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9mm (0.354331) is a little bit less than 3/8" so don't cut 3/8" wide slots. If the plywood was a bit thicker (12mm or better) I'd say your layout was OK, but to better support the thin unsupported halves I'd flip the middle vertical board slots back to front. If you are skilled with your circular saw lay out (draw) the slots carefully, extend the blade as far as it will go, then use it to cut both sides of each slot stopping the cut at the middle of each board. Then use a chisel or jig saw or both to clean out the center/ends. I'll OK 2 pairs per cubby if you flip the center board and no one is wearing the shoes when they are put on the shelf. ;) If your final half lap (aka halving) slots are a good snug fit you may not need a brace. If there is some slop in the slots then a back panel or some cross bracing will be needed. If you plan to move it frequently maybe extend the center vertical board and cut a hand hold slot in it. Maybe extend the left and right verticals down to become the feet of the design.

I apologize if my suggestions are too much. I just retired from teaching furniture design to college kids.
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If you find a good deal on a router then look around for a 9mm straight router bit with 1/4" shank. Likely online as I've never found metric bits at my local sources. Otherwise a 1/4" straight bit and two passes can make a 9mm wide slot. Something to clamp down a straight edge (a couple of small C-clamps) and place to set your coffee while you work. Before I had a shop space I bought a Black and Decker Workmate folding bench. I still have it and use it often. Great work holding built in.
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Yep. Plywood thickness is a guessing game. I had a student several years ago make a chair from stacked layers of
3/4" Baltic Birch Plywood. We hadn't come across much metric material then and the student sliced his 3D model into 3/4" slabs that we cut out on our new (at that time) CNC. Of course all the BB was 18mm thick and when stacked together left his final prototype only 94.5ish% as thick front to back as he expected. The last semester I taught a student of mine wanted to apply oak veneer to both sides of BB plywood before we cut out her project parts. I needed to know the actual thickness to set up the toolpaths. She proudly informed me that it would be 19mm. Having learned this lesson several times before I took a micrometer to her panels and found the average thickness with veneer was closer to 17.5mm. So you can't rely on 18mm baltic birch to be 18mm thick. You likely can't rely on 9mm plywood to be 9mm thick consistently. If the plywood is close to 9mm, then cut a 9mm slot. You can sand the surface a little where the slots intersect if they seem too snug.
 

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Looks great! Definitely nice when you can find a bit that matches the thickness of the material.
I'll add that for such a tall bookshelf you might want to add a bracket between top shelf and wall to prevent any chance of the shelf tipping forward.
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On a related note, at the college I taught at we had a large CNC which students often used to cut out plywood parts. I came up with a CNC version of halving joints that would slide together easily but tighten up when finally closed. An example here: Tapered Halving Joint. This eliminated any frustration students had putting their crisscrossing parts together.
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A link or a book I've often searched for. The creative environment of a design college and teaching furniture design to creative students is what inspired my pursuit for original joinery solutions. I have my own shop full of tools including a couple small CNCs to experiment with. I often was challenged to solve the impossible and would come home to sleep on the challenge. Usually early the next morning I'd have an idea to test out before meeting my classes that day. I'd bring my sample solution(s) in to show the students. I retired at the end of the Spring semester last year and my blog has become my repository for the repeatable useful ideas. There might be a book in there among the posts. The solutions that were unique to a specific project left the building when the student graduated. My "Process" evolved from a natural engineering/architecture aptitude, mechanical drafting courses taken in high school, a couple of design degrees and 4 decades of that creative interaction with students and other faculty. Our Workshop 2 furniture class project was to design something that could be compressed down to 1/3 of its assembled volume for storage or shipping. The knock-down ideas came from teaching those classes. Using a CNC it easy to create a pocket to embed a square nut in. Machine screws/bolts are the best repeatable strategy for assembling furniture that you want to disassemble later. The better looking furniture connection bolts out there make this potentially an attractive detail.
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I admire the leaning shelves on your blog and love the PHdesk! I was trying to comment on the blog post but it wouldn't allow it (said 403 forbidden) - that desk is a little beauty and proven itself out in the real world, so what more could be asked of it? Superb work.
Thanks. I'm not sure why you weren't able to comment on my blog unless you weren't signed into a google account when you tried.

