Sunday, 28 February 2016

Camellias and Geometry

"Camellia japonica cultivar", drawing by Sarah "Sallie" Thayer, 2016

Certain flowers and plants easily reveal their geometric patterns -- like the Camellias above.

Golden Spiral Collage -- Source Unknown
I have always been fascinated by those bits of the natural world that we see around us each day that easily reveal their "template". For example, we can quickly think of any number of spirals that are part of our daily lives such as the Nautilus shell, a hurricane, a spiral Aloe plant, an egg or even the human ear.

By the way, for those who might not know, this spiral that I am referring to in today's column is known as the "Golden Spiral" and you can't say much about the golden spiral without bringing in Fibonacci numbers which provide us with the Golden Ratio. The Fibonacci Number sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics. No need to worry, however, I will not be giving you a lesson in geometry or the Fibonacci Sequence of numbers! Rather, I am going to post a drawing which demonstrates the relationships found in these naturally occurring spirals. (See diagram below).

Golden Spiral inside box formed by
Fibonacci numbers relationships 
Here is a brief paragraph on how the Golden Spiral is described mathematically along with a bit of information about the Golden Ratio. If any of this information interests you, then, by all means, pursue it farther. 

"In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. Approximate logarithmic spirals can occur in nature and golden spirals are one special case of these logarithmic spirals.  [Just to let you know, the particular Camellia featured today is a Japanese cultivar named "Phi" which is the way the name of the Greek letter "φ" is pronounced].
Johannes Kepler (1571–1630) proves that the Golden Ratio is the limit of the ratio of consecutive Fibonacci numbers, and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel."  

Why mention all of this? Well, while working on today's featured drawing, I could not help but be aware of how visible the curves, spaces and spirals are in these particular flowers.  So, when I finished the drawing, I made a sepia and white copy on which I quickly traced the Golden Spiral and then tried placing the rectangles formed by the Fibonacci in their proper places.  (See drawing below).

"Camellia japonica with Golden Spiral overlay", by Sarah "Sallie" Thayer, 2016

Of course, I was doing this freehand, without measuring, and so the placement of my lines is far from perfect, but I am able to give you some idea of how these spirals and ratios can be applied in art. (Also see examples below).


As you can see, my drawing does not sit as perfectly as it might within the frame formed. However, while a drawing will appear more pleasing to the eye the closer its arrangement falls somewhere within these patterns, Nature does nothing perfectly and the wise, repsentational-type artist never forgets this.  In real life, you may find a flower whose petals appear to form one perfect spiral after another and then, suddenly, you come across some petals that don't form any shape at all. So, while there may be universal templates, remember that every single thing in creation is unique -- thus there will be naturally occurring variations in every created thing.

Certain sentences and images above were taken from various Internet sources


"Suki Drinking from the Lily Pond", drawing by Sarah "Sallie" Thayer, 2016

I  have no idea why on earth I decided to put Suki into a recent drawing I did of Water Lilies.  A moment of whimsy, I suppose.

I actually did the above drawing a few weeks ago while working on a mandala composed of Water Lilies -- but, then, decided not to use it in the mandala.  So, as I was putting the file away in its proper folder, I suddenly had a thought: "Hey, Sallie, why don't you add Suki to your drawing?  You can show her drinking from the Lily pond."

This idea in itself is rather whimsical as Suki would never drink from a Lily Pond or any kind of pond for that matter.  Even if she were an outdoor cat, I am sure she would continue to do what she has done all her life -- get her "water" from the liquid in her food! However, as you can see, I went ahead and included Suki in the drawing anyway.  Perhaps I was feeling very affectionate towards her at that moment!

To be honest, I have never known a cat that was so uninterested in water. Over the years, I have often tried putting out a water dish for Suki and every time she refuses to go anywhere near it. Instead, it sits on the floor gathering dust.  Obviously, she needs some liquid in her diet and that is probably why she turns up her nose at any food that doesn't contain some sort of gravy. As well, I always top up any gravy the canned food contains with a spoonful of water (what she doesn't know can only help her!).

The only exception to Suki's general attitude towards water seems to be water dripping from the tap in the bathroom sink.  I discovered this years ago when I once failed to turn off the tap completely.  I came back a few minutes later to find Suki sitting on the counter-top, gleefully splashing water everywhere.  Sadly, she wasn't drinking any of it -- just making a big mess for me to clean up.  

Now, of course, she can no longer jump up on the counter due to the arthritis in her hip.  Every so often, though, I turn the bathroom sink tap on just a tiny bit, pick Suki up and place her on the counter, letting her play with the water for a while.  She may not drink any but she does get an extra bit of moisture into her system when she thoroughly grooms her wet fur after her watery play time.

As for me, I have had a to deal with additional pain problems this past week so I am back to experimenting with my medication regimen again.  I did have one medical test this past Friday, but won't know the results until sometime later this week.  Otherwise, things are much the same -- sadly.



"Christ Teaching the Parable of the Fig Tree", drawing by Sarah "Sallie" Thayer, 2016

At that very time there were some present who told him about the Galileans whose blood Pilate had mingled with their sacrifices. He asked them, “Do you think that because these Galileans suffered in this way they were worse sinners than all other Galileans? No, I tell you; but unless you repent, you will all perish as they did. Or those eighteen who were killed when the tower of Siloam fell on them—do you think that they were worse offenders than all the others living in Jerusalem? I tell you; unless you repent, you will all perish just as they did.” 
Then he told this parable: “A man had a fig tree planted in his vineyard; and he came looking for fruit on it and found none. So he said to the gardener, ‘See here! For three years I have come looking for fruit on this fig tree, and still I find none. Cut it down! Why should it be wasting the soil?’ He replied, ‘Sir, let it alone for one more year, until I dig around it and put manure on it. If it bears fruit next year, well and good; but if not, you can cut it down.’ ” Luke 13: 5-9

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