Sunday, June 15, 2014

Fibonacci Masonry

Today I’m taking a look at an esoteric topic relative to masonry: the Fibonacci sequence, the golden mean and some of the resulting geometry.   I will attempt to describe some of my own thoughts on this topic which are not yet fully formed, but seem to hold some promise nonetheless.  I beg the reader’s indulgence if I am overly speculative, but this is the nature of this particular beast.  I hope that someone else out there may be able to add to my speculation and –perhaps- provide additional insight into this curious realm of mathematics and geometry.

In mathematics,  the Fibonacci numbers or Fibonacci sequence are the numbers in the following integer sequence:
1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\;
or (often, in modern usage):
0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; 
By definition, the first two numbers in the Fibonacci sequence are 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. (taken from Wikipedia)
The earliest occurrence of this numerical sequence is found in Indian mathematics, in the context of  Sanskrit prose structure.  In the oral tradition of Sanskrit, great emphasis was placed on how long syllables (L) mix with short (S) and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonacci number Fm + 1.   This prose structure is first traced back to Pingala, at around 200 BC.  It was later more fully described by Virahanka, at around 700 AD: Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations]. (taken from Wikipedia)
For the past 4 years on this blog I have spent the month of April writing poems about masonry for National Poetry Writing Month (NaPoWriMo) after being dared to do so by a poet friend.  It might be interesting to attempt a poem about masonry using the Sanskrit tradition of employing a Fibonacci sequence relative to variations of meter.  I wander…
In the West, the Fibonacci sequence was first realized and articulated by Leonardo of Pisa (known as Fibonacci) who described it in his book Liber Abaci (1202 AD).  He described the numerical sequence bearing his name by describing the idealized growth of a rabbit population over time.
As the Fibonacci sequence gets longer and the numbers get larger, the ratio between 2 adjacent numbers in this sequence approaches the golden ratio, or golden mean.  This is mathematically expressed as:  \psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\cdots
Put another way, the golden ratio itself is approximately 1.6180339887…
Geometrically, the golden ratio may be expressed as a rectangle, one side being equal to 1.0, the other side being equal to 1.6180339887…

This rectangle can be used to generate a Fibonacci spiral:

I recently used this relationship to create a series of “Fibonacci spiral bowls” from clay on my potter’s wheel.  This was a fun experiment, I may make some more and play with this form a bit more.






It occurred to me that it would be a simple thing to make a masonry Fibonacci spiral structure using my triangular blocks to build cylindrical sections of varying radii, just as I have done with bowls of different radii.  Such a structure could be aesthetically interesting, and it could perhaps create an interesting interior space.  It may also possess unique characteristics which may serve some functional purpose: perhaps acoustic, or wave attenuating, or even structurally stronger.  I would like to build a Fibonacci masonry spiral and see what it’s like.
I also built a curious sculpture which relates the golden rectangle to an icosahedron.  If 3 golden rectangles are assembled at right angles to each other (orthogonally: x,y and z axes) then the corners of these 3 rectangles describe the corners of an icosahedron.  These pictures describe it better than words.





Finally, I would like to create a series of rectangular masonry bricks which employ the golden ratio in a way which I have not seen done by others.  Each rectangular brick would possess the edge lengths of 0. 6180339887… (depth), 1.0 (width), and 1. 6180339887… (length).  I would like to create a whole series of these bricks, with many different sizes.  Each size would be scaled by the golden ratio; each would get larger (or smaller) by a factor of 1. 6180339887… or 0.6180339887…   It seems to me that these bricks could be arranged in some very interesting patterns.  It would be necessary to dry stack them in order to realize the curious geometric relationships, since mortar would change the geometric patterns between bricks.  I have done some crude hand sketches which illustrate the curious possibilities of such a modular series of rectangular bricks which employ the golden ratio, but I am not sharing this for now. I am curious what anyone else out there might come up with.  If you have any ideas, let’s share them!  Show me yours and I’ll show you mine. 

I made this bowl a few days later.  I tried to include helicity to the Fibonacci spiral; I used 8 bowls scaled by Fibonacci - the largest is 24 inches, 61 cm diameter.


I hope to hear from someone out there, let’s do Fibonacci masonry!

Thursday, May 1, 2014

The mollusk, the arch and conjugate shearing

I’ve written repeatedly on this blog about nature’s masons.  Nature is the ultimate inspiration for design;  evolution showcases many masonry techniques.

