Here’s another math problem inspired by real life! Each correct answer could earn you five dollars ($5) off any of our products or services (to redeem in person, just print and show your certificate). Good luck!

You are only allowed one try and have already submitted your quiz.

I recently posed the question “What’s Wrong With This Picture”^{blog} about a modified landscape photograph of a foggy sunrise in Ten Thousand Islands National Wildlife Refuge in Goodland, Florida. It turns out Deborah Gray Mitchell, one of the commenters, was right; the image was upside down.

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Ms. Mithcell has her own website (www.dgmfoto.com), but several other sites have information about her. Just Google “Deborah Gray Mitchell”.

To be more precise, I flipped the image vertically and took steps to remove ripples in the reflection and such so that the answer wouldn’t be so obvious. The original picture can be seen at “Foggy Sunrise” on our website. Now I’d like to discuss reflections and the clues that should have given the answer away.

Reflections

. . . Of Your Subject

First of all, the reflected image should NOT look like a mirror copy of the unreflected image, because the photographer has a different perspective or viewing angle of the reflection. As your high school physics teacher may have told you, in reflections, the angle of incidence (e.g. α_{2} in Figure 1) equals the angle of reflection (α_{1}), so the view you have of the reflected image would be the same as if the subject had been flipped below the reflecting surface, as shown in Figure 1 above. I know that may sound like I just contradicted myself, but it is the subject itself I just flipped, not the direct image of the subject. Notice in Figure 1 that in the reflection, the two trees appear the same height, as depicted with red sightline C, while in the direct image the far tree looks higher as shown by green sightlines B_{1} and B_{2}. The further away the subject is, the less of a difference this makes.

. . . Of Celestial Bodies

Here’s another way to look at the effects of reflection; it is as if you had been flipped below the reflecting surface, as shown in Figure 2, instead of flipping the subject. Although possibly less intuitive, this interpretation yields the same results, as shown by lines B_{1}, B_{2}, & C, but makes the effects of the reflection of the sun more apparent. In the image under consideration, as in most cases, the sun would have been your biggest clue. The sun is 93 million miles from us, but even our closest celestial body, the moon, at under a quarter of a million miles (say 238,900 miles), is much further than what your lens considers to be infinity. All light rays from the sun are virtually parallel (or come in at the exact same angle), no matter where you are.

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This detail helped Eratosthenes figure out how large the Earth was 2,260 years ago^{explained} and was crucial to celestial navigation. It is also important in the creation of rainbows. I might be addressing that aspect in an article about my quest for a midnight rainbow. Stay tuned!

This means that the sun will always be higher in the direct view than it appears in the reflection (compare the angle between sun ray A and line B_{1} to the difference between comparable sightlines D and C).

So There You Have It

I hope that clears things up. This information should make you better at spotting fake reflections, or as a photographer, help you create better forgeries by knowing what mistakes to avoid. Good luck!

Of course, you may share your reflections on this or any related material (or questions) in the comment section below. Thanks for stopping by.

In this article, the first of the “Weird Wood” series^{intro}, we show how to build a picture frame using four strips of moulding that aren’t all the same width. Although Figure 1 uses a different width for each piece of moulding, we used three different sizes in our test frames (only because I couldn’t find four different sizes in the same moulding family).

First The Math

Warning: This discussion includes a little trigonometry. Do Not Panic! It’s not as bad as it sounds.

Definition of “Tangent” (skip ahead To Next paragraph if you still remember this):

There are three sides to any right triangle (a triangle with a 90° corner), which I will call the height and the width, which both touch the right (90°) angle and the hypotenuse, which is opposite the right angle and is the triangle’s longest side. You can use the ratio of the lengths of any two of those sides to find the size of the other two angles. Each possible ratio has a name, but we are only interested in one of them today. Probably the most common ratio and the one we will be using is called the tangent. The tangent is defined as the ratio between the height (the length of the side opposite the angle you are interested in) and the width (the length of the shorter of the two sides that create that corner that you are interested in). If you want to know the angle of corner α in the above drawing (Figure 2), for instance, you would calculate its tangent by dividing the height (3 inches in this case) by the width (1¼ inches), which is 2.4 this time. Then you would use your calculator (or phone app – I use RealCalc Plus by Quartic Software (even though it cost $3.50)) to find the angle corresponding to that tangent. On your calculator, the tangent is abbreviated “tan”. If you enter 45 (degrees are assumed) and hit the “tan” button, you will get 1 because for a 45° angle the height is the same as the width, so their ratio is 1. To go the other way (to find the angle), like we are trying to do, we need the inverse of the tangent. Look for the “tan^{-1}” button (it could be the same button, in which case you may need to hit a (yellow) shift or second-function key, and then hit the “tan” button). In this case, once we have the tangent of 2.4, we hit the inverse tangent button(s) to get 67.380135…. (the calculator is obligated to give you 8 or more digits – that doesn’t mean they mean anything. In Figure 2, I rounded that answer to 67.4 degrees and even that third digit should be suspicious.)

