Maths – Me on the net

More ‘Spirograph (®) plots using ‘matplotlib’

‘Spirograph’ includes toothed rings which can be used with small disks to draw Spirograph patterns inside the ring. The small toothed rolls around the inside of the fixed ring, producing the typical Spirograph spirals.

The diagram above shows how this works. The small circle rolls around inside the larger circle (which represents the inside of the ring). If there is no slippage, the small circle rotates in the opposite direction to its movement around the larger circle.

I modified my Python program to simulate this situation. As before, the rate of rotation of the smaller circle can be controlled by varying the ‘mult2’ variable in the program. To start with I used a small value for the variable as shown below.

The diagram has five ‘lobes’ and an empty patch in the middle. The curve can never go through the centre. If the ratio of the sizes of the circles is less than one half, then smaller circle is not big enough to reach the centre of the larger circle. If the ratio of the sizes of the circles is greater than one half, then the point on the circumference of the smaller circle that is drawing the curve (point D on the diagram above) never falls on the centre of the larger circle. If the ratio of sizes is exactly one half, all the loops will pass through the centre of the larger circle.

As the multiplier ‘mult2’ is increased, the curve comes to resemble the circular shape of a typical Spirograph. I think that it looks a bit like a torus or a donut.

I’m not going to try to emulate the behaviour of the non-circular Spirograph components. In fact, I have only emulated the circular ones, but not completely. I’ve only plotted the curves that would result if the point P (the pen) corresponded to the point D (on the smaller circle) in the diagram above.

Here’s one final plot, where the ratio of the smaller circle to the larger circle is exactly one half. The program draws a pleasant chrysanthemum or dahlia shape.

Finally, here’s the Python program that I used to draw these diagrams. To get the different figures, I’ve changed the multipliers, mainly ‘mult2’. I’ve also changed the sizes of the circles (variables ‘large’ and ‘small’) to get some of them.


import numpy as np
import matplotlib.pyplot as plt

# Dimensions of two circles
r1 = 100
r2 = 50
# c0 = [0,0]
# c1 = [1,0]

# Multipliers
mult = 10   # Used to generate the number of points plotted.
mult2 = 25  # Controls speed of rotation of the smaller circle, relative to the first.

ax = plt.subplot()
ax.set_aspect( 1 )

# Parametric array for the larger circle
t1 = np.linspace(-2 * np.pi, 2 * np.pi, mult * 360)
# Parametric array for the smaller circle
t2 = t1 * mult2

# Calculation of X/y cordinates using the parametric arrays.
# x0 and y0 are the coordinates of the tangential point, B.
x0 = r1 * np.cos(t1)
y0 = r1 * np.sin(t1)
# plt.plot(x0, y0, label=("Large circle"), color = 'r', linewidth = 2, linestyle = '--')
# x1 and y1 are the cordinates of the centre of the small circle, C.
x1 = (r1 - r2) * np.cos(t1)
y1 = (r1 - r2) * np.sin(t1)
# x1a and y1a are the coordinates of D relative to C.
x1a = r2 * np.cos(t2)
y1a = r2 * np.sin(t2)
# plt.plot(x1a, y1a, label=("Small circle"), c = 'r', linewidth = 2, linestyle = '--')
# x2 and y2 are the coordinates of the desired point on the curve, D.
plt.plot(x0[0],y0[0], color='r')
# The small circle rolls in the opposite direction if it is inside
x2 = r2 * np.cos(-t2)
y2 = r2 * np.sin(-t2)

# Plot the Curve
# plt.plot(x1 - 1.5 * x2, y1 - 1.5 * y2)
plt.plot(x1 + x2, y1 + y2, label=("'Spirograph' curve"), c = 'b', linewidth = 1)

plt.title("Inside, mult = {}, mult2 = {} ".format(mult, mult2))
plt.legend(loc="upper right")
plt.show()

Drawing ‘Spirograph'(®)-like Curves with matplotlib

‘Spirograph’® is a set of devices which can be used to draw some interesting looping curves. It consists of various toothed wheels and other toothed shapes with small holes them. A wheel or other shape is chosen, and fixed to the drawing surface. A second wheel is placed so that its teeth interlock with the teeth on the second wheel and a pen is inserted into a hole in the second wheel. The pen is moved to keep the two gear wheels enmeshed at all times. As a result the pen traces out a curve.

