# Golden Ratio Calipers

I made a set of "Golden Ratio Calipers".

I like to use the "Golden Ratio" (1.618…) when creating hollow forms and wide rim plates. It allows me to create things with pleasing proportions found in nature.

The golden ratio tells me where the widest diameter of a hollow form should be relative to the top and bottom. It also tells we how big the bowl in the center of wide rim bowl or plate should be.

The golden ratio can be used on salad bowl shape bowls to tell you how big the foot on the bottom of the bowl should be. However, it is NOT useful for the salad bowl shape or height vs diameter. A hanging chain catenary is better for bowl shapes.

The proportions of the golden ratio pop-up in all sorts of places in nature and man-made things since early times. It is a handy shortcut to make something we seem to find pleasingly proportioned. In photography, art, or nature, we just keep being drawn to it.

Note: See "The Golden Ratio" section at the end for a bunch of mathematical stuff you DO NOT need if you have a set of Golden Ratio Calipers.

## Hollow Forms and Vases

Here are some examples of how I use Golden Ratio Calipers to design Hollow Forms and Vases..

Hollow Forms or Vases

Set the outside legs of the Golden Ratio Calipers to the height of the hollow form or vase.

Then use the "short" golden ratio size to estimate where the widest part of the hollow form or vase should be. See photo.

## Plates or Wide Rim Bowls

Here are some examples of how I use Golden Ratio Calipers to design Wide Rim Plates and Wide Rim Bowls with pleasing proportions.

Plates (with Wide Rims)

Set the outside legs of the Golden Ratio Calipers to the outside width of the plate.

Then use the "long" golden ratio size to estimate how big the bowl in the center of wide rim plate should be. See photo.

Wide Rim Bowls

Set the outside legs of the Golden Ratio Calipers to the outside width of the wide rim bowl.

Then use the "short" golden ratio size to estimate how big the bowl in the center of wide rim bowl should be. See photo.

The golden ratio can be used on salad bowl shape bowls to tell you how big the foot on the bottom of the bowl should be.

However, it is NOT useful for the salad bowl shape or height vs diameter. A hanging chain catenary is better for bowl shapes.

Note: "Salad Bowls" is my generic term for bowls that are not Wide Rim Bowls, like the ones shown above.

Set the outside legs of the Golden Ratio Calipers to the outside width of the bowl.

Then use the "short" golden ratio size to estimate how big the foot on the bottom should be. See photo.

Note: It sort of works ok. It really kind of depends on your salad bowl shape. If you want something functional then the foot should probably be larger. If you want something that is really artist and sells for an outrageous price then the foot should probably be a lot smaller.

In my not so humble opinion the Golden Ratio is NOT useful for salad bowl shape or height vs diameter. A hanging chain catenary is better for bowl shapes.

Or check out the bowl shape diagrams in Richard Raffan’s book “The Art of Turned Bowls: Designing Bowls with a World Class Turner“.

## Make Your Own “Golden Ratio Calipers”

Here is how you can make your own #12 size Golden Ratio Calipers.

My #12 size calipers are a good size for woodturning. They work good on things 3" to 18" wide or tall.

I call them "#12 size" rather than 12" because they are a little less than 12" long (11-3/4"). I choose to make my calipers a little less than 12" long because my design fits in a 36" (3 foot) long piece of 1/8" thick x 1/2" wide aluminum bar from local hardware store.

Golden Ratio Calipers Plan

My plan really tells you everything you need to known to make your own. It has all the cut sizes and hole locations you need. Etc.

Click here or on the image on the right and print out the PDF file.

Notes:

1. Golden Ratio Calipers are all over Amazon. I decided to make my own because the ones on Amazon are to expensive or to small (for eyebrows, etc).
2. Plans for Golden Ratio Calipers are all over the internet. I decided to draw up my own because the existing ones on the web are to small, to big, or to confusing.
3. There are tons of Golden Ratio Calipers videos on the web. I hate videos! I like real numbers on paper! A picture (blue print) is worth a thousand words. A video is often 10,000 useless words with no real content!
4. Note: Use the "Formula" in table on my plan if you want to make calipers some other size. Start with AF = 11-3/4" – 10 mm. Change the 11-3/4" to the size you want and redo the calculations.

Materials for 1 Set of Golden Ratio Calipers

Updated: 4/1/2022. Some times you feel like a fool. You can just use Pop Rivets from the local hardware store rather than binding barrels or nuts and bolts.

1/8″ diameter by 1/4″ long or 3/16″ diameter by 1/4″ long pop rivets. And a pop rivet gun. You will need to change the hole sizes on my plan to match your pop rivets.

You will need to the following materials from local hardware store or McMaster-Carr. All # below are www.mcmaster.com part numbers.

```MATERIALS
36" of 1/8" thick by 1/2" wide aluminum bar.  # 8975K577
4 of Binding Barrels & Screws (8-32 Thread Size, for 1/4" - 3/8" Material Thickness)  # 93121A325
Or 4 of #10 Bolts & Nuts from local store
4 of Plastic Washers (#12 Screw Size, 0.24" ID, 0.456" OD) # 90295A430
Or live with out
```

BEWARE! I recommend getting aluminum bar from local hardware store or order two of 2 foot length bars from www.mcmaster.com. 2 foot long items fit cross ways in McMaster's standard 12" x 18" box and thus DO NOT add to shipping cost. If you order 3 feet of anything from McMaster it comes in a 3 foot tube that costs at least \$12 extra!

Note: You don't need to use Aluminum Bar. You could use 1/8" or 1/4" thick plywood or wood that is 3/4" wide.

Cut Aluminum Bar to Length

Start with 1/8" x 1/2" x 36" aluminum bar from local hardware store.

