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.

4D

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.

4D

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? 

4D

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

4D

A Board Stretcher Joint. Just for Fun.

These are renderings from my CNC software of a zipper joint I came up with. A simple way to connect the ends of two boards. Easy to cut using a 3/16" end mill with the boards clamped flat.  This joint could even be used across the center of a table top. For furniture buyers with trouble deciding which woods they want their furniture to made from.  

Mix wood types

Both sides use the same toolpath

Are there schisms between members of your family?  Sport fans who don't agree on the teams they support?  Hardwoods are available in many colors, and two different colors of wood could be used to make a zippered table with both team colors.  

Just for fun. Very possible. 

4D 

Rotary Cut Radius Box Corners

You can cut a rounded corner for a box as a two sided job on your CNC as shown in a previous post:  HERE

You can also cut the rounded corner on the rotary axis of your CNC from the same rectangular starting block. 

16" Long 90 Degree Rounded Corner

What the CNC thinks it is cutting

Actual ends shown in the first image

The centers on the block ends must be a little lower than actual when mounting the block on the radial axis. The profile shape must be placed so all parts of the inner curve can be reached by the bit as it rotates around the center. Very little material is taken off the rectangular block compared to what would need to be removed from a cylinder of wood for the same result. No roughing pass is needed given the small stepover taken by the bit when cutting this shape from a block. For this example I used a 1/4" diameter ball nosed bit. 

Vectors Used

The starting block for a rounded corner with a 1" internal radius and 1.75" external radius is 2.5" wide and 1.0625" thick. Adjust the starting block size to encompass different sizes of rounded corners.  I recommend making the block 2" longer than the final part you need to keep the router/spindle chuck away from the rotary axis chuck and tailstock. 

An example Aspire CRV3D file for this specific project can be found on Vectric's forum here:  Forum.Vectric.com

The actual vector used for the final cut was created with the "Unwraps selected Object" tool you can only access while in a rotary job. 

I helped students with rounded corners for their cabinet projects several times over my teaching career. This is one creative application for the rotary axis of the CNC. 

Question, critiques, and comments are as always encouraged.

4D  

A Little Pedestal Table

This project was inspired by a thread posted on a woodworking forum about a 3-legged  pedestal table. It was missing a top and had been discarded. It was found and restored with a new top by the woodworker. 

The detailed center post of the table had considerable aesthetic appeal. The 3 legs that dovetailed into it showed a classic way to connect pedestal legs. 

I've had a 3" x 3" block of oak wood mounted on the rotary axis of my CNC for the last 3 months or so.   It was waiting to discover what it wanted to be.  Becoming the center post of a new table seemed to agree with the wood block, so this is where this project started.  The 4th axis toolpaths I cut using my CNC would result in this shape from that block:

Render from Aspire

Actual cut time took a few hours.  The square block had to be reduce to a cylinder. 

First reduce square into a cylinder.

There is a gadget in Aspire that creates a toolpath specifically to remove the corners of square blocks to leave a cylinder. 

From a cylinder, then roughly shaped.

A roughing pass was done to remove most of the waste area from the cylinder. 

Rough cut contours

3/4 done

A final pass with a 1/8" diameter ball nosed bit works its way slowly down the cylinder to produce the final smooth shape. 

Final Shape

Flat areas on the bottom section will be where the feet join into the post. These areas of the bottom section are where dovetail slots were cut.

I flipped the post over end to end so the area where dovetail cuts were needed was at the top (right in the photo) and approachable by the spinning dovetail bit. 

First came pocket clearance cuts done with a 1/4" end mill to remove all the area except where the dovetail undercut would be.  Less work for the dovetail bit to do. 

1/4" end mill set for clearance cuts.

The pocket cut was repeated around the post every 120 degrees. 

Clearance pockets done.

Next came a bit change to a dovetail bit.  This one is 9/16" diameter and has 8 degree sloped sides. 

Setup for a dovetail pass.

The dovetail pass was repeated every 120 degrees around the post.  Once done the post was removed from the rotary axis mount. 

Dovetail cuts done.

With the post done it was time for the legs.  A few sketches of the table with legs were productive. Sketches pointed out that the wood needed to make the legs from needed to be found before the final shape of the legs could be determined.  I had in my scrap bin a couple of oak stretchers left from an abandoned chair design I never finished. They were 3" wide, 1" thick, and 25 inches long. Unfortunately their edges had been rounded over.  I planed them down to 7/8" thick, taking the same amount of wood off of each side. This left about 2.5" of width that was flat in the center of each stretcher.  The leg design I came up with is a simple curve with a hooked foot. Three were cut from that 2.5" width of wood.  

A challenge with the legs was to cut the male dovetail end that would fit into the center post. 

Jigged up for the dovetail cut.

I took care to clamp the first leg so the area for the dovetail was horizontal and sticking up where the router bit could reach it.  I then added supports and stops around it so repeating the cut on the next two legs could use the same setup. 

Dovetail bit zeroed out on the leg top. 

To start the cut I needed to zero the dovetail bit against the left and front edges and the top of the leg. 

 

Dovetail cut.

First dovetail looked centered and correct.  It was time to check if it fit in the post slots.

Fit is good. 

The dovetail slot is tapered to match the tapered shape of this leg end.  The leg slips in easily and snugs up tight.  With a good fit all I needed to do was repeat the cut on two more legs.  With all three cut I could stand up the table base for aesthetic evaluation and detail contemplation. 

Three legs fit. 

The top was yet to come.  I spent a few days contemplating the top.  Final shape. Edge detail?  Inlays maybe?  How the top ends up influences the final details I add to the legs.

Often it takes building an initial design idea to reveal the reality of it compared to renderings on a PC.  That is true for this table. It is a prototype.  A design in progress. The center post was the inspiration for making a table.  Initially the "fancy" post seemed to beg for a fancy top to hold up.  Eventually I realize I was wrong.  

It needed a simple top. Just a nice circle with some subtle detail and curve. My goal was to bring focus to the post.  

Simple round top.

I made this round top to enforce detail continuity.  The legs now stand out as the last parts with minimized relationship to the post.   Below is some iterative playtime with the leg design.

First idea was to taper the leg from top to bottom. Remove some of the visual mass. 

The round top has a subtle elliptical curve on the bottom, and to relate the leg an elliptical taper could be applied.  Perhaps too subtle.
 

A completely new leg profile was tried. A section of an ellipse.

Redirecting the groove to better define the foot.  I may have to make at least one to evaluate how well it adds to the aesthetic composition of the pedestal table. 

The groove on the side of the legs could continue from leg to leg around the section of the post where the legs attach.  Linking the legs specifically to the  post. 

Perimeter groove to continue grooves on legs around the post.

Actual making of that last leg design would present a challenge. It has to be cut from both sides, and the only area around the shape that isn't cut is the back of the dovetail section.  I would need to divide the toolpaths into two sessions, to allow at least two clamps to hold it down.  Cut one side of the edges, then move the clamp to that side before cutting the other side.  Flip the board over and repeat.  Extra effort for a leg that may not be the right leg for this table. 

I used a 3/16" radius round-over bit to soften the round top edge and also the edges of the original legs. My initial instincts to focus on the post and de-focus on the legs and top did the job.  I've concluded it is time to wrap this project up.   The initial legs will do. 

Final less finish.

Some cherry stain on the red oak, followed with a clear finish will match the color of this table to other furniture projects I have in my house.  

Feel free to comment, criticize, or post any questions you may have. 

4D