# Shadow Casting by Usonian Perforated Boards

### Abstract

This article explores whether the design of Usonian perforated boards might indicate an intention to cast shadows when illuminated by sunlight. In order to create recognizable shadows, openings or masking elements must have dimensions larger that 0.0093 x distance the shadow is cast. Shadows will become too blurry to recognize if the casting distance is greater than 100x the size of the feature that creates them. The Bachman-Wilson perforated board is analyzed to show its design and installation are within this limit. It is suspected this will be found true for other designs, but further work is required.

### Introduction

Frank Lloyd Wright referred the shadows cast by his architecture as “eye music” [1][2]. His Usonian homes often feature perforated board designs that cast beautiful shadows. The fretwork in these design is instantly recognizable as belonging to Wright. They use angular lines to form openings in well-balanced visual designs. Compared to other types of perforated boards, like Japanese ranma, Wright’s design have a more modern, minimalistic style.

While these designs provide elegant frames for looking out the clerestory windows towards the sky, they also can let in direct sunlight to cast shadows in the home’s interior that move throughout the day. So first, the source of light creating these shadows must be considered.

### The Sun as a Light Source

The star that sustains life on Earth has two important properties that impact how shadows are cast: the parallel nature of its rays and that it is an extended light source.

#### Parallel Rays

It’s hard to grasp how far away the Sun is from Earth based on the scales at which we live: 92.96 million miles [3], represented by “D” in the above diagram. What is the angle “b” if we moved 20 feet (“w” in the diagram) from the sun ray following the arrow labeled “D”? The angle “b” is approximately 0.00379/92960000 radians- essentially zero. The light rays at both points are effectively parallel. This is why the Sun seems to follow you when you drive at right angles to its rays- the parallel rays make the source seem fixed in space.

It also means that shadow of an object does not grow as you move it closer or further from the Sun in a room.

#### Extended Source

It is also hard to grasp how large the Sun is: its diameter is about 218 Earth diameters or 864,576 miles [3]. So even though it is 92.96M miles away, the angular width [4] “a” from Earth is 0.53 degrees. As before, because “D” is so large, moving closer or farther from the Sun in a room doesn’t noticeably change its angular width.

For this article, masks and the planes where shadows fall are assumed to be a right angles to the Sun’s rays. Tilt of either will skew the resulting shadow image.

The diagram above shows what each point in a plane sees when looking towards the Sun. As described above, the Sun seems to follow an observer at each point due to the parallel nature of its rays. The solar disk is projected onto the plane of the mask, represented as a yellow line. The red lines depict the angular width of the solar disk.

Point P1 does not see the Sun due to the mask. This point is fully shaded and referred to as the “umbra” of the shadow.

Point P2 is partially shaded by the mask and is referred to as the “penumbra”. This point is brighter than P1, but still not at full brightness.

Point P3 is clear of the mask entirely and receives the full illumination, making it the brightest of the three points.

Next, consider what happens when the mask is moved from position B to position A. Due to the parallel nature of the Sun’s rays, the overall shadow size stays the same. Don’t be fooled by the depiction of the Sun in the diagram- it is very different than a light bulb in the room. The light bulb is too close to have parallel rays so the shadow size would increase when the mask moves from position A to position B. This not the case with the Sun.

From point P, the angular width of the mask increase (blue lines versus green lines). As noted earlier, the angular width of the Sun remains constant since it is so far away. Thus, the projected solar disk at position A is smaller than that at position B.

The consequence is that as the observation point moves out of the umbra, the transition to full illumination will happen over a shorter distance in the plane of the shadow. In other words, the penumbra region will shrink at position A versus position B. This matches our experience: the shadow becomes sharper as the mask moves closer to the plane on which the shadow is cast.

The above points to how to simulate the shadows from a mask:

1. Calculate the projected solar disk size on the mask based on the distance from the mask to the plane where the shadow is cast.
2. For each point in the plane where the shadow is cast, place the solar disk directly behind the corresponding mask section and sum up how much light gets through. Repeat this for each point in the plane.

To illustrate, consider a vertically oriented grating with equal opening and bars. Using arbitrary units, suppose the grating had a pitch of 200 units (opening and bars are 100 units). If it was located such that the projected solar disk was 50 units, applying step (2) would give (at a horizontal cross section):

The blue line represents the grating where a value of “1” is an opening and “0” is a bar. The orange line is the resulting shadow normalized to full illumination as “1”, full shadow (umbra) as “0”. The sloped lines represent the penumbra region.

If the grating were moved further from the plane where the shadow was cast, the projected solar disk size would increase. Supposing it doubled to a solar disk size of 100 units, the shadow now changes:

Note when the projected solar disk size equals the opening and bar width, there is just penumbra- the shadow is very blurred. If the grating pitch was shrunk from 200 to 160 units (opening are now 80 units relative to the projected solar disk size of 100 units), the overall contrast of the shadow region decreases.

Rule of thumb: Mask features must be bigger than the projected solar disk to avoid excessive blurring in the resulting shadow.