Cut the following with a hack saw. I recommend cutting everything 3 mm LONGER than numbers on plan. Then belt sand or file the pieces to final length AFTER taping them together. See photos below.

```CUT LIST
2 of 295 mm (+3 mm oversize = 298 mm)
1 of 185 mm (+3 mm oversize = 188 mm)
1 of 124 mm (+3 mm oversize = 127 mm)
```

Note: The +7 mm or +14mm in the table on my plan is the extra you need beyond the bolt hole. You need to add 3 mm on top of this then belt sand or file the pieces to finial length.

BEWARE! The 4 pieces just barely fit in 36".

I STRONGLY recommend measuring, marking and cutting each piece ONE AT A TIME. Because, the width of the saw cut may screw you if you mark them all at once.

I like inches! Why metric? Because 295 mm is easier to measure than 11-39/64”.

Stack Up & Masking Tape Together

Stacking up the pieces is the best way to end up with holes that line up.

Masking tape the pieces together after you stack them up. Ensure the ends align exactly when taping (or at least they come real close). See photo.

Note: In the photo I have already drill the holes. I forgot to take photo before drilling. You just can't get good help these days.

Drill Starter Hole

See photo. I like to use a “Center Drill” to create a starting hole for twist drills when I want to drill a precise hole.

Center drills are cheap on Amazon. See my "Center Drill Set" blog entry.

Drill Holes

Drill 13/64″ holes for www.mcmaster.com # 93121A325 Binding Barrells & Screws. Or 3/16″ holes for #10 bolts from local store.

BEWARE! The binding barrells or bolts should fit SNUGLY in holes you drill! Avoid a sloppy fit that results in sloppy calipers.

Updated: 4/1/2022. If you are using Pop Rivets then you will need to change the hole sizes on my plan to match your pop rivets.

Line up Ends

Line up the ends that will be pointed.

1. Remove masking tape after drilling holes.
2. Insert binding barrel or bolt in holes B and C. This will line up F, G, and H points. See my drawing and/or photo on right.
3. Masking tape together the 3 pieces with F, G, and H points aligned. See photo on right.
4. Cut the point ends all to the same SQUARE length with a file, grinder or belt sander. See photo on right.

The distance from center of hole B to point F should be 178 mm.

Cut Points

Cut points on end with with a file, grinder or belt sander. See photo.

Clean Up & Deburr

Clean up and deburr things with a file. See photo.

Assemble Calipers

Assemble with Binding Barrels & Screws, with Plastic Washers between the pieces.

From local hardware store or www.mcmaster.com # 93121A325 and # 90295A430.

Tip: A drop of super glue down the hole on the barrel side will lock the screw.

Test Calipers

When the distance between the outside two points is 100 mm. The distance from one point to the center point should be 62 mm. See my plan and photo on right.

## Good for 3″ to 18″

My #12 size calipers are good for things 3" to 18" wide.

I call them “#12 size” rather than 12″ because they are a little less than 12″ long (11-3/4″).

Good for 3″

The calipers are good for things 3" to 18" wide. See photo on right for 3" case.

Good for 18″

The calipers are good for things 3" to 18" wide. See photo on right for 18" case.

## The Golden Ratio

This section is for those who need all the nitty gritty details. You DO NOT need any of this if you have a set of Golden Ratio Calipers.

All over the web it says the "The golden ratio is 1.618…" then there is often a lot of USELESS mathematical stuff. Here is how I make it useful.

```Given a line that looks like this:
|------------- Long -------------|---- Short -----|
They say:
Long / Short  =  Long + Short / Long = 1.618 ...
What they mean is:
Long / Short  = 1.618 ...
OR
( Long + Short) / Long = 1.618 ...
Yea that is true. But, it is hard to use. If you solve "(Long + Short) / ..." with high school algebra. You get:
(Long + Short)  /  Long = 1.618   (Example (178 mm + 110 mm) / 178 mm = 1.618)
(Long + Short) = Long * 1.618
Long =  Long + Short  / 1.618     (Example 178 mm = (178 mm + 110 mm) / 1.618)
Now that is useful. It means that if the total length (Long + Short) of a line is 10" then:
Long  = 10" / 1.618 = 6.180" = 6-1/8"
Short = 10" - 6-1/8" = 3-7/8"     (From Short = (Long + Short) - Long)
```

Thus if a wide rim bowl is 10" wide and you want it to pleasing to the eye then the bowl in the middle should be 3-7/8" wide.

If a hollow form vase is 10" tall then the wide point on the vase should be 3-7/8" below the top OR 3-7/8" up from the bottom. The vase should be 6-1/8" diameter. Note: I personally don’t always follow the diameter rule.

You can also solve "Long /Short = 1.618"

```   Long / Short  = 1.618 ...  (Example 178 mm / 110 mm = 1.618)
Long = Short * 1.618       (Example 178 mm = 110 mm * 1.618)
Short = Long / 1.618       (Example 110 mm = 178 mm / 1.618)
```

Thus the important take away from ALL of the above is:

```   Long =  Long + Short  / 1.618   (Example 178 mm = (178 mm + 110 mm) / 1.618)
Long = Short * 1.618            (Example 178 mm = 110 mm * 1.618)
Short = Long / 1.618            (Example 110 mm = 178 mm / 1.618)
```

## Rule of Thirds vs Golden Ratio

The Golden Ratio is close to 1/3 and 2/3. It is a little less than 2/3.

The red lines in photo at right are thirds.

If we take 2/3 of 10" it is 6.66" = 6-5/8"

Which is a little less than the 6-1/8" predicted by the Golden Ratio of 10" / 1.618.

Thus, before I made a set of Golden Ratio Calipers, I often used the rule of thirds to approximate the Golden Ratio when turning things. Make it a little less than 2/3.