see that 33215 in the microscope you’d need something like a million leaves.
32:40 Definitely my mac is not happy when I ask for that number of leaves
32:45 and responds with a different kind of spiral
32:47 infused circle:):):) But, as I already mentioned,
32:50 since pi is irrational there is no end
32:52 to this sequence of magical fractions:) And, of course,
32:55 you can play this game for any
32:57 number to pin down those amazing rational approximations.
32:59 Here is another example.
33:01 Here is what you get for root 5 when you zoom in on its fractional part.
33:10 682/305, giving a six digit coincidence, also pretty good.
33:15 And do you see that eye?
33:18 “Eye and SIX” inspired Mathologer junior to make a totally different connection:
33:23 Gojo’s six eyes in Jujutsu Kaisen:) Hmm, well I don’t know:) Alright,
33:28 so what the microscope does in the first
33:31 instance is to find great rational approximations of numbers.
33:36 In turn, these rational approximations give
33:38 valuable insights into the nature of numbers.
33:41 They also have a lot of real-world applications, for example,
33:44 designing fancy calendars to ensure Christmas lands on the right day
33:47 for the next million years:) and designing
33:50 optimal gear systems for clocks and orreries.
33:53 Also on my to do list for Mathologer:) Anyway, to finish this video,
33:58 let’s use what we’ve found so far to build some intuition
34:02 for why nature is so biased towards golden ratio divergence angles.
34:09 What does it actually mean for 22/7 to be a great approximation of pi?
34:15 Right, not obvious!
34:17 No doubt 22/7 is an approximation of pi
34:20 that works well enough for many practical purposes.
34:23 But then, as we’ve seen, 355/113 is a lot better and that third
34:28 monster fraction over there is even better.
34:30 So, why do we say that 22/7 is
34:33 a great rational approximation since there are much better ones?
34:38 In fact, sticking with approximating pi for the moment,
34:41 it’s a no-brainer to find rational approximations based on the decimal
34:45 expansion of pi that are as close to pi as we wish.
34:49 Just to illustrate, say you are looking for a fraction
34:52 that differs from pi by less than one thousands.
34:55 Easy:) Just make the first 4 digits of pi
34:59 into the 4-digit number 3141 and divide by one thousand.
35:04 Obviously, that’s a fraction that differs from pi by less than one thousands.
35:08 And if you want the difference to be one
35:10 billionth just use the first 10 digits of pi.
35:13 Hmm, so, really, why do we get so excited about fractions like 22/7 and 355/113?
35:19 Well, let me explain.
35:22 Let’s say we fix an integer like 7
35:26 and consider all the fractions with denominator 7.
35:29 Then these fractions correspond to points
35:31 on the number line that are 1/7th apart.
35:34 Since pi is also somewhere on the number line (chop in) this means that among
35:40 these fractions the one closest to pi must differ from pi by less than 1/7th.
35:46 Obvious, right?
35:47 But that then means that overall we can
35:50 expect better approximations from fractions with larger denominators.
35:54 Right?
35:54 Of course, in the case of denominator 7 the closest fraction to pi IS 22/7.
36:00 But then 22/7 is special in that none
36:04 of the fractions with denominator 8 are better approximations to pi,
36:08 none of the fractions with denominator 9 are better,
36:11 and none with denominator 10 are better.
36:13 In fact, very surprisingly, we have to go all the way up
36:16 to denominator 57 to find a fraction that beats 22/7.
36:20 Have a look at this table which
36:23 lists the BEST rational approximations by denominators.
36:26 So, there is the no-brainer approximation 3 at the top,
36:30 corresponding to the first possible denominator 1.
36:32 Then denominator 2 and 3 don’t give anything better.
36:35 The first denominator giving a better approximation is 4.
36:38 Then 5 gives something better than 4.
36:39 6 gives something better than 5.
36:40 And 7 gives something better than 6.
36:42 And then there is that big jump to 57.
36:47 Let’s add a column that also
36:51 lists the differences between consecutive denominators.
36:54 Very interesting, isn’t it?
