Nautilus in Black and White—The interior of the nautilus shell is roughly logarithmic. Each nautilus shell maintains the same proportions in the spiral throughout the lifetime of the cephalopod that creates the chambers within the shell.
In the wild, no scientist has ever documented a nautilus from hatching to maturity. So a great deal is unknown about these wondrous animals.
What is known is that they can live for 15 to 20 years and reach maturity when they are about 12 years old. Nautilus eggs are huge and are among the biggest eggs relative to adult size of any animal. As a cephalopod, the nautilus is a member of the same family as octopi and squid.
I photographed this cross-section of a chambered nautilus shell on a black background and paid special attention to lighting to bring out the luminescent quality of the internal shell.
Nikon D200, 50mm Zeiss macro, 8 seconds at f/32 and ISO 100, tripod mounted.
A spiral is a three-dimensional curve rotating around a fixed position that is receding or approaching the viewer—think of a spiral staircase or the spiral arms of a galaxy. Spirals range in size from the tiny spiral in a flower to the vast reaches of intergalactic space; the spiral is a universal shape that occurs over and over again.
A fractal image is an image where small areas of the image are replicated across the larger image in its entirety.
There a number of different kinds of fractal images, ranging from those generated mathematically (take a look at the note on page 126 about fractal art) to photographs of fractals in nature. With natural fractals, examples include the pattern of an ice crystal that mimics the pattern of a frozen pond (page 121), branches in a complex tree that resemble the tree itself (page 123), and portions of a landscape that look like the landscape as a whole (page 174–175).
Keep in mind that visual uses of spirals and fractals are not intended to be precisely mathematical in many images. Most representations of spirals and fractals are not exact: a spiral staircase is often irregular to fit the needs of the architecture, and a portion of a fractal image can be reminiscent of the image as a whole without precisely duplicating the image. Outside of the world of pure mathematics, precise fractal replication is aspirational rather than exact.
A spiral is a shape—think of a chambered nautilus shell such as the one on page 108—and also a template for a pattern within an image. For more about working with pattern templates, turn to “Design Patterns” on page 104.
A fractal is not a shape in and of itself, but represents a family of shapes. Since each fractal contains information about how the fractal was created and about the fractal as a whole, a fractal can also be thought of as a kind of blueprint for life and a way of creation.
Inside a Red Vein Indian Mallow Blossom—The Red Vein Indian Mallow blossom of the species Abutilon striatum is beloved by hummingbirds. The flower is veined and shaped like a calyx that dangles off branches. If you look inside a blossom, you’ll see a deep cup with the petals swirling in a spiral around the sigma, ova, and anthers, which itself appears spiral.
When I looked inside this hummingbird magnet, it seemed like a great place to try out my Laowa Macro Probe lens. This lens is a unique macro wide-angle lens on a stalk that is over a foot long. You might think it is an ideal lens for photographing venomous small critters without getting too close to them, and you’d be right! It’s also a great lens for getting inside an Abutilon striatum and other concave blossoms.
The macro probe lens includes an LED ring light at the end of the stalk. The wide-open aperture for this lens is f/14, but I found that it needs to be stopped down all the way to its smallest aperture of f/45 to get even minimally adequate depth of field. Focus stacking is not an option here due to the ergonomics of the lens because it is constantly slightly vibrating and slipping.
I started with the camera and probe on a tripod and the probe lens inserted into the flower blossom. I wanted to photograph the flower stopped down at f/45 and use the natural backlighting to create an attractive image. But at f/45, using natural light, nothing can be seen through this lens. What to do?
My workaround technique was to turn on the LCD ring light and focus with the aperture wide open at f/14. When everything looked good to me, I turned the ring light off and stopped the lens all the way down to f/45, and created the image.
Nikon D850, 24mm Laowa Macro Probe, 24mm extension tube, 2 minutes at f/40 and ISO 64, tripod mounted.
Both fractals and spirals are often found in nature. A spiral can be a fractal, and a fractal image often contains spirals; the two shapes are related.
Fractals are “cousins” of the spiral, often accompany spirals, and are a design pattern that is invaluable to learn to work with. The fractal way of life, which is closely related to our sense of the infinite, is always significant.
Spirals are, themselves, powerful shapes with emotional and spiritual connotations.