Mollusks have recently been investigated by two researchers at MIT, graduate student Ling Li and Professor Christine Ortiz.  Their research findings were published in the journal ‘Nature Materials’ (March, 2014) and focused on the mollusk Placuna placenta



This mollusk’s shell exhibits very tough qualities (resistant to crack propagation) while simultaneously remaining optically transparent.  When subject to extreme focused stress -such as may be encountered by its predators- the calcite material of Placuna placenta’s shell demonstrated very efficient energy dissipation and the ability to localize deformation, limiting damage to the area directly impacted and preventing crack propagation.

The mollusk’s shell is comprised of around 99% calcite and around 1% organic material which bind the calcite crystals together.  This is somewhat similar to the sharp defensive spikes found in sea urchins (as discussed here) which are also made primarily of calcite with small amount of organic binder material present.  Pure calcite (without organic binder) is a brittle crystalline material which easily cracks.

The mechanism wherein the type of deformation in Placuna placenta shell occurs was studied by using an indentation apparatus consisting of a diamond tip which is forced into the mollusk shell.  The resulting damage to the indent region was then visually recorded using electron microscopy and diffraction techniques to characterize the resulting damage.



This research cleverly showed that the deformation (or strain) of the mollusk shell was a crystallographic ‘twinning’ response to the applied stress.  Crystal twinning occurs when two separate crystals share some of the same crystal lattice points in a symmetrical manner. The result is an intergrowth of two separate crystals in a variety of specific configurations. A twin boundary or composition surface separates the two crystals.



Part of the crystal shifts its position in a predictable way, leaving two regions with the same orientation as before, but with one portion shifted relative to the other. This twinning process occurs all around the stressed region, helping to form a kind of boundary that keeps the damage from spreading outward (preventing crack propagation).



This twinning mechanism provides for conjugate shearing.   The conjugate shearing mechanism has significance in terms of a toughened structure and is better than a conventional masonry arch structural response to an applied stress of voussoirs forming hinges.



Conjugate shearing was initially employed by geologists as a term to describe shear fractures in rocks subject to compressive stress.  The context and scale of this geologic feature have kept it from being analyzed, utilized or realized in the context of microscopic analysis or in the context of masonry design and modular structural systems.   Similarly, it is apparent that biologists and engineers have failed to fully appreciate the conjugate shearing mechanism demonstrated by the Placuna placenta’s calcite shell structure in response to applied stresses such as the indentation tests done by researchers at MIT.



The force required to cause conjugate shearing to occur (in an architectural arch or in a mollusk shell) is much higher than the force required to create a hinging mechanism as occurs in a conventional masonry arch comprised of wedge-shaped voussoirs.  For example, Thor’s hero shrew’s spine is configured in such a manner that it is disposed to conjugate shearing instead of creating a hinging mechanism which leads to buckling and collapse of the spine.  An adult human can stand upon and be supported by the tiny Thor’s hero shrew’s spine without breaking the poor animal’s back.  Conversely, a common shrew does not have the interlocking triangular design of Thor’s hero shrew’s vertebrae; its spine would buckle and collapse in the hinging mechanism of a conventional masonry arch if an adult human stood on top of it: the back would simply and easily be broken (poor regular shrew).




The calcite shell of Placuna placenta and its unique crystallographic twinning response to applied stress is another of Nature’s exemplars of exquisite design which incorporates the structural response of conjugate shearing to create a toughened structure which will blunt and stop crack propagation in an otherwise brittle material. 

Monday, April 28, 2014

Oh Block, why interlock?

If a block in an arch can budge
then it just needs a nudge
for the arch to get weak
you’ll not hear a creak
or some warning snaps
before the collapse.
So what’s to be done?
How safe’s anyone?
If the blocks in an arch interlock
they can’t move, anchored block
no unnerving fragility
the arch itself is stability.


Sunday, April 27, 2014

Trust

I can’t think of a trade I’d trust
more than the trade you really must
believe in to actually do it right
trust a mason all day and night.


Saturday, April 26, 2014

Anchor bricks

Bricks made of clay
have holes or recesses
to anchor the mortar.
It’s the best way
for mortar impresses
itself in the border.


Friday, April 25, 2014

A boy named Mason

He could’ve been a smith
or a sawyer or a cooper
or named for a trade with
skills so super,
he could’ve written apps,
been an Argonaut, Jason
too modest perhaps:
Mom calls him Mason.

Thursday, April 24, 2014

Mason's line

If you’re laying blocks or bricks
in a straight-lined wall with mortar
there are a few quick mason’s tricks
to keep your blocks and bricks in order.

Rather than place blocks askew
you can line them up just fine
all you really have to do
is guide your bricks with mason’s line.