36:57 So you see, 3 and 22/7 and their denominators 1
37:01 and 7 really stand out in two different ways not just one.
37:06 On the one hand, it’s those two denominator
37:09 1 and 7 where the most significant jumps occur.
37:12 But then we also see 1 and 7 over and over as the differences
37:17 next to THOSE best rational approximations
37:20 that don’t stick around for very long.
37:22 So cool, isn’t it?
37:24 Okay, now for which denominator does the next significant jump occur?
37:28 I think you can all guess by now.
37:32 113.
37:32 Aha!
37:33 Have a look at the fraction just above.
37:36 333/106.
37:37 Remember, we also stumbled across that one before.
37:42 There again is 113.
37:44 There is 106.
37:46 But then notice that you also see the denominators further up.
37:50 There 99.
37:51 And 92, and so on.
37:52 And, actually, these special denominators also jump
37:55 out as spirally spirals in the microscope,
37:58 just as very twisted spirally spirals.
38:00 And, why are 113, 106, 99, 92 and so on nicely lined up like this?
38:07 Well, of course, because being seven apart they are
38:11 all part of one of the 7 standout spirals.
38:14 Ponder that for a moment.
38:16 113, 106, 99, 92 and so on are all part of one of the seven spirals.
38:22 All good?
38:23 Great!
38:23 Anyway, now you know in what sense 22/7
38:28 and 355/113 are extra special best rational approximations of pi,
38:32 and I think it’s also intuitively clear why the denominators of these special
38:37 fractions should stand out when we zoom in on a number in our microscope.
38:43 It’s just that, as we zoom in, because of the big jumps,
38:47 these denominators stay visible for very long and end up being
38:52 the by far straightest standout spirals at different levels of magnification.
38:57 If you are interested in a nifty purely algebraic way
39:00 of generating the standout best rational
39:02 approximations based on infinite continued fractions,
39:04 I recommend this Mathologer video from a couple of years ago.
39:08 Alright, what we encountered so far can also give us some clues
39:13 as to why it is the golden ratio divergence angle that is prominent in nature.
39:19 Let’s say nature has decided to build
39:22 a plant like a 1-branch per level helicone.
39:25 Nature can choose among infinitely many different divergence angles.
39:28 Why does it choose the golden ratio?
39:30 Why is this distribution of leaves or seeds or florets corresponding
39:34 to the golden ratio superior to this one corresponding to root 5,
39:38 or this one corresponding to pi.
39:40 Well, if you look at all this from nature’s point of view,
39:44 then anything that tends to settle down into essentially straight arms
39:47 as in the case of root 5 and pi is not great,
39:51 as that would translate into plants like this tree
39:54 with the leaves stacked one on top of the other,
39:57 thereby preventing sunlight to reach the leaves.
40:00 As mother nature I don’t want this:) Also,
40:03 in the case of flowers heads this sort
40:06 stacking along arms would never happen because the growth
40:09 mechanism in flower heads ensures a constant divergence angle
40:13 while simultaneously arranging the florets to achieve maximum density.
40:17 You just don’t get a dense packing along straight arms.
40:22 And to prevent leaves to settle
40:24 into straight arms you really want the denominators
40:27 in your sequence of super duper best rational
40:30 approximations to be as close together as possible.
40:33 And this turns out to be exactly the case when nature chooses
40:37 the divergence angle to be the fractional part of the golden ratio.
40:41 In fact, when you zoom in on the golden distribution
40:45 of leaves by upping the number of leaves in the microscope,
40:48 you’ll always see not just one but two types of spirals,
40:52 corresponding to two consecutive Fibonacci numbers fighting
40:55 for dominance and violently twisting in opposite directions.
40:58 Have a look.
40:59 There just a few leaves and all you can see on the outside are
41:04 two set of spirals dominating 8 shallow blue ones and 13 steep red ones.
41:08 Add a few more leaves and blink your eyes
41:12 and 13 and 21 are now dominating on the outside.
41:16 And so on.
41:17 All super neat and magical, don’t you think?
41:20 And this definitely gives some intuition as to why
41:22 the golden angle works well in nature.
41:25 Having said that, it’s definitely not a full explanation.