In this chapter, we’ll explore how to use spirals and fractals in your compositions and how to take advantage of their power.
Working with Spirals
When working photographically with spirals, there are three common points of view. These refer to the camera and focal plane position in relationship to the spiral subject matter. The spatial relationship of camera position to subject is discussed on page 61.
- Perhaps the most common configuration is a head-on photo of a spiral that is in front of the camera; for example, the photo of the nautilus shown on page 108.
- Another common camera-subject relationship involving spirals is to look down at the spiral. Examples include the mallow blossom shown on page 111 and the shell on page 114.
- Looking up and through the spiral (for example, the staircase shown opposite) is somewhat less common. But gazing up a spiral is quite powerful because seeing to the end of the spiral brings a sense of completion and viewing the infinite.
It is important to identify which of these three scenarios your image will use before you undertake a detailed setup, as they are quite different in implication, meaning, and how you want to position the camera.
The first configuration, a frontal, head-on spiral, leads to a relatively flat composition that does not require much depth of field. The key point here is to position the focal plane of the camera as parallel as possible to the subject.
Twisted—The city of Santiago de Compostela in Galicia, Spain, is where the main route of the Camino de Santiago pilgrimage trail ends. It is a city of many wonders and a World Heritage Site. Not least among these wonders is the triple-spiral staircase in the Convent of Santo Domingo de Bonaval. Built by Domingo de Andrade in the 1700s, this is one of the very few triple-spiral staircases in the world.
To photograph this incredible triple spiral, I laid down on the cold stone floor beneath my tripod and used an extreme wide-angle lens (a 15mm rectilinear fisheye) to photograph straight up the spiral.
Nikon D850, 8-15mm Nikkor fisheye zoom at 15mm, six exposures with shutter speeds ranging from 0.5 of a second to 13 seconds, each exposure at f/22 and ISO 200, tripod mounted.
Architectonica perspectiva—The common name for this shell is the “perspective sundial” shell. It is a kind of sea snail, a marine gastropod mollusk of the genus Architectonica. This shell is also sometimes called a staircase shell. The snail is found in Asian coral reefs mostly near the Indian subcontinent.
Looking at this shell, I saw an almost perfect spiral. An issue was that the center of the shell rose like a cone toward the lens. It was clear that depth of field was going to be an issue. To resolve this, I captured a number of exposures using different focal points and combined them in Photoshop, creating the final composition.
Nikon D810, 100mm Zeiss Makro-Planar, 15 seconds at f/2 and ISO 64, tripod mounted; twelve exposures, stacked for extended depth of field.
When you are looking down on a spiral, it’s important to think about how far down the tunnel of the spiral you are going to go to frame the image. There are always compositional choices and options, starting with changing your position.
Depending upon your equipment, you can also change the focal length of the lens. Particularly with spirals photographed close up (such as the shell, opposite), your point of focus and aperture can make a great deal of difference to the composition. Changing the point of focus when you are up close can actually alter the framing of the image. (If you don’t believe me, give it a try!)
As a result of these choices, you can frame the spiral at a distance or close up.
Peering through the camera down a spiral, there is always a center point. A compositional question becomes how much of a spiral around the center point will your image show?
The closer to the center point you are optically, the less your image will reveal of the spiral. However, if you pull back to show more of the spiral, then the spiral can become “tunnel-like” and you lose the sense of grandeur inherent in the spiral.
There’s a balance in this decision about whether to get close or capture more of the spiral from far away, and often an optimum point. This choice largely determines the kind of image you make.
The key thing to think about is symmetrical closure. In other words, the composition should be attractive at the edges of the spiral as well as in the center, regardless of where you choose to crop the composition.
When choosing to photograph a spiral looking up from underneath, the key thing to keep in mind is that you are looking up. Most upward spiral photos are captured with a very wide-angle lens, because there is no other way to capture the full extent of a widening spiral. Therefore, as a matter of geometry, you will need to be below your camera (unless you want a body part to appear in the image). This means lying or crouching on the ground under your tripod.
Whether you’re looking at your subject head on, up, or down, a spiral, like any other subject, needs to fit into the frame boundary of your image (framing is discussed in more detail starting on page 54). This means that as you previsualize a spiral composition, you should consider how the outer edges of the spiral interact with the frame to create an interesting composition.