This can, of course, be plotted mathematically, using a tool like ‘matplotlib’. The above diagram shows how the Spirograph system works. The circle representing the first wheel is centred on point A and is fixed. The other circle is centred on point C, rolls around the first circle, and is always in contact with the first circle at point B. The point P is a point inside the smaller circle, and the line CD is the radius which passes through point B.

It is easy to see that the centre of the smaller circle (C) travels in a larger circle around the static circle. This larger circle (not drawn) has a radius is equal to the sum of the radii of the two circles. This radius corresponds to the line AC. As C travels around the circle, the smaller circle, CD, rotates. Meanwhile, the point P on CD traces out the curve we are interested in.

In the ‘Spirograph’ case, the two circles are linked by the teeth on the gears wheels. This connection ensures that the smaller circle rotates at a fixed speed. The Wikipedia page on the ‘Spirograph’ system mentions that the teeth prevent any slipping.

In the program generated case, smaller wheel can be programmed to rotate at any rate that is desired, with ‘slippage’. It can, for example, be programmed to move around the large circle without rotating at all. The next image shows what happens then. [Note: For simplicity, the point P is assumed to be located on the circumference of the small circle.]

The plot shows large fixed circle drawn with a dashed line. The small circle is not visible because it is actually a single point. The ‘Spirograph’ curve is just a larger circle offset from the centre of the large circle.

If the small circle rolls around the larger circle without slipping, then the ratio of small circle to the large should be exactly in the ratio of the radius of the small circle to the radius of the large circle. So in this next image, the size of the small circle is 75, and the size of the large circle is 100. The smaller circle rotates 100/75 (4/3) times every time it goes around the large circle.

I’m going to finish up with a couple more images that look more like a standard ‘Spirograph’. The point P is still on the circumference of the smaller circle, but this makes little difference to the plots.

This show a typical curve that a ‘Spirograph’ would plot. The ‘mult2’ parameter controls how many lobes the figure has. In this case, with a value of 25, there should be 24 lobes.

Finally, this image, with a mult2 value of 67 will have 66 lobes. When a Spirograph Wheel rotates around a fixed wheel, the pattern typically shows a rosette of loops, with a hole in the middle where the fixed wheel is located. I’m going to experiment with a moving wheel inside and if it is interesting I might do another post on the topic of ‘Spirographs’.

?Below I have pasted the program that I used to draw these images. Feel free to take it and alter it as much as you like.

import numpy as np
import matplotlib.pyplot as plt

# Dimensions of two circles
r1 = 100
r2 = 75
# c0 = [0,0]
# c1 = [1,0]

# Multipliers
mult = 10
mult2 = 67  # was 11

ax = plt.subplot()
ax.set_aspect( 1 )

# Parametric array for the larger circle
t1 = np.linspace(-2 * np.pi, 2 * np.pi, mult * 360)
# Parametric array for the smaller circle
t2 = t1 * mult2

# Calculation of X/y coordinates using the parametric arrays.
# x0 and y0 are the coordinates of the tangential point, B.
x0 = r1 * np.cos(t1)
y0 = r1 * np.sin(t1)
# plt.plot(x0, y0, label=("Large circle"), color = 'g', linewidth = 1, linestyle = '--')
# x1 and y1 are the cordinates of the centre of the small circle, C.
x1 = (r1 + r2) * np.cos(t1)
y1 = (r1 + r2) * np.sin(t1)
# x1a and y1a are the coordinates of D relative to C.
x1a = r2 * np.cos(t2)
y1a = r2 * np.sin(t2)
# plt.plot(x1a, y1a, label=("Small circle"), c = 'r', linewidth = 3, linestyle = '--')
# x2 and y2 are the coordinates of the desired point on the curve, D.

x2 = r2 * np.cos(t2)
y2 = r2 * np.sin(t2)

# Plot the Curve
# plt.plot(x1 - 1.5 * x2, y1 - 1.5 * y2)
plt.plot(x1 + x2, y1 + y2, label=("'Spirograph' curve"), c = 'b', linewidth = 1)

plt.title("mult = {}, mult2 = {} ".format(mult, mult2))
plt.legend(loc="upper right")
plt.show()

Plotting with ‘matplotlib’

What is ‘matplotlib’? It’s a Python package which can be used to plot everything from a simple parabola or sine wave right up to complex statistical data. Of course, I use it just to print interesting curves, like the one above. I’ll discuss the program I wrote below.

