Day and night is present in our lives since the moment of our birth. Place a lit candle in this cup and a world of reflection opens, giving a fantastic ambience. Light is all about sensation, math, physics and some mysticism at the mixture. Light has been studied by many geniuses from Albert Einstein to Richard Feynman giving birth to fantastic ideias about the universe around us, but a side from the speed of light, it’s diffraction the common human knows little about it.

Many activities explore light, modelled it and play with it. Photography explores light to an outstanding level. Texture, highlights, color, black and white, grey toned images are just a few examples.

Unfortunately I don’t have much answers regarding light simulation and what can be done with it. Interestingly games play with light from rendering shadows, light glare, and more effects. Photo editing software also create lens flare effects, maybe these can be considered to be some sort of light modelling.

Either by exploring light on a nice summer day or by taking that photo at sun down, light remains the source of inspirations that holds still many secrets to tell.

I always like to play around with my models and see what happens next. Don’t you? Problem is that most of the time we don’t know how to do it and with what resources… Let me show you how to model a Fibonacci spiral easy.

First we will use Python. Don’t worry. Install Python or just install Anaconda and open Spider.

Once in spider paste the following code to the editor and hit play.

The result should be something like this:

There is a nice page that explains how the python script works and gives a deeper insight into the Fibonacci series:

# chromoSpirals.py
# ----------------
# Code written by Peter Derlien, University of Sheffield, March 2013
# Draws spiralling patterns of circles using the Golden Angle.
# ----------------
# Import from the numpy and matplotlib packages.
import numpy as np
import matplotlib.pyplot as plt
import matplotlib
from matplotlib.collections import PatchCollection
import matplotlib.patches as mpatches
ox=0.5; oy=0.4 # centre of plot
ndiscs=300
ndiscs=input('No. of discs (e.g. 300)? ')
ndiscs=int(ndiscs)
ncols=input('no. of colours (1 to 34)? ')
ncols=int(ncols)
offset=0.0
offset=input('offset (in radians) from golden angle? ')
offset = float(offset)
tau=(1+5**0.5)/2.0 # golden ratio approx = 1.618033989
#(2-tau)*2*np.pi is golden angle = c. 2.39996323 radians, or c. 137.5 degrees
inc = (2-tau)*2*np.pi + offset
theta=0
k=0.1 # scale factor
drad=k*(1+5**0.5)/4.0 # radius of each disc
minv=maxv=0 # minv and maxv will be used later to display inputs chosen
# now collect in list 'patches' the locations of all the discs
patches = []
for j in range(1,ndiscs+1):
r = k*j**0.5
theta += inc
x = ox + r*np.cos(theta)
y = oy + r*np.sin(theta)
if y > maxv:
maxv=y
elif y < minv:
minv=y
disc = mpatches.Circle((x,y),drad)
patches.append(disc)
# start building the plot
fig = plt.figure()
ax = plt.axes([0,0,1,1])
# create text to show which inputs the user has chosen
font = "sans-serif"
maxv=maxv*0.95
nd = 'ndiscs: '+ str(ndiscs)
plt.text(minv, maxv, nd, ha="center",family=font, size=14)
setting = 'angle offset: '+ str(offset)
plt.text(minv, minv, setting, ha="center",family=font, size=14)
nc = 'ncols: '+ str(ncols)
plt.text(maxv, maxv, nc, ha="left",family=font, size=14)
# build colour cycle, using a number between 0 and 100 for each colour
colcycle=[]
s=100/ncols
for j in range(ndiscs):
colcycle.append((j%ncols)*s)
# bring together the information for locations and colours of discs
collection = PatchCollection(patches, cmap=matplotlib.cm.jet, alpha=1.0)
collection.set_array(np.array(colcycle))
ax.add_collection(collection)
ax.set_xticks([]); ax.set_yticks([]) # suppress display of axes
plt.axis('equal')
plt.show() # display the plot we have built

It is really beautiful when we can see math in the world around us! How can a flower find the most optimised structure to balance growth, sunlight and moisture in each seasonal energy burst? I guess millions and millions of year and we get the golden ratio in the fibonacci series.

Today we use super computers and special algorithms for complex problems, however in the past mathematicians could model problems and solve them with very little resources. It is always good to get to the basics and see art in the form of math.

Leonardo of Pisa nicknamed “Fibonacci”, brought state of the art Arab mathematics to medieval Europe. One of his contribution was his book Liber Abacci where it described a series that we commonly see in nature. It’s all about spirals but they follow a rule…

The Fibonacci series can be seen as a numeric sequence starting with two ones where each subsequent number is equal to the sum of the preceding two numbers: 1, 1, 2, 3, 5, 8, 13, and so on.

