Drop Leaf Table Design and the Square Root of 3.

The previous post of a table showed a top that when folded in half remained the same length to width ratio.  When folded up the table top took up 50% of the area it did when unfolded.  

This alternate design pushes the concept to a practical limit by using the square root of 3 to make a table that is 1/3 of it's unfolded size when folded up.  A table base that is inset just a little more than 20% but not more than 25%  will support 3 table sections when unfolded but allow the two outer sections to fold down when rotated 90 degrees.  The base length to width ratio is the same as the unfolded table top above it. 

This 1/3 design starts with a wood panel that is 24" x 96" (or 2' x 8').  Using 1/3 of the length as the maximum width of our table top, we can determine the largest possible table top we can make using the square root of 3 ratio.

1/3 of 96 is 32. The square root of 3 is 1.732. 32/1.732 = 18.48".  So the dimensions of each table top section will be 32" x 18.48" .  This will also be the maximum footprint of the table base. Three of these sections will make up the unfolded table top at 32" x (18.48 x 3) or 32" x 55.4".  This table top should seat 6. 

Top ends folded down

Top ends lifted up before rotating

Rotated over the legs for support

Square Rt. of 3 Proportions.

The top pivots around the center of the base.  The pinwheel stretcher configuration makes room for a center plate where the pivot can be placed.

Obviously the legs could be more interesting. The top would have a nice edge profile. A classic drop leaf connection and hinges would be used to allow the top ends to hinge down after rotating 90 degrees. Tops of the legs have a teflon t-peg in them to allow the top to rotate smoothly over them. The center section of the top has a metal post that slips into a bronze bushing in the wood plate between the stretchers. It is held in with a spring, washer and wing nut to allow some tensioned lift when rotating.

Final details needed can only be determined if I ever make such a table. This post is simply a parking place for ideas I've had over the past 4 decades.

Question?  Leave a comment.

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Another useful Geometry Application. Furniture Using The Square Root of 2.

When I taught furniture design to college students one project assignment was to design a table that could be reduced by 50% and still be useful when the starting dimension wasn't needed. 

This is one useful solution that my students have used several times over the years.  It uses the Square Root of 2 (1.414213562373095) to create a rectangle that remains the same width to length ratio when folded in half. 

In this example I have a wood panel to use that is 24" x 36" x 3/4" thick.  If I use the 24" side as the width of my table top and multiply it by 1.41... I get 33.94....  So with 36" of potential length I have room for the kerf of a table saw or CNC 1/4" profile cut. 

The Whole Top

One half of this 24 x 33.94" top is 16.95 x 24. When rotate 90 degrees and centered over the base it is the same ratio as the starting dimension.

Half the Top

Soss hinges between the table halves are hidden when unfolded, but will be visible on the inner edge when folded in half.

Rotated and Centered

One challenge was to find a single point of rotation for the top that would keep the top centered over the base after being folded over and rotated 90 degrees. 

Project a line from the center of 1/2

To find a common point that will be in the same position between unfolded and folded/rotated draw a line 45 degree from the center of the side that will be on the bottom and rotated.  

Project the same line, but from the rotated/centered outline. 

Draw the same 45 degree line from the rotated view of the bottom half.  Where those lines intersect is the only point the top can be rotated about to remain centered over the base in both positions.

The inner outline shown in the 2 views above is the outline of a base that would be inset 20% from the large perimeter. You can see it is still inset a small amount under the folded top. 

Here is one option for a base.  The legs extend to the inner edge of the 20% inset outline. A bracket attaches  to one stretcher to provide a pivot position for the top to rotate about. 

Under the folded top.

Under the open top.

Need a larger table?  What is the largest rectangle of that same sq.rt. of 2 ratio that can be made from a 24" x 60" panel? Account for a 1/8" wide kerf on any cuts.  How much scrap will be left after cutting the two table halves?

As the table base is a little smaller than 1/2  of the table top, you could also make the folded half a drop leaf rather than just double the smaller top thickness. This side would hang down and block access to one side of the base, but if that is not important the table top will benefit from being the same height above the floor in both unfolded and dropped side positions.  

Leave a comment if you need any more information.

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Dividing Any Line into 3rds or 5ths Using a Triangle.