Since it is a Python package it helps if you know how to write Python programs, but I don’t dive too deep into Python. Anyway, the above curves were printed by a program that is only eleven lines long!

At the top of the Python code, we need to tell it to use ‘matplotlib’ and I’ve included a line to tell the code that we need to use the ‘numpy’ maths package. Here’s the first bit of the program:

import matplotlib.pyplot as plt  # This is the plotting library
import numpy as np               # A common Python package

Plotting something involves matching one set of data against another. Commonly these sets of data are named ‘x’ and ‘y’, but they can have any names that the plotter desires. The relevant code in this program is as follows:

max_range = 1000
x = np.linspace(-2*np.pi, 2*np.pi, max_range)
y = np.sin(x)

The line that starts ‘x =’ uses the numpy ‘linspace’ routine to generate a numpy array of one thousand (max_range) elements evenly spaced between -2 time pi and 2 times pi. The line that starts ‘y =’ then creates another numpy array, taking the individual elements of the array x, applying the numpy sine function and appending the result to the end of the array y.

In the earliest version of this program, I created the two numpy arrays by means of a loop, but by creating numpy arrays using this method, I can do it in two lines. Like all good ideas I got it from someone else’s program on the Internet. There is only one program, the old joke goes, and that is the “Hello World!” program, and all other programs are descendants of that single archetype.

OK, so, I created two more arrays using the same technique, and here are the two lines of code:

z = np.sin(2*x)
zz = np.sin(4*x)

So, I could have drawn three plots, one for each array, but that wouldn’t have been that interesting, so I decide to plot the sum of all three arrays (against the x array), and also the product of all three arrays (again, against the x array). Numpy arrays make this easy.

# y = np.sin(x) * np.sin(2*x) * np.sin(4*x)
plt.plot(x, y * z * zz, label='product')
# y = np.sin(x) + np.sin(2*x) + np.sin(4*x)
plt.plot(x, y + z + zz, label='sum')

The ‘plot’ statements each draw a line in the final figure, and generate a label for the line. The comments describe the curve in an x/y format, in a simple mathematical style .

The final couple of lines (see below) are needed to show the labels and the complete figure. Below is the whole Python program, comments and all.

import matplotlib.pyplot as plt
import numpy as np

# Basic Figures
# sine wave
max_range = 1000
x = np.linspace(-2*np.pi, 2*np.pi, max_range)
y = np.sin(x)

# sine wave, freq * 2
# x = np.linspace(-2*np.pi, 2*np.pi, max_range)
z = np.sin(2*x)
zz = np.sin(4*x)

# y = np.sin(x) * np.sin(2*x) * np.sin(4*x)
plt.plot(x, y * z * zz, label='product')
# y = np.sin(x) + np.sin(2*x) + np.sin(4*x)
plt.plot(x, y + z + zz, label='sum')

plt.legend()
plt.show()

To close the post, I’m going present a more complex example. This one combines the plots described above, with a separate subplot of the three waves that were used to create the first subplot. At the very end is the source code.

import matplotlib.pyplot as plt
import numpy as np

# Basic Figures
# sine wave
max_range = 1000

fig, (ax1, ax2) = plt.subplots(2, 1)

# The x array runs from -2*pi to +2*pi
x = np.linspace(-2*np.pi, 2*np.pi, max_range)

# One component of the final plot
y = np.sin(x)

# sine wave, freq * 2
z = np.sin(2*x)
# Freq * 4
zz = np.sin(4*x)import matplotlib.pyplot as plt
import numpy as np

# Basic Figures
# sine wave
max_range = 1000

fig, (ax1, ax2) = plt.subplots(2, 1)