Positioning the Center
In addition to the interaction between the spiral and the frame, the positioning of the spiral’s center is extremely significant. The most common compositional approach is to position the spiral’s center at or near the center of the photographic composition. You’ll find that roughly 99 percent of spiral photos take this approach.
The advantage of centering is that it leads to a sense of completeness and allows the viewer to see the whole spiral. A possible disadvantage is that placing the spiral center in the center of the composition can lead to an unexciting, static image.
If you decide to break the normal pattern either by omitting the spiral center—as in Caixa Forum Stairs on page 118—or by skewing the composition so the spiral center is close to one edge—as in Wright Stairs on page 119—do so carefully.
If you play visual games with the location of the center of the spiral, you need to know that you are doing so. You should provide clues so the viewer can clearly extrapolate the location of the spiral center, as well as visualize the completion of the spiral.
Spirals are everywhere and are a universally recognized shape and pattern. It’s great to internalize the use of the spiral as a design pattern in your work, and to understand that when you encounter a spiral there’s a good chance of making an exciting composition.
Gem of the Drakensberg—Aloe polyphylla, commonly called the “Gem of the Drakensberg,” is native to the higher elevations of the Drakensberg mountains in the landlocked southern African Kingdom of Lesotho. While many succulents exhibit fractal-spiral patterns, the Drakensberg is unusual in its regularity and obvious spiral.
The plant will grow its spiral in only one direction, either clockwise or counterclockwise. When fully mature there will be at least five fractal-spiralling rows.
While the Drakensberg is considered rare, it has become a popular succulent in the gardens near where I live, so it is possible to find great specimens to photograph like this one!
Nikon D810, 85mm Nikkor macro, 2.5 seconds at an effective aperture of f/51 and ISO 64, tripod mounted.
CaixaForum Stairs—The CaixaForum in Madrid, Spain, an art center on Madrid’s “museum row,” boasts a spectacular, almost rectangular, internal spiral staircase. In this image looking down the CaixaForum stairs, I omitted the center of the spiral from the frame.
My idea was that the viewer of this image would complete the pattern unconsciously—and mentally finish the spiral—even though it’s not fully shown in the image.
Nikon D850, 28-300mm Nikkor zoom at 68mm, 1/20 of a second at f/9 and ISO 1250, hand held.
Wright Stairs—The vast Marin Center in San Rafael, California, is the largest public project designed by Frank Lloyd Wright. The building contains a number of Wright’s distinctive spiral stairs, reminiscent of the shapes in the Guggenheim Museum.
This photo looking up a back stair at the Marin Center is intentionally framed off-kilter with the spiral center toward the top of the frame.
Nikon D300, 12-24mm Nikkor zoom at 15mm, 10 seconds at f/22 and ISO 100, tripod mounted.
Fractals the Natural Way
The gist of the fractal—whether natural or mathematical—is to create complexity from simplicity. In other words, a simple transformation is used on a portion of the image to eventually create a highly complex overall image. In mathematics, this transformation is accomplished through recursion (repetition of a formula). Leaving math out of it, in the real world this transformation is accomplished primarily through a natural process that repeats itself.
In nature, fractals include all manner of branching patterns, such as trees, river deltas, lightning, and blood vessels, as well as spiral patterns like sea shells, hurricanes, and galaxies. In fact, the largest known fractal is a spiral galaxy.
Since all fractals are formed by simple repetition, if you combine repetition with rotation and resizing, it’s easy to create your own spiral-shaped fractals. You can also create your own universes and images using fractal repetition whether or not spirals are involved.
Many fractals can be found in the plant family, with examples including pine cones, sunflowers, aloes, and the agave cactus. Aloes and agaves are typical and interesting because they form a spiral by rotating a fixed amount before growing each new piece. The angle of rotation is determined by each plant’s best angle to the sun. So the angle of rotation of the fractal created by the plant’s spiral growth depends on the sun’s angle in the location where it is planted. Turn to page 117 to see a Gem of the Drakensberg aloe that grows in a spiral-fractal shape.
It is astounding and true that the incredible complexity of organic forms around us comes largely from simple repetition.
This is because most apparently complex fractal patterns start with a simple sub-element. The sub-element is modified slightly with rotations, resizing, mirroring, and so on. This modification occurs consistently and is repeated indefinitely, with the newly modified elements added to the original pattern.