The Process

All you have to do is take the ratio between the widths of your two moulding pieces and take the inverse or arc-tangent to get the angle. Here are a few things you need to remember:

Which angle – the tangent gives you the angle that was touching the side whose length was used for the denominator (the width, which would be the second number in the division). The simplest way (but certainly not the only way) to get the other non-90° angle is to just subtract the first from 90° (since the two angles are complementary). Also remember that if the tangent was greater than one, the angle will be larger than 45°; if it was supposed to be a smaller angle (less than 45°), then you may have divided the two lengths in the ratio backward. Don’t worry, you just found the complementary angle and all you have to do to get the right answer is subtract what you got from 90.

It is up to you to keep track of whether that angle you are cutting should be to the left or the right. Making a drawing of your frame design might help. To be useful, the drawing doesn’t even need to be that good. This should also tell you if you calculated the complement (the other angle in that corner (for the other piece of moulding)).

Your saw may be measuring angle backward. My miter saw calls a cut perpendicular across the board 0°, not 90°. If that’s the case, just subtract the angle you calculated from 90.

As an exercise, go ahead and check the rest of my calculations in Figure 1. 😁

Make The Cuts

There is more than one way to make these cuts and more than one set of tools to help you. Which set of tools you should use will depend on such factors as how much of this work you intend to do, your skill set, what your budget is, and what tools you already have on hand.

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Looking through the Framers’ Corner, the forum of the Professional Picture Framers Association, I found recommendations for the following tools for this application:

You would only need one of these (if any), not both.

(The Amazon.com descriptions are only used here as a reference. Although frequently competitive, Amazon isn’t always the only or the best place to buy something.)

Our workshop includes all of the tools listed in www.BeeHappyGraphics.com/about.html#BruceEquip, along with a number of other regular hand & power woodworking tools that Nancy has accumulated over the last several decades. For this project, I used our compound miter saw, but not without complications.

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The precision on this saw looked fine; you should be able to get within ¼° of your target. The first picture (Image A) shows me trying for 22.6° (which would be one of the angles between a 3″ and a 1¼” moulding).

After cutting the 3″ piece, I ran into problems trying to cut the complementary angle (67.4°) on the 1¼” piece, as shown in Image B.

I am not claiming that mine was the best path to reach our goal. In fact, I would love to see your ideas in the comment section about how to improve my techniques.

How I Did It

Working with one corner at a time, I cut both pieces of moulding square just a tad longer than their overall/outside measurement according to your diagram (you will see why in Step 4). If you don’t already have one, this is also when you would put a perfectly square cut on the alignment block you’ll see in Figure 5 to the left of the moulding. I grabbed a 2″ by 4″, but the wider the better.

I set the saw for the smaller of the two complementary angles, rechecking my diagram to confirm whether it should be to the left or right. In the setup shown below, the 3″ moulding would be clamped to the right of the blade.

I made the cut.

Without adjusting the angle of the saw, I set up the second cut. I positioned my (newly cut) alignment block to the left (opposite the side we placed the moulding for the cut (in Step 2)) so that I could also place the 2″ moulding to the left of the blade and perpendicular (at a right (90°) angle) to the miter saw fence. After clamping down the alignment block, I added a support block to the right of the moulding to keep it in place. I could still move the moulding in or out to position the cut. You can see why I needed to precut this piece of moulding.

I made the cut.

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For those who noticed that the color of the moulding in Figure 4 was different than in Figure 5, I had to make two different frames while doing research for this article 1) to confirm and refine my techniques and 2) because I didn’t get enough pictures the first time.

Always check your work. If, when you put the two pieces of moulding together, the miter edge on one piece is longer than the other, that is the angle that should have been larger. The angle on the other piece of moulding should have been smaller (by the same amount).

Figure 7 shows the second setup from the right side. If you look close, you might notice that I didn’t cut enough to make a sharp corner and needed to recut.

Moving to the next corner, I precut at least one more piece of moulding and repeated Steps 2 through 6.

I repeated Step 7 two more times. The second time (when working on the last corner), I used the first two pieces of moulding I just finished cutting to mark the next cut by matching the inside edges, as shown in Figure 8.

Finishing

As with my normal (45° miter) frames, I would next need to make sure the inner lengths on opposite pieces of moulding matched, and the outer lengths as well. Figure 9 shows a way to check to see if the outside and inside corners of the opposite sides match using two carpenter squares (or equivalent).

Some of the tools we normally use next to finish putting the frame together, namely our Logan Precision Sander and Logan Pro Joiner, are worthless for this application. After gluing (and clamping the pieces together until dry) we had to pound the V-nails in by hand (interestingly, the simpler Logan Studio Joiner can be adapted).