# The x array runs from -2*pi to +2*pi
x = np.linspace(-2*np.pi, 2*np.pi, max_range)

# One component of the final plot
y = np.sin(x)

# sine wave, freq * 2
z = np.sin(2*x)
# Freq * 4
zz = np.sin(4*x)
ax1.plot(x, y, label='sin(x)')
ax1.plot(x, z, label='sin(2*x)')
ax1.plot(x, zz, label='sin(4*x)')

legend1 = ax1.legend()

# y = sin(x) * sin(2x) * sin(4x)
ax2.plot(x, y * z * zz, label='product')
# y = sin(x) + sin(2x) + sin(4x)
ax2.plot(x, y + z + zz, label='sum')

legend2 = ax2.legend()

plt.show()
ax1.plot(x, y, label='sin(x)')
ax1.plot(x, z, label='sin(2*x)')
ax1.plot(x, zz, label='sin(4*x)')

legend1 = ax1.legend()

# y = sin(x) * sin(2x) * sin(4x)
ax2.plot(x, y * z * zz, label='product')
# y = sin(x) + sin(2x) + sin(4x)
ax2.plot(x, y + z + zz, label='sum')

legend2 = ax2.legend()

plt.show()

A Simple KmPlot Example

A simple sine wave is not that interesting. However adding sine waves together can produce something more aesthetically satisfying. I could, for example, add another sine wave with a higher frequency to the above simple sine wave, as shown below.

The above diagram shows the result of adding a sine wave of three times the original frequency to the first wave. Below I show the result of adding a third sine wave of five times the original fequency to the first.

A pattern is beginning to emerge from this process. I’ll add a few more terms to the summation.

This shows the sum of the terms up to the the term for the sine wave which is 13 times the original frequency. Those who have done some maths probably recognise that this is tending towards a square wave. Here’s a link to a MatLab version of what I’ve done here.

Here’s a depiction of all the individual waves that have been added together, plus the final result.

Here’s an image of just a few of the waves. It’s probably more aesthetic than the full set!

A couple of times when I was creating the above series of plots, I made mistakes, and ended up with some more interesting graphs, but I discarded those ‘mistakes’ and stuck closely to my original idea, which was to show how simply adding a few sine waves together could result in an approximate square wave.

In fact the Fourier series for a square wave can be expressed as an infinite sum of sine waves, as shown here. A fairly complex mathematical fact can be approximated by a few simple plots.

Now I’m going to go back to the ‘mistakes’ that I made and rejected earlier when developing and writing this post.

What’s the answer?

The Maths is out There

Last year I posted an article which discussed some simple arithmetic and algebraic expressionss that people are posting on the Internet. I mentioned in that post that the questions are not posted to be answered. They are posted to incite discussion. Since then, there has been an upsurge of such postings, and I am going to address this topic further and expand on it.

Most people have left school with the impression that there is only one way to solve an arithmetic or algebraic question, and there is only ever one correct answer.

This is true, in a way. The rules of maths are rigid. However, when we write a problem down on paper, we write down a representation of the maths, in symbols. In the simplest case, we write down the symbol for a number, eg the symbol ‘7’. However in Ancient Rome, a mathematician would have written the symbols ‘VII’ to represent the same number. Even when we say the word ‘seven’, the word is only a name for the number.

Of course, our minds cut out all these distractions, and we, inaccurately, conflate the number and the symbol or name for the number. “This is the number seven,” we say, pointing to the symbol, but it isn’t.

When we add mathematical operations into the mix, things get more complicated of course. We write “7 + 3”, and think “Seven plus three”, mentally come up with the sum, the number 10, and think everything is fine.

But what if we were from a culture that reads and writes from right to left? We might possibly think “Three plus seven”, and would come up with the sum, the number 10.

However the language that we would use is unlikely to be the same as the one that I am using here, and the symbols would be different. Nevertheless, in some eldritch way, the maths is identical.

In this rather long preamble I have suggested that mathematics, specifically arithmetic and algebra, is independent of the symbols on the paper, and, probably, independent of the way our brains think of maths.