Yep, that desk has been the most repeated build of mine. I made them all before I had a CNC to use, and the design is pretty, well, obvious. Legs 2" wide and 1.25" thick. The hook in the legs is what allows the desk to fold up nice and flat. Top of the desk was 12mm baltic birch. Bottom was 1/4" masonite hardboard. Furniture connection bolts and nylon washers between moving parts. A 1" radius roundover bit on a router table rounded the front and back of the desktop. I've used maple, oak, and cherry for different builds. Added veneer to the bb top to match whatever wood I had used. If you try to make one and have any questions just ask here.

The only halving joint project I've made for myself was a bed platform with drawers that stayed with my first ex-wife in 1980ish. Plenty of student projects over the years though did inspire the tapered halving CNC cut joint.
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How can it be so obvious to me now? I guess that after cutting 500 or so slots, I might have figured that out on my own (or maybe not?) :)
Don't fret. 99% of things done successfully don't ever find a reason to evolve. You already succeeded in making all the slots you needed to. You had and used the tools needed. The project works. Make another one though and take advantage of the great suggestions posted above. Post it here and I'll bet you still get 3 or 4 new ways to do that job easier, more precisely, and even in less time. :)
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I've been looking for a project to build besides shop items, and as the PhDesk always sells quickly I've drawn up a fresh detailed drawing and ordered some quarter sawn white oak veneer for the top(s). I'll make a trip to my closest hardwood source (Roberson's Lumber in Silver Lake KS) to pick up some 12mm baltic birch ply, white oak for the top frames and leg sets.

Four of them to make will be incentive to actually work on selling them, and should keep me busy for a few weeks at least. Four also makes good use of the sheet material needed as I can get 4 tops from a 4'x8' sheet and 4 bottoms from a 4'x8' sheet of 1/4" hardboard. No part of the design is longer than 4', so I'll have the lumberyard cut all the wood into 4' long pieces to fit inside my car or Ford Connect van if I can get it started.

I've attached a PDF side view that should be 1/4 scale when printed out. Desk height 29". Desk depth 24". Desk length variable but I usually make mine 44 inches. Feel free to make your own.

As I have a CNC I'll likely do some creative joinery between desk top parts and embed square nuts for the leg top bolt to screw into. Dowels or dominos would be fine between frame sides and front/back. A threaded insert is what I used before I had a CNC. I put a desk drawer in the middle, and may also rethink how it mounts and slides this time. I like the soft-close slides that are pretty common these days.

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Why didn't you post this as a new discussion, so you didn’t dilute the OP?
The OP, LumberJoe, had already brought up my desk design in this thread, and mentioned he might be interested in making one for himself. I'll start a new thread when I have materials on hand and actually start making a few more to sell. Just trying to follow up the side story that also started in this thread.
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Is it possible to explain how that drawer is constructed please? It's very neat but I can't really 'see' what is going on there
Possible for sure. Drawer front is a section of the front half round you can see a section view of in the PDF I posted. It is modified to receive drawer sides notched into the side corners, and a 1/4" thick drawer bottom into the half inch section. I'll draw up a top view and section of just the drawer later today and post it here.
 

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Attached is a plan view with most of the construction details shown. Bottom of the drawer is 1/4" thick hardboard inset with a 1/8" rabbet on the edges tucked into a 1/8" x 1/8" wide slot 1/8" up from the bottom edges. My actual drawer has a couple dividers in it. Once for pencils and such 2" from the front, and another dividing the rest of the space into 1/4-3/4 sections. This plan shows the width of the desk to be 36", but mine is 44, and you could make yours any width you want up to 48". My former department head made one 36" wide to fit inside a bedroom closet space he had as a small office where he hid himself from his kids when he needed to concentrate on doing taxes or paying bills.
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The joint connector bolts and barrel nuts, as well as the nylon washer for the 4 desks I'll be building should arrive today. I already have some veneer for the tops and drawer slides that will fit. I need to make a trip to a town 35 miles away where a good hardwood seller resides to get the hardwood I'll use.
 
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