Mathematically, the operation described in step (2) is call convolution. A convenient way of performing this calculation is to use Fast Fourier Transforms (FFT) [5]. The FFT quickly computes the spatial frequency spectrum of both the solar disk and the mask. Multiplying these and taking the inverse FFT is mathematically equivalent to convolution. The following section demonstrates this with a test case.

### Case Study: Simple Mask

As a test case, a simple 2D mask was created (units are in pixels):

A range of projected solar disks (units of pixels) were then examined. For comparison, two solar disk cases are shown next to the test mask:

Examples of the spatial power spectrum of the projected solar disk and test mask are shown below, where the center of the image is zero frequency. Spatial frequency increases moving away from the center.

The test mask spectrum shows high vertical and horizontal spatial frequency content.

As the solar disk size increases, its peak power spectrum size decreases (since it has less high frequency content). Multiplying the solar disk spectrum with the test mask spectrum is a form of filtering. A large solar disk has a low frequency spectrum, which in turn leads to a lower frequency output spectrum. This elimination the higher frequency components of the mask spectrum is the cause of the blur in the shadows.

Why not just use the blurring function in most photo touch-up tools? A comparison is given below:

Note that the convolution method (labeled “Shadow” above) has additional structure, seen as a star-like center in the smaller openings. This is because the spectrum of the solar disk has structure, leading to some interesting effects.

The series of images below are the resulting shadows for different projected solar disk sizes (denoted in pixels). Some of these images exhibit “bridging” of light between openings or even reversal of opening brightness.

In order to examine the accuracy of these predicted shadows, a small experiment was carried out using a physical mask of the same proportions, constructed from black cardstock. The smaller openings were 0.375” and the bars between openings were 0.25”. The large opening is 0.75” by 1”.

At approximately 46” from the mask (projected solar disk is ~0.43”), the resulting image is compared to the simulation:

It was a bit difficult to get exposure correct to show the detailed structure that could be observed by eye.

At approximately 76” from the mask (projected solar disk is ~0.70”):

At approximately 102” from the mask (projected solar disk is ~0.94”):

This small demonstration confirmed the ability to accurately simulate shadows from a given mask.

### Case Study: Bachman-Wilson Perforated Boards

The Bachman-Wilson house was relocated from New Jersey to Crystal Bridges Arkansas. Its floorplan and living room dimension indicate shadows are cast approximately 20.4 feet [6]. Lacking a print of the perforated board design, several online sources [7][8] provide photographs used to estimate dimensions and approximate the layout.

From the above photograph of a perforated board panel assembly, dimensions can be estimated assuming 4” from the knuckle to fingertip of the person holding the panel.

Using tracing techniques on photographs, the perforated board fretwork design was captured:

The spatial power spectrum is also shown above (this might be the first time this has been calculated for a Usonian perforated board!). The strong vertical component in the spectrum represents the horizontal lines. The angled components in the spectrum represent angled lines. The spread in some of the angled frequency components likely indicates inaccuracies in the tracing!

Using the above information, shadows cast onto the back wall of the living room with a low angle sun were simulated. The simulation assumed a projected solar disk diameter of 2.25”. The following compares the result to an actual image. Note that the actual image also includes glass, thick boards, and potentially some other background shadows. Despite this, and the approximate nature of the simulated design due to lack of drawings, the result is a good match.

Estimating from photographs, the width of the top edge of the small triangle is approximately 80% of the projected solar disk. As shown earlier, this leads to reduced contrast and significant blurring. This can be seen in both the actual and simulated shadows. For this particular installation, features with dimensions above ~2” have shadows that are still recognizable.

### Conclusion

If shadow casting of the fretwork pattern was desired to be recognizable, the minimum feature dimension (Fmin) needs to be greater than the projected solar disk size:

`Fmin > 0.0093 * S`

where S is the spacing between the perforated board and the surface with the shadow. The factor of 0.0093 is the tangent of 0.53 degrees (the Sun’s angular width).

This requirement creates a limitation in the design detail, hence would favor a minimalistic style.

Of course, shadow casting is not the only function of perforated boards so example are easily found where features can be smaller. However, as a group, its seems that there may be some linkage with detail level and shadow casting performance.

### References

[1] Frank Lloyd Wright: An Autobiography. Reference to “eye music”: https://books.google.com/books?id=S8zlZcJjNEMC&lpg=PA419&ots=hywypY2j49&dq=frank%20lloyd%20wright%20eye%20music&pg=PA419#v=onepage&q=frank%20lloyd%20wright%20eye%20music&f=false

[3] Wikipedia. https://en.wikipedia.org/wiki/Sun

[4] Wikipedia. https://en.wikipedia.org/wiki/Angular_diameter

[5] “Fast Calculation of Soft Shadow Textures Using Convolution”. Cyril Soler and Francois Sillon. Siggraph ’98 Conference Proceedings.

[6] Bachman-Wilson House floorplan and dimensions. http://www.tarantinostudio.com/FLLW_BW_LocationScout_DL/FLLW_BW_LocationScout_DL/Plans.html

[7] Bachman-Wilson shadows from perforated boards. https://www.pinterest.com/pin/116249234105647280/

[8] Bachman-Wilson reconstruction.https://crystalbridges.org/blog/bachman-wilson-house-update/