 I discovered this method to divide any line into 3 or 5 equal sections by just playing around with the drafting features of my CNC software.  It has come in handy many times while creating joinery for furniture connections.  Sizing a tenon for the end of a stretcher there is a 1/3 rule which suggests the tenon thickness should be 1/3 the thickness of the stretcher.  When I'm connecting a narrow board into a wider board I like to increase the relative size of the tenon to 3/5ths of the stretcher thickness. One 5th for the perimeter shoulder and three 5ths for the tenon.  

I haven't seen this method used anywhere on the web or in print.  It may exist out there some where.  When I look for methods to divide any line by 3 or 5 equal sections I find most use an adjacent connected line with marked off known segments, projected parallel across to the starting vector to divide it.  I used that trick when drafting was done with a parallel bar, triangles, compass and straight edges. There is another easy trick to dividing the width of a board using a ruler/scale held diagonally across the board at marks equal to the number of divisions you want. 

This triangle method starts by adding two lines to your starting vector to make a triangle. Now find the center points of all three sides of the triangle.  This works with any size or shape of triangle, although the density of lines created is easier to draw and see on a large triangle closer to equilateral. Be accurate.  Mistakes here propagate through following steps. 

Find Centers of Each Side.

Next connect the centers to each other.

Connect Centers

Now draw in the medians by connecting the corners to the opposing center points.

Corners to Centers

You can see where each center line intersects the center triangle lines.  If you draw a line from a corner through the opposing intersections, it will cross the opposing face at 1/3 over from a corner. 

Thirds, in Red

Do this for all 3 sides.

3rds on all 3 sides.

The new vectors intersect the previous vectors, and provide useful nodes we can use to divide each side into 5 equal segments. 

Project to find 5ths, in Green.

It is easy to get confused about which intersections to use.  Basically for each face there is a red/red, a red/black. a red/black. and a red/red intersection to pass through. 

5 Equal Segments.

I usually only need to go far enough to find 5ths on one side, but it is easy to project the other two sides to divide them both into 5 segments.

5ths on all 3 Sides.

There are further divisions that can be found with the new intersections created. I rarely need 7ths or 9ths though and have a much more automated way of dividing line segments in my CNC software.  

For example:

7ths. In Orange. 

9ths. In Purple

I suspect this method will work for odd numbers up to infinity so long as you have infinite patience and a very very fine pencil. I extended it to 11ths with no trouble but the density of vectors is making it harder with each step up.

11ths

Of course since 13 is a lucky number I went ahead and proved that 13 segments of a vector can also be found. 

13ths.  Just Because.

I dedicate this post to my Sister Vickie's husband Chris Z. A useful application of geometry. 3rds and 5ths (and 7 and 9 and 11 and 13), with no parallel lines needed.  If no one else before me has discovered this method, I name it the D.A.Brown Method. Published here first on 7/9/2023. 

As a final teaser, you can also divide the triangle sides by even numbered segments using a variation of projected lines through intersections of previous vectors used.   Although this works with any shape of triangles, you can also do the same with 4 sided polygons. With rectangles, trapezoids, or parallelograms this is easiest to see. How about 5 sided polygons? 

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Zero Clearance Fence for my Miter Saw

 The normal gap between fence sides of most miter saws is wide enough to allow rotating the blade angle or tilting the blade for angled or compound miter cuts.  This gap width is a bit dangerous for making small cutoffs.  The small cutoff part can get kicked back and thrown by the blade. 

I made this zero-clearance fence add-on to add safety to my Bosch Glide miter saw. 

Zero Clearance

The saw already had holes in the fence side for bolts.  A quick measure of height off the bed and location side-to-side of all the holes and I drew of patterns for these parts in my CNC software.   What the CNC thought it would be cutting out:

Cut from 1/2" plywood

The slots allow me to slide the plywood back one inch.  The top half of the fence itself can be loosened and slid back to allow miter cuts with the blade tipped down. I can open up the gap a small amount for angled miter cuts with the blade remaining vertical. 

1/4-20 T-bolts are used with wing nuts on the back side to hold the add-ons in place. 

I've posted a .CRV3D (Aspire) and a .CRV (VCarve) file on Vectric's forum for anyone using their software. 

Comments welcomed

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