So ultimately, a fractal is a larger pattern using repetition to create a complex and pleasing larger shape that is often symmetrical in part, and all based upon a simple element and method of growth.
Skim Ice—In early spring in Yosemite Valley, California, the surface of still water was covered with a thin layer of ice. The water beneath the ice was partially thickened from the frost and snow, and crystalline drops were embedded in this intermediate layer.
Snowflakes and ice crystals are ideal places to look for fractal-like compositions. If you take a detail of this close-up of skim ice on water, the detail itself could be the entire composition. Out of the simplicity of the ice formation comes incredibly complex patterns and compositions that can be repeated and enlarged as much as the artist would like.
Nikon D850, 28-300mm Nikkor at 300mm, 13 seconds at f/36 and ISO 64, tripod mounted.
California Live Oak—In the hidden nooks and valleys of California’s Coastal Range, ancient gnarled and twisted California live oaks (Quercus agrifolia) thrive. When I photographed this imposing specimen near Walnut Creek, California, in early spring, the tree looked almost as if it might be dead. But it was merely dormant for the winter and would soon be putting forth ample greenery.
Before the foliage camouflaged things, it was very easy to see the fractal nature of the tree. Natural fractal patterns are formed by repeating a simple branching process. While the ultimate fractal can become unbelievably complex, the process of creation is simple and always tells the story of the pattern.
This magnificent tree grew by simple, repetitive branching that led quickly to complexity. The fractals tell the story of the processes that created it.
While many of the tree branches can be taken as fractal pieces, one is shown here to give you an idea of how fractal-like branches from this old tree combine to form the fractal whole.
Can you see where the magnification on the right came from in the photo of the entire tree on the left?
Nikon D850, 28-300mm Nikkor at 210mm, five exposures with shutter speeds ranging from 1/100 to 1/8 of a second, each exposure at f/14 and ISO 64, tripod mounted.
Dandelion Inversion—A dandelion seedpod contains an entire universe within a small, puffy sphere. In this sense, it’s a fractal image: when you look at each seed node within the pod at high magnification, the node itself becomes an entire pod.
When I photographed this image outside, the seedpod was white against a dark background in a field. Back in my studio, I used Photoshop to invert the dark background to white, and presented the design of the dandelion in black against a white background to create the image you see here.
It always fills me with awe and wonder to contemplate the beautiful symmetry and repetition in a modest dandelion seedpod.
Nikon D850, 50mm Zeiss macro, 1/13 of a second at f/20 and ISO 200, tripod mounted.
Romanesco Broccoli—Browsing in the produce section of a local green market, I was transfixed by the spiral nature of Romanesco broccoli, an edible flower in the broccoli family. The flower’s form approximates a natural fractal because each flower is composed of a series of smaller flowers, each one arranged in a logarithmic spiral.
This pattern is repeated in smaller sizes at different levels on the Romanesco broccoli flower. The pattern only approximates a fractal because the pattern eventually ends when the flower size becomes really small—in contrast, a mathematically-generated fractal would go on forever. Incidentally, the number of rotations in a spiral found on a head of Romanesco broccoli is always a Fibonacci number. Fascinating!
Nikon D810, 85mm Nikkor macro, 24mm extension tube, five exposures with shutter speeds ranging from 2.5 to 25 seconds, each exposure at an effective aperture of f/64 and ISO 64, tripod mounted.
Zabriskie View—One of the world’s great destinations for landscape photography is Death Valley National Park in California. Within Death Valley, no one should miss Zabriskie Point, a short distance from the parking lot and possessed of views that are always changing depending on the light. There is seldom a sunrise or sunset when photographers aren’t lined up with their tripods on the viewing platform above Zabriskie Point.
In making this image, I was mindful of the way the folds in the land repeated, with slight variations in the angles of the bluffs. It seemed to me that to make the most visual sense of this landscape meant emphasizing the fractal nature of the scene in front of me.
Nikon D850, 28-300mm Nikkor at 78mm, six exposures with shutter speeds ranging from 1/8 of a second to 3 seconds, each exposure at f/22 and ISO 64, tripod mounted.