The Need for an ‘Order of Operations’

From now on I’m going to assume that that we are not from a culture that reads and writes from right to left, but the following reasoning can be expressed in a similar way for those who do write that manner.

Now, if I multiply two numbers I get a third number. I’m not too concerned about what multiplication actually is, but essentially I am taking two (or more) number, doing something with them and ending up with a single, different number. I can communicate this idea to someone else by writing down the two numbers with a multiplication symbol between them. I might then write a ‘=’ symbol next, and then the result.

5 x 4 = 20

A string of multiplications is easy:

5 x 4 x 2 = 40

Similarly with addition:

5 + 4 + 2 = 11

So far so good. Note that it makes no difference which order we perform the multiplications or additions. If we decide to start with the ‘2’, rather than the ‘5’, we get the same answer at the end. ‘5 x 4′ gives us ’20’ and so does ‘4 x 5’.

Problems arise when we introduce division and subtraction into the mix. Firstly, you can’t reverse the order in which you perform a subtraction or division:

5 - 4 = 1, but 4 - 5 = -1
Worse, 5 - 4 - 2 = 1 - 2 = -1, but 5 - 4 - 2 = 5 - 2 = 3

Even worse things happen when you mix addition and subtraction with multiplication and division. It’s important to note that the order that we should perform operations on numbers is not an essential part of arithmetic and algebra, but is related to the way we portray numbers and operations on paper.

So, mathematicians introduced the concept of the order of operations. This simply means that there are conventions for writing down arithmetical and other mathematical processes, in a way that someone else can take the expression, evaluate it and reach the same result as the originator of the expression.

There are ways of writing mathematical expressions which do not need an order of operations (or brackets, see later), indicating that the issues are not something within mathematics itself, but I’ll leave that for another time.

There are two ways of of writing an arithmetic or algebraic expressions in a way that can be communicated to someone else. Firstly, an order of operations rule can be applied to writing such expressions. Then expressions such as

1 - 21 / 7 + 4 * 2 =  6

can be written and understood unequivocally. The order of operation rules that are most often used are usually referred by the mnemonic PEMDAS or BODMAS.

Secondly, brackets can be used to specify the order that operations should be performed. The previous example could have been written as

(1 - (21/7)) + (4 * 2) = 6

You can always add brackets to an expression, even if they are not essential, as above.

So, if we agree to use PEMDAS, we can pass equations to and fro with a good chance of agreeing on the results, but there are some edge cases.

Is PEMDAS enough?

Some people write things like

2 / 3(5 -2)

I think that this is badly formed, and one should not write ‘…3(5…’, but people do it all the time. That said, there is an implied multiplication symbol between the ‘3’ and ‘(‘, so the same thing, with the implicit multiplication made explicit is

2 / 3 * (5 - 2)

Now there are two schools of thought on how this expression should be processed. The first group say that there is nothing special about the multiplication sign and the rules should be applied as usual. That is, from left to right as each symbol is reached.

2 /  3 * (5 - 2) = 2 / 3 * 3 = 2

The other group say that because the multiplication sign is implicit, the multiplication between the ‘3’ and the bracketed item should be processed as soon as the bracket is evaluated, before the usual left to right processing. In other words

2 / 3 * (5 - 2) = 2 / 3 * 3 = 2 / 9 = 2/9

The two schools of thought result in different answers, so which is correct? The answer is either or both. It depends on what the writer intended, and he/she could have used a set of brackets and made certain that the idea was communicated properly.

Why is it such a big thing on the Internet?

People like to cause arguments! So they post ‘quizzes’ on the Internet which are ambiguous, and then sit back and watch the fireworks. Other people read the posts and provide answers which are informed or not informed as the case may be.

Then those who think that they know, mock those who they think are wrong. Neither side acknowledge the fact that it is a communications problem. There are those who try to point this out, but they are often ignored.

Those who know of BODMAS/PEMDAS are usually adamant that they are right because it’s maths and they were taught it, right? Those who don’t are ridiculed. The BODMAS/PEMDAS believers are certain that they are correct because, it’s maths, isn’t it, and maths is never wrong. They lose track of the fact that PEMDAS/BODMAS is just a convention , and other conventions, such ‘strict left to right’ or SADMEP/SAMDOB (the reverse of BODMAS/PEMDAS) could be used. Providing both parties agree on a convention, they can communicate mathematical (well, arithmetical or algebraic) ideas successfully to each other.

