1. Why Light Spots Under Trees Are Round Images of the Sun, Not the Shape of the Gaps
In summer, look closely at the spots of light falling on the ground under trees. Even though the small gaps between leaves are square or jagged, all the light spots have the same round shape as the sun [1]. What is being projected is not the shape of the gap, but the shape of the sun.
The small gap acts like a pinhole, forming an image of the sun on the ground. It is explained that this works on the same principle as a pinhole camera, where light passing through a hole creates an image [1]. A pinhole camera forms an image by letting only the light from each point of an object reach one specific point in the hole. To get a clear image, the hole needs to be quite small [2].
In the case of dappled sunlight (komorebi), when the gap is sufficiently small, the influence of the gap's shape becomes negligible, making it easier to see the round image of the sun. This discussion applies specifically to cases where the gap is small.
So, what about the edge of these round spots? If you look closely, they are not sharply defined. The same situation that affects shadow edges is hidden here.
2. Why Shadows Form Behind Objects: Light Travels in Straight Lines
Shadows are created because light travels in straight lines [3]. When an object blocks light, a place where light cannot reach is created behind it. That place is the shadow. The shape of the shadow is the outline of the blocking object projected onto the ground or wall [3]. The sun is on the opposite side of the shadow.
There is a condition to "light travels straight." It is explained that light moves in a straight line when traveling through a substance with uniform properties [4]. A phenomenon where light bends slightly around obstacles is called diffraction, and strictly speaking, the path of light can also bend in places with strong gravity [4]. However, when thinking about everyday shadows, it is fine to assume that light travels straight.
However, if you look at the edge of an actual shadow, it does not look like a line drawn with a ruler. The further you hold your hand from the ground, the more blurred the edge becomes. This blurring of the edge can be explained by the straight-line travel of light alone. We will see why in the next section.
3. Why Shadow Edges Look Blurry: The Sun Is a Light Source with Size
The reason for blurred edges is that the sun is not a point source, but a light source with size [6]. Light coming from the right edge of the sun and light coming from the left edge hit an object at slightly different angles.
When a light source has size, the area where an object completely blocks the light source is called the umbra, and the area where it partially blocks the light source is called the penumbra [5]. The penumbra becomes darker closer to the umbra [5]. If the light source were a point, no penumbra would form, and the entire shadow would be an umbra [5].
The further the object is from the surface where the shadow falls, the more slight light from the edges of the sun enters behind the object, creating a faint penumbra at the edge [6]. This can be explained by the straight-line travel of light alone; it is not due to diffraction [6].
The same thinking applies when there are multiple light sources. The area where no light comes from any source is the umbra, and the area where light comes from only some sources is the penumbra [3]. If a room has two lights, you see thick shadows and thin shadows overlapping.
The width of this blur can be estimated numerically based on the apparent size of the sun.
4. How Wide the Blur Is: About 1/107th of the Distance to the Surface
The apparent size of the sun (angular diameter) is about 32 arcminutes, or roughly 0.5 degrees [7]. Precisely, it is about 0.53 degrees. Converted to radians, this is about 0.0093, which means it is approximately 1/107th.
When the surface receiving the shadow is at a distance d from the object, the width of the penumbra is roughly d × 0.0093, or about 1/107th of d [6]. If the distance to the surface is 1 meter (about 3.3 feet), the blur is about 9 to 10 millimeters (about 0.4 inches). If it is 5 meters (about 16 feet), it is about 5 centimeters (about 2 inches). This is a rough estimate calculated mathematically; in reality, it changes depending on the shape of the object's edge.
This calculation can also be applied to dappled sunlight. A small gap located 5 meters above the ground projects an image of the sun that is roughly 5 centimeters wide. In actual light spots, the size of the gap itself adds to this. Light passing through higher branches creates larger circles.
If you place your hand just above the ground, the shadow edge is sharp. If you lift it about 1 meter, the edge blurs by about 1 centimeter. This number matches the calculation of approximately 1/107th.
Try this: On a sunny day, look at the edge of your hand's shadow on the ground. Place your hand just above the ground, then at 30 centimeters (about 1 foot), and then at 1 meter. Observe how the width of the blur changes. Measure the width with your fingers. Look only at the shadow; do not look directly at the sun.
5. Why Light Spots Become Crescent-Shaped During a Solar Eclipse
During a solar eclipse, when the moon covers part of the sun, the round spots of dappled sunlight become crescent-shaped or partial circles, matching the shape of the eclipsed sun [9]. Since dappled light is an image of the sun, this happens.
The phenomenon of a solar eclipse is also a story about shadows. From outside the Earth, a solar eclipse is when the moon's shadow reaches the Earth [8]. For people inside the moon's shadow, the sun appears hidden by the moon.
The moon's shadow also has an umbra and a penumbra. When the moon's penumbra begins to cross the Earth, a partial eclipse starts. When the umbra crosses, it is a total eclipse. When the extension of the umbra (antumbra) crosses, it is an annular eclipse [9]. Total eclipses are visible only in a very narrow area, while partial eclipses are visible over a wide area [8].
You must not look at the sun directly. When viewing an eclipse, use special solar filters. You do not need to look up at the sky to see dappled light on the ground.
6. How to Check the Theory with Your Hand's Shadow and Future Eclipses
There are two things you can verify. One is the blurring of your hand's shadow edge when you move it away from the ground. For example, at 1 meter, the blur is about 1 centimeter [6].
The other is the shape of dappled light on the day of the next solar eclipse. If you are in a location where an eclipse is visible and it is sunny enough to create dappled light, seeing crescent-shaped spots on the ground would be a natural verification that dappled light is an image of the sun.
If you cannot encounter an eclipse, you can try the same method with everyday dappled light or by making a small hole in paper and passing light through it. You do not need to look up at the sky. When looking at the sun, always use special solar filters.
Shadows have also been clues for knowing the tilt of the sun. One source introduces that Eratosthenes, around the 3rd century BCE, estimated the Earth's circumference as about 46,000 kilometers (about 28,582 miles) based on a difference of about 7 degrees in the sun's apparent angle at noon on the summer solstice between a place where the sun was directly overhead and a place about 900 kilometers (about 559 miles) north [10]. This differs from the currently known value of about 40,000 kilometers by about 15%. Values vary depending on the source.
It might be interesting to measure the size of dappled light circles at different branch heights and check if they match the formula.