Dr. Edward Teller

In 1957 Edward Teller, the "father" of the H-Bomb, made a set of recordings for the General Dynamics Corporation.  Teller discussed the size and nature of the universe and Einstein's Theory of Relativity.  He closed his talk about Einstein with an amusing comment about science and science fiction.  

Leaf Curl

In his classic essay, Autumnal Tints, Henry David Thoreau wrote of fallen leaves: "How beautifully they go to their graves!...They teach us how to die."  Thoreau was forever drawing moral lessons out of nature.  For most suburban home owners, dead leaves are tree trash. Fall in my neighborhood means the loathsome sound of leaf blowers.  The people next door have a lawn service.  I think the lawn service gets its leaf blowers from an aircraft engine factory. I rake my leaves, but I can do that because I am retired and can spend parts of several days swishing up piles of leaves and trundling them in a large plastic tarp to the curb, where they are vacuumed up by large trucks and taken to a county mulch pile.  The county will recycle the leaves and deliver mulch in the spring.

Thoreau did not dwell on his death metaphor.  Autumnal Tints records his close observation of the turning of the trees in and around Concord, Massachusetts. He extolled the vivid reds, brilliant yellows, and mellow oranges of the maples, oaks, and elms that made up the majority of the trees in his dwelling spaces. He even exclaimed over the colorfully variegated mixtures of leaves fallen from a several different, close standing species of trees.  Thoreau's eyes in this essay were on the tints of tree leaves, and this is their most readily noticeable autumnal feature.  When they fall to the ground, they soon lose their coloration and lie on the ground, inert and brown.  But pick up a leaf and turn it around in your hand.  Fallen leaves shrivel as their lose their water content.  They sometimes curl in exquisitely sculptural forms.  Below are three different Chestnut Oak leaves.  I've arrayed them to display the various complexities of form manifested by leaf curl.  First is a simple folding over from the tip of the leaf.  Next is a double, asymmetric curl along the length of the leaf's sides.  Finally, there is an elaborate twist that I think is dynamically appealing.  Leaf curl is a subtle dimension of the aesthetic pleasure of looking closely at leaves, but sculpturally curled leaves are rare and it's been my experience that you have to wade through a lot of fallen leaves to find something worth putting up on your blogsite.

Fold

Lenghtwise Curl

 

Twist

Fractal Images

Factals are geometric figures that are self-similar at all scales. The branchings of trees and lightning are fractal forms. Benoit Mandelbrot, a mathematician working at IBM, discovered a way of producing fractal images by means of iterating equations and plotting the results on a computer terminal. During the 1990s many computer buffs wrote software to make fractal images. Such images can be strikingly beautiful, if a bit bizarre. They are often somewhat askew in shape, although they can also be perfectly symmetrical. The intricacy of many images recalls the complicated designs of Celtic knot patterns, the illuminations of the Lindisfarne Gospel and the Book of Kells, Islamic tilings, and other ornamental designs that exhibit what art historian/psychologist E.H. Gombrich has termed amor infiniti--a love of the infinite. In The Sense of Order Gombrich provides an illustration of what he calls decorative filling. The crosses in this illustration can be repeated at smaller and smaller scales. Utimately, the repetitions will fill the originally blank space. Crosses within crosses within crosses....

Nearly all books or web pages about fractals have an image of the Sierpinski Triangle, which fills its space in a manner similar to Gombrich's crosses.

Triangles within triangles within triangles....

Below is a sampling of some of the images I produced with software that I wrote.

Here spirals extend from spirals as far as pixel resolution allows. By this I mean that if we blow up a small section of any of these four images, a pattern of sprials on spirals appears. The blow-up resembles the larger pattern we began with. Mathematically a fractal is infinitely extensible. Although we can see only a part of the infinity of self-similar patterns, this glimpse is sublime.

I call this one "Firedance." The spiky shapes come from a method of fractal image production called orbit trapping. Orbit traps greatly enhance the weird visual aspects of fractals. "Firedance" is derived from a Julia fractal. Here is what an unenhanced Julia fractal looks like.

This image is quite striking, but it does not have the globular, or ovate, or tubular, or spiky appendages that the orbit trapped images above possess. 

Double Tipped Leaves

Leaves grow by producing cells from two special germinative parts called the apical meristem and the lateral meristem. The apical meristem is at the tip of the leaf. Usually, each leaf has a single tip, but I have come across a few leaves with double tips. This is like finding a four-leaf clover. It is hard to tell whether there are two collateral meristems or an apically driven leaf tip and a lateral growth that happens to parallel the dominant tip. Here are some examples. What do you think is happening in these departures from the standard, single tipped leaf?

I don't think the four leaves above are particularly good looking. I have found only a few double tipped leaves that have aesthetic appeal.

This is a small White Oak leaf. It's symmetry and leaf lobe elaboration are particularly eye-catching.

Below are two more small White Oak leaves. The leaf on the left appears to split into two growing tips. The one on the right seems to exhibit a side lobe (on the left side) which closely parallels the main portion of the leaf blade.

My favorite double tipped leaf is a Pin Oak, which has a lovely, nearly symmetric shape.

Below are two Tulip Poplar leaves. The one on the right is a standard shape. The one on the left appears to be a double tipped leaf, but it is so undeveloped that it might be a deformity. It also might not be a Poplar left. I found them close to one another, but the doubled leaf bears so little of the usual identifying characteristics that it's hard to tell. Call it a leaf oddity.

Trees as Individuals

Trees of different species are partly identified by the distinctive shape of their leaves.  But leaves from different trees of the same species often differ in remarkable ways.  Observing such intra-species differences is one way of recognizing the individuality of trees.

The six leaves in the row above come from the end of a single branch of a White Oak tree. The lobes of these leaves are simple structures with uniformly rounded ends.  Minor varations in midrib curvature and in the number of fuly formed pairs of lobes do not obscure the overall resemblance among the leaves.  The rounded, simple lobes characterize White Oak trees and it is this shape that most tree identification images use.  Unelaborated, smoothly rounded lobes constitute a shape norm for the White Oak leaf.

But White Oak leaves vary enormously.  The leaves in the group below come from the same tree.  They resemble one another, but differ noticeably from the leaves of the first group.  Their primary lobes tend to form smaller, secondary lobes. In addition, the lobes tend to square off at the tips.

It is probably not known how many varieties of White Oak leaf shapes there are.