Although this number series was known in India previously, Leonardo helped spread this knowledge worldwide.

The most interesting thing is that he stumbled upon it trying to solve a rabbit problem! I think the book proposed the following:

‘A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits are produced from that pair in a year, if it is supposed that every month each pair produces a new pair, which from the second month onwards becomes productive?’

On my previous post I got to discover that the new design for the 2017 Brompton bike model is much more robust. Didn’t get to understand if it was designed on purpose or was a technical difficulty. My hunch was that they enjoyed the design and it made construction simple. However with experience they concluded that it was a problematic design and went for a new design. They could have just reduced the central bar length but instead almost removed it to a more challenging design at least for manufacturing.

FreeCAD produces nice images and this one is an undeformed vs deformed shape. Here I think it is visible that near the stem is where most of the stresses will be generated. So if you have a handlebar from a bellow 2017 model take special care to not pressure in an exaggerated manner the handlebar down.

Special thanks: I would like to thank the FreeCAD forum team and in particular Bernd for providing troubleshooting for FreeCAD and showing how to do the deformed and undeformed image.

While browsing the Internet I found some past references to Brompton Handlebar cracks near the bending elbows. I noticed my colleague Brompton had a different design and thus concluded that probably the Brompton bike team, decided to turn the handlebar more robust.

Check out my previous post to know what I am talking about. Click here

I notice that I make some force to the ground direction when pedalling and curving, special when the road has bumps.

In wich elbow will the stress higher and the cracks be formed? I will be continuing on this subject, so please stay tuned.

Going back in time, I just remembered about a game that gave me quite a good understanding of stress distribution. Bridge builder. Using simple truss combinations to allow passage of vehicles through it. Some times the most chalenging designs required the use of circular shapes to transfer stress from one point to another, distributing it through as many bars as possible…

This sun lounger supporting legs are made of a single component and has one main function, to support the weight of the occupier and a bonus function to support the arms. Stresses are evenly distributed throughout the structure and the design feels more organic.

I started my week looking at my Brompton bike handle bar and noticing that a friends bike had a different one. Same model type but with a different design shape.

My Brompton version is a 2017 model however my friends model is a 2015 model.

Notice the length of the central U shape arms. Smaller?

In my bike version there was a reduction of the length of the straight tube and the two bending zones are pretty near .

I have to mention that this changes a bit how the Trigo Brompton bike acessory fits. Why? On the first design (2015 version) the accessory at the bike folded state will be positioned much higher far away from the wheel axel giving space. In my version it gets aligned/touches making my life just a bit more difficult to fold the bike with the go pro attached.

I am wondering why they made the change. Was it due to reduced stiffness? Cracking of the handlebar?

I really liked the biking handling the new handlebar offers it feels great and allows me lean a bit when curving.

I had the magnificent privilege to stare at the moon on a deeply dark night. I went initially to see the milky way since the observatory is located on a Dark Sky reserve. A Dark Sky reserve is a an area which guaranties low illumination at nigh allowing for high quality star gazing at night. Unfortunately the Moon out shined many starts and reduced star gazing conditions. Although I did’t have great conditions the moon was positioned just right for some good Moon photos.

I normally focus my attention to more standard questions “will the bridge withstand the train?” or “what is the temperature in my room if i turn on the heater” or even “will this shape have good aerodynamics?”. But do we often ask our serves “will light reflect?”, “How does light reflect?”. (Maybe optics specialists look at this) Simulating light. I know this subject isn’t new. What is this nowadays? Still this affects our glasses, telescopes, data connections through finer optics.

Since I know little about this subject, I will investigate deeper what is out there regarding simulating light.

Previously I discussed two design approaches, a tilted leg design and a straight leg design. These designs were very so, much different. The tilted legs showed some harmony on the design and looks great, the straight legs log resistant but no so much beautiful.

Today I will analyse the side leg shape seat.

From all this approach looks more stressed. The problem is that the table is supported by two legs instead of four plus it is taking all the torsional stresses generated by the table and weight. In regards to the design it looks that it will fall apart and challenges hour notion of safety. From all the designs this will be the most demanding regarding the material strengths.

Using a steel shaft unibody to support the upper seat table. The seat table is where we sit on. In order to make this type of approach it’s very important to use a very resistant material that will not break, can flex and is light weight. Not any wood could be used for this design since it is heavily stressed compared to the other two alternatives.