Figure 10: Nancy pounding V-nails.

The Back Side

For completeness, the left figure below shows what the backside of the lower left corner would look like. The gray section represents the rabbet, the equal-width (¼”) cut-out that holds the glass, mats, image, and backing of the picture inside the frame. Some of you might be surprised to see that there is a triangular notch in this rabbet in the corner along the miter cut. This notch has no effect on the functionality of the rabbet. To solve this “problem”, however, you could make a compound cut 45° in from the inner edge to the edge of the rabbet and 79.7° in from the outer edge to the same point, as shown in the right figure below (as an exercise, you can check my math on these angles also). But there is really no need to make these cuts. If the gray were to represent an equal-width feature on the front of the moulding, it might be worthwhile to take the extra trouble. Otherwise, don’t even think about it.

The End

Congratulations, you now have a fancy new picture frame. Of course, you still need to find a picture, cut mat(s) and backing, mount picture to same, cut glass, assemble the pieces without showing any annoying little specks, and apply a dust cover and hanging hardware, but all of that is beyond the scope of this article. Good luck!

As mentioned, this article is just the beginning of a series about “Weird Wood” that I announced months ago. Up next, we will look at handling moulding that is not of uniform width. You won’t find this moulding in any store; it is only an exercise to prepare you for our final project. But if it stimulates your creativity, that’s not always a bad thing. Stay tuned, and thanks for reading! Your comments are welcome and appreciated.

The easiest and most common mat layout is one with the widths of all four borders equal. If you are forcing a picture into a standard-sized frame, however, that’s not always possible. And then there’s the matter of bottom-weighted mats.

Bottom-Weighted Mats

Bottom-weighted mats, or mats with the bottom edge wider than the others, were introduced long, long ago. Some say that pictures centuries ago were hung very high on the wall and the bottom width of the mat was increased to compensate for the ‘distortion’ of that perspective. Unfortunately, that story makes no sense; top-weighting would be required to correct for the top being further from the viewer than the bottom. Another explanation involves the notion of a difference between the visual or optical center and the geometric center. Yet others claim it is to compensate for the drop of the mat in the frame due to tolerances necessary to account for expansion, etc. For whatever reason, bottom weighting could be seen as an attempt to fool your audience or overcome optical perceptions, whichever you prefer. As commonly practiced in “finer frame shops everywhere”, the bottom width is generally increased ¼” to 1″, depending on the size of the picture^{ref}.

Using Standard Mats

But how would one incorporate bottom weighting while fitting an image into a standard-sized mat? For example, if the vertical difference between the hole size and mat size is greater than the horizontal difference, and assuming the left and right borders will be the same width, is it better to:

A

Make the top and bottom borders equal,

B

Make the top the same size as the left and the right and put all of the extra width on the bottom,

C

Make the bottom larger than the top by some fixed amount,

D

Make the differences even more subtle by making the difference between the top border and the side borders the same as the difference between the top and bottom borders?

Let’s clarify your choices with an example. Suppose you want a 4″-high hole that’s 7″ wide in a standard 8″-high by 10″ mat. The horizontal difference between the mat size and the hole size is 10″ – 7″ = 3″, so if you want the left and right borders to be the same, each will be 3″ ÷ 2 = 1½”. The vertical difference between mat and hole size is 8″ – 4″ = 4″.

Choice A

Would make the top and bottom borders the same, making them each 4″ ÷ 2 = 2″.

Choice B

Would make the top 1½” like the left and right borders, leaving 4″ – 1½” = 2½” for the bottom border.

Choice C

Uses the customary bottom weighting, which the one reference I give above lists as ¼” for an 8″x10″ mat (personally, a ¼” bottom weight isn’t worth the trouble). That means the top border would be (4″ – ¼”) ÷ 2 = 1⅞” and the bottom would be ¼” more, or 2⅛” (notice as you check your work that 1⅞” + 2⅛” = 4″). Finally,

Choice D

Is a tad more complicated. Let’s call the difference between the left or right border width and the top border width “d”, such that

1½” + d = T (for top border width).

Then the bottom border (B) would be

T + d or (substituting the last expression for T) (1½” + d) + d = 1½” + 2⋅d.

Since T + B = 4″, then (substituting for T and B)

(1½” + d) + (1½” + 2⋅d) = 4″, meaning 3″ + 3⋅d = 4″ or 3⋅d = 1″, meaning d = ⅓”,

so (substituting back into our equations for T and B)

T = 1½” + ⅓” = 1^{5}/_{6}” and B =1^{5}/_{6}” + ⅓” = 2^{1}/_{6}”

(again noting that 1^{5}/_{6}” + 2^{1}/_{6}” = 4″) .