41:28 If you are after a fairly full explanation,
41:30 check out this video, again from a couple of years ago.
41:34 Let me finish with two remarks.
41:36 First, a nice surprise awaits in the sequence
41:39 of best approximations for the golden ratio divergence angle.
41:45 Definitely an obvious question to ask at this point:
41:51 What are the best rational approximations of the golden ratio?
41:56 Well, turns out that these are exactly
41:58 the fractions corresponding to the Fibonacci number denominators.
42:02 Here are those fractions.
42:04 3/2, 5/3, 8 over 5….
42:07 But wait a minute.
42:08 Do you see a pattern here?
42:10 Yes, denominator and numerator always
42:12 appear to be consecutive Fibonacci numbers.
42:14 Let’s check a couple more cases.
42:16 yep.
42:17 yep.
42:17 yep.
42:18 And it’s actually not hard to prove
42:20 that the sequence of best rational approximations of the golden ratio really is
42:25 the sequence of ratios of consecutive Fibonacci numbers.
42:28 Also, did you notice just how BAD these best rational
42:33 approximations of the golden ratio are compared to those of pi?
42:38 That one up there is already the 7th best rational approximation
42:41 of phi and it only coincides in the first three digits of phi.
42:46 So in a sense it seems hard
42:48 to find good rational approximations of the golden ratio.
42:52 Phi is an irrational number that is
42:54 hard to approximate with rational numbers and so,
42:56 in a way, is a very irrational irrational number:) In fact,
43:00 in a very precise sense the golden ratio is characterised
43:04 by the fact that it is the most irrational of all irrational numbers.
43:09 This is also evident in the microscope, where none of the prominent spirals
43:12 at any level of magnification appear perfectly straight.
43:15 Again in a very precise sense it is true to say that Nature prefers
43:20 golden ratio divergence angles because the golden
43:23 ratio is the most irrational irrational number.
43:26 And again, check out this video for more about this interesting observation.
43:36 And now let me really finish off with a bit of a golden ratio puzzle
43:42 that arises when you play with the microscope
43:44 that corresponds to the original two-leaves per level microscope.
43:48 Two leaves per level means two leaves labelled 0.
43:52 Then there are also two leaves labelled 1.
43:56 One I’ve written the right side up, and the other one upside down.
44:00 Then 2 2s Again, one written the right side up, one upside down.
44:04 2 3s.
44:05 And so on.
44:07 Alright, here are our three double-spirals.
44:09 We construct the first connected spiral used in the microscope in the usual
44:14 way by starting with the 0s and adding 3s over and over.
44:18 And so 0, 3, 6, etc.
44:21 as well as upside down 0, 3, 6, are strung together into a double spiral.
44:26 Here is what you get when you switch to 5 colours.
44:31 But then drawing in the connected spirals by adding
44:35 5 over and over results in a surprise.
44:38 Where did this abomination come from?
44:41 Turns out that what you have to do to string together the double
44:45 spirals that visually jump out is
44:47 to alternate between numbers and upside down numbers.
44:51 There, normal 0, 0+5= upside down 5 plus 5= normal 10 plus 5= upside down 15.
44:59 With 8 we also have to string the leaves together in this alternating way.
45:05 Then 13 requires all right side up stringing again.
45:09 Checking what sort of stringing needs to be
45:12 done for the first couple of Fibonacci
45:14 number suggests a simple pattern Here blue stands
45:17 for alternating stringing and red for normal stringing.
45:21 Can you explain this pattern?
45:23 What pattern do you get for trefoil Helicones?
45:25 Leave your answers in the comments.
45:28 Also, I’ve really just touched on some of the secrets
45:31 becoming manifest in those spectacular
45:33 images produced by the Helicone microscope.
45:35 If you end up playing with the Helicone
45:38 microscope yourself and discover anything interesting,
45:40 please also let the rest of us know in the comments.
45:44 And that’s all for today.
45:46 I hope that you enjoyed today’s crazy excursion into Helicone
45:49 maths:) If there is anything you did not understand,
45:53 please ask me to clarify in the comments:)