Kinds of Fractal Compositions
In my work as an artist, I have created fractal images in a number of ways. Like some other visual artists with a background in mathematics (M.C. Escher comes to mind), besides intuitive visual approaches, I am always interested in artistic methods that involve computation. Of course, the proof of the art is in the pudding, as it were. Good art transcends whatever technique may have been used to make it.
Using computers and mathematical formulas to create fractal art is a significant trend in and of itself. I’ve played with these possibilities over the years. For more about mathematical fractal art, turn to page 126.
Back in the film days early in my career, I played with prismatic auxiliary lenses to create fractal-like effects in-camera. Maybe you’ve seen this effect in 1930s films such as those created by Busby Berkeley. Back in the day, I bought my prismatic lens in New York at a camera-gadget store called Spiratone. These days, this type of special-purpose, auxiliary lens is sometimes referred to as a “fractal lens.”
My favorite approach is to photograph fractals in nature or create them in the studio using specialized photographic techniques. Fractal Face 2, opposite, is an example of a studio fractal creation using in-camera multiple exposures.
Many of the photographs that illustrate this chapter are of fractal-like subjects that I have found in nature. To photograph fractals in nature requires identifying fractal structures when you see them, and understanding a coherent way to relate the fractal composition to the edge of the image.
Fractal Face 2—I used multiple in-camera exposures to create a psychological portrait of a model with a fractal appearance. The multiple exposures lead to an interesting composition with the “fractal” hands around a face with a repeated gaze looking at the camera.
It is interesting that repeating elements in a figure study have such a profound effect on the way we view the portrait. It’s almost as if by breaking up a person into a repetitive pattern, we are making a suggestion of a personality pattern or emotional state.
Consider the impact of the cubist painters such as Pablo Picasso and Juan Gris: largely the impact of their paintings was to break up subject matter into fractal-like pieces. This was most shocking when the subject was people, such as Picasso’s famous Portrait of Gertrude Stein or Les Demoiselles d’Avignon. In fact, Demoiselles was so shocking at the time it was first exhibited in 1916 that it was considered bizarre and immoral and one critic exclaimed, “the end of the world is upon us!”
Nikon D810, 28-300mm Nikkor at 122mm, five images combined via in-camera multiple exposure, each exposure at 1/160 of a second at f/13 and ISO 400, tripod mounted, studio strobe lighting.
Calling Alice—“Curiouser and curiouser!” cried Alice, just as did I after about 24 hours straight creating this image in Photoshop. Working on an image like this containing multiple optical illusions was a little like going down the rabbit hole in Lewis Carroll’s Alice in Wonderland.
The primary image that I used to create this composite was a spiral staircase in a less-frequented area of San Francisco’s Embarcadero Center. The spiral staircase was dark and poorly lit.
To get as far out as I could at the top of the staircase, I spread two of the legs of my tripod, hooking them into the bottom railing. Next, I laid the third leg flat across the top of the railing. Then, I used my tripod to lever myself up and hook my feet into the top of the railing, so I was pretty secure. I used a fisheye lens to maximize the extent of the spiral in this mostly forgotten staircase.
Besides Lewis Carroll, my inspiration was also the artist M.C. Escher. In the spirit of Carroll and Escher, I wanted the “path” in the image to seem to go on forever. Once the viewer enters the image, there is no obvious exit. To accomplish this, I needed to be able to fold copies of the staircase on themselves and to create a context where gravity was irrelevant. Parts of this image, if it depicted the real world, would be impossible under the known laws of physics.
Nikon D300, 10.5mm Nikkor fisheye, 10 seconds at f/22 and ISO 100, tripod mounted; image composited with itself in Photoshop.
Framing Fractals
Many, if not most, spirals are also fractal. Generally, the framing issue with fractal-spirals is straightforward. Whatever your position in relation to the spiral (see page 112 regarding the camera’s focal plane orientation to the spiral), the spiral should take up most of the space of the composition and have an apparent relationship to the boundary frame of the image. So this quickly becomes an issue of positioning the spiral.
Compositional challenges become more intriguing with a complex fractal such as California Live Oak, shown on pages 122–123. With a natural fractal, such as this tree, in some sense the composition has no beginning and no end. How do you put a border around a subject that is inherently borderless and constantly expanding?