Our Four Mat Choices (drawn to scale)

The choice you make would be an artistic decision, but I think A is the most common answer. Choice C could be used for traditional bottom-weighting, as in our example, or could be used for some other more artistic value. Technically, both Choices B and D are possible results of that equation. B would be exactly what you get when you want bottom-weighting and are not restricted to standard mats; it would work best if the resulting difference between the top and bottom borders is not too much greater than the customary bottom-weighting distances mentioned above. In our example, it yields 2½” for the bottom border, which is an inch larger than the other three borders and may just be too much. In our example, C and D are very close, and remain close when we change the amount of weight in C from ¼” to ½” (as shown by the lighter blue opening). D is more subtle than C, but may only be worth the effort when the difference between the left and top borders is small enough to fool somebody. In other cases with different numbers, results may vary.

With Larger Side Borders

If the horizontal difference between the hole size and the mat size is greater than the vertical difference, you could face up to the same number of choices as above, but you are working with less material for the top and bottom borders and I think it is usually better to keep things simple and make those borders equal.

Differing Left And Right Borders?

Do the vertical borders always need be the same size? Although I can’t say I’ve ever seen or read about different-sized side borders, I’m not convinced that uniformity is strictly required. For example, in photography, as in older art forms, there a “rule” of space^{ref} that says, among other things, that there should be plenty of space on the side of the subject into which it is looking. If you have a “perfectly” centered and close-cropped picture of your mother looking to your left, could a mat with a wider border on the left side create the space that’s lacking in the image? Maybe you could even choose a mat color that is a pastel version of the background to her right (your left)? Maybe a contrasting outer mat could be added with traditional (identical) vertical borders.

I present the above thoughts to give some background and (more importantly) stimulate your own creativity. If you think of other possibilities, I’d be thrilled to have you add them to the comments. Thank you!

There are mathematical or drafting programs that may do a better job of finding areas of all sorts of seemingly random two-dimensional shapes, and I may have used one or two of these as a student, but I haven’t had any of them on my computer for many moons. So when I recently needed to compare the size of the visible sun at different times during a solar eclipse so I could compare exposure levels, I was out of luck. But then there was Photoshop. I just finished this article about how to find an object’s area, and put it on our website at www.beehappygraphics.com/find-area.html, mainly because I mentioned the technique in an earlier blog post, and was about to mention it again in an article I promised about the challenges of our newest eclipse image. This probably isn’t the most common task you will be doing, but when you need it, this can be handy. Enjoy!

Sadly, we had no winners to this contest. Here is a solution to that math problem:

There is more than one way to solve this problem, but we will be exploiting three different relationships. First, in preserving the aspect ratio, the length of the image (we’ll call L) is 1^{1}/_{2} times the width (W). . Then, adding up the components making up the overall width of the mat, the image width (less two overlaps of ^{1}/_{8}“) plus two mat widths (M) would equal 16 inches. By the same token, the image length (less same overlaps) plus two mat widths would be 20 inches.

If you replace the L in the last equation with its W equivalent from the first equation, and then add ^{1}/_{4}” to both sides of both equations to combine constants, you are left with the following two equations to solve with two unknown variables:

From here you can use linear algebra (matrices) or algebraic manipulation to simplify until you are left with just one variable. For example, just subtracting the bottom equation from the top (subtracting the left sides separately from the right sides of each equation), you will wind up with

which means the image width is eight inches, which means its length is twelve inches, and the mat guide would be set to 4^{1}/_{8}“.

What’s Next

I’ve come up with one more printing-inspired math problem, which I will share as soon as I master a new plug-in for this blog. After that, I’m not sure. Response has been weak, but the former teacher in me feels a need to keep pointing out opportunities to use some of this stuff you learned in school (or is it just to torment those students who were the most difficult – I’m not telling). This isn’t really costing anything, and I give enough warning for the math-averse to stay clear. Stay tuned.

OK, here’s another problem inspired by matting pictures. Suppose you have an image that you want to put in a standard 16″ by 20″ mat. You can print the image any size, but want to keep the original 2:3 aspect ratio (meaning that the length will always be 50% longer than the width so you won’t lose any of the image due to cropping). You want the mat to be the same width on all four sides. Although standard mats overlap the image by 1/4″, this is not a standard hole so I like to use a 1/8″ overlap (which would be riskier with borderless prints). The first question is “How large should you print the picture?” Mathematically, there is only one correct answer to this question. Once you figure it out, how wide should I cut the mat (where do I set the mat guide on the mat cutter)?

Another Mat Problem

The Prize

Email your answer to blogger@BeeHappyGraphics.com. The first three correct answers will receive $7 off any print and another $7 off if you choose to frame (or gallery-wrap) the image. As before, I will publish some responses, but obviously not immediately. So that nobody dies from the suspense, we will put a one-month deadline on this offer. Prizes may be redeemed any time after the winners are announced. Good luck!