Those who have been taught about implied multiplication by juxtaposition dispute the answer with those who haven’t heard of it, but really, the answer to the poser is irrelevant. The reason for that people post these dubious and intriguing equations is not to seek a value for equation. It’s not to find ‘the answer’. It’s usually to stir discussion, and the rowdier the discussion is, the better from the poster’s point of view.

A Long Long Way to Go

River valley (Blender) — My first attempt at a landscape with river valley in Blender 2.8.

I’m pressing on with Blender, using the 2.8 beta, rather than the current 2.79 version. I figure that if I’m going to learn Blender, I might as well use the version that will be the current version when I get up to speed.

I’ll get to Blender in a minute but I’d just like to mention that I have a few writing ideas floating around in my head. I’ve completed one story and I’m mulling one story line in my head, but the characters want to do something that I’m not completely happy with. Yes, that is how it seems to me. I have little control over the characters and what they do!

The way that they want to act is within character, so I suppose I will put pen to paper, or rather, pound the keyboard, eventually. Sometime soon I’ll drop Blender for a while and switch over to my word processor, but for now I’m still learning Blender.

The image is from FREEIMAGES (https://www.freeimages.com/photo/television-studio-1-1171607), supplied to them by Joseph Hoban.

Blender is complex because real life 3-D objects which artists are trying to draw are complex. For instance, in real life there is rarely a single source of light. We may say that a an object in a scene is lit by the sun, but really it is lit by the sun, by the ambient light from the sky and the surroundings, and by reflections from other objects in the scene.

The image above shows a studio set up for television, and it shows multiple sources of light and the stage on which the scene, whatever it is, is to be set. The stage is illuminated by a lamp, but there is also a shadow on it. The shadowed part can be seen because it is illuminated by other lamps. Things in the unlit areas around the stage can be seen dimly because some light is reflected from the stage or it spills over from other lamps.

Lighting is complex and one of the things that Blender does is emulate some of this complexity. When someone builds something in Blender the object that is modelled is colourless and textureless. It’s like one of those dressmaker dummies that is merely a shape and a framework for the clothes to be are created on it.

Some of the lighting has been set up for this object, but not enough. No materials have been applied.

The object in the above image is illuminated by a light source (or rather the object is drawn as if it is illuminated by a light source). It stands on nothing, is surrounded by grey, and is visually it is not that impressive.

For a number of reasons I recreated the tower and this time I gave it some colour by adding colour to the background and another light source, and some ambient lighting.

Cartoon tower — Cartoonish tower, with background, something to stand on and some ambient lighting. There is some other lighting, but it is a bit dim and hard to see.

That still looks cartoonish! It’s still decidedly unimpressive, but I am, I hope, making progress. I’ve learned a lot and hope to put some of it into practise soon. I’ve watched dozens of videos, some of which helped and some of which didn’t. Some of the people making the videos are very good. Others click rapidly on the various windows and menus, leaving me confused.

Some videos seem to refer to earlier versions of the software, which means I have to figure out some of the steps that the video creator just quickly clicked over. Some show how to do specific tasks, like create a floating island, or realistic landscapes, and I have to extract from those the techniques that I need. But I believe that I’m making progress. Maybe by 2020 or early 2021…

I forgot to mention that once the model has been built and been decorated with materials and textures, it is finished but only exists as a Blender file. To produce images, like those above, the object needs to be rendered. The 3-D object needs to be converted to a 2-D image file. Rendering of complex Blender files can take hours as it is a complex mathematical task. The above images, and the one below that I am going to leave you with, took only a minute or less.

The last image is an early version of the river valley that I started this post with. Since then I have dialled back the ambient lighting and added some “materials” to colourise it. It’s probable that I will rebuild the river valley completely.

River valley (early version) — An early version of the valley and river.

Please read my books. The paperback versions can be found Amazon, and the eBooks can be found there or at your favourite eBook store. Just search for my name, Cliff Pratt. I mainly write fantasy fiction.