I don’t have a simple answer to this conundrum other than to say it is a little bit like Edward Weston’s take on the study of composition and gravity. The response should be intuitive. Compositional framing of a large fractal can probably never be perfect, but it can be graceful. This kind of framing should not seem arbitrary. In other words, there should be a reason for drawing the boundary and border.
It’s a good idea to plan to include enough of the subject matter so that its fractal nature becomes clear, but not so much that the viewer becomes lost in the details of the composition. As with any composition, there needs to be a guiding, underlying principle even if this principle isn’t clear to the viewer. For example, the tree itself organizes the composition in California Live Oak.
With fractal compositions, considering how the viewer will enter the composition, the visual path the viewer will take, and how the viewer will exit the composition is particularly important. See “Entering & Exiting,” starting on page 136.
Epic Stairs—To create this photo collage, I used postproduction techniques that are essentially fractal-like (see text). Calling Alice, shown on page 130, was the base image.
I took a portion of Calling Alice and used it as my repeating element to create the fractal-like effect. The repetitions were modified in a regular way—meaning each repetition was modified using the same angle of rotation and the same proportional scaling. Eventually, I put together the various pieces that I had created, making this collage. The final image shows many elements that are themselves modified sub-pieces of the entire image.
Photo collage based on Calling Alice, created using Photoshop LAB inversions, transformations, and compositing. As a final step, the image was converted to black and white.
Creating Fractal Composites
Fractal compositions can be created in postproduction using compositing with surprisingly interesting results. Post-production fractals created in Photoshop (or with other comparable software) use the techniques for their creation that mimic the mathematical or natural approaches to fractal formation. As in mathematics and nature, this is a process that creates complexity from simplicity.
If you are going to invoke complexity from simplicity, this can be accomplished by creating some simple rules for processing the component parts of your image. There’s no magic formula for what these rules should be, and you most likely will need to find them using trial and error. Examples of the kinds of transformations that you can use to create an iterative rule include replication, rotation, inversion, and resizing.
You do want to consistently apply the rules once you have found them. Consistency is what gives a fractal-like composite the sense of being a regular pattern that is comprehensible.
To create the fractal-like image, you need to repeat the steps—meaning iterate your rules—many times. The way this works is that whatever the rule is, it gets applied and reapplied to the work that has already been altered. It’s this repeated reapplication that serves to create the fractal-like effect. The rule needs to be repeated regularly on the image that has already been modified with your rule.
For example, suppose I make a rule that a portion of the image will be resized smaller by 10 percent and rotated 90 degrees (most rules will be more complex). I apply this a first time, and now I have an image with a portion of itself that is smaller and rotated. Next, I could take the entire modified image with the changes and apply the rule once again to either the entire image or a portion of it.
Crystal Spiral—I spent the morning setting up parfait glasses on my light box. I tried various lenses and angles but nothing really came together. Then, I got really close with my wide-angle, macro-probe lens. Looking through the viewfinder, a very surprising spiral, fractal-like pattern was revealed.
Nikon D850, 24mm Laowa Macro Probe, 0.8 of a second at f/40 and ISO 64, tripod mounted.
Rinse and repeat, then rinse and repeat!
A number of kinds of transformations can work as rules, but you will need to find these by experimentation. Also, the image that the rule operates on must be chosen carefully. Not every kind of image will work for fractal replication in postproduction. Successful composite subject matter usually calls to mind architectural or natural fractal processes (for an example, take a look at Epic Stairs on pages 132–133).
Creative Use of Spirals and Fractals
If I only had two shapes or patterns in my artistic palette, I would likely choose the spiral and the fractal. For good reason, most folks love spirals. Fractals are best thought of as a pattern for creating patterns, rather than a pattern in and of themselves.
Images that make good use of spirals or fractals are always interesting, so I recommend that you look for these shapes and patterns in nature and in your work. The more you study spirals and fractals, the more likely you are to find creative approaches to working with them. The examples in this chapter show some of the ways I have worked with fractals. I hope you find them helpful.
Spirals echo many of the fundamental shapes of the universe around us and are great representations for our life voyage. Fractals demonstrate the building blocks of life and show how complex forms, shapes, and ways of being can come together using simple mechanisms.
Together, spirals and fractals are well worth study and experimentation. Artists and photographers who work creatively with spiral and fractal shapes and patterns will find their work enhanced and enriched.