Background
The parallel projection of a four dimensional Hypercube onto three dimensional space is a Rhombic Dodecahedron.
From Cube to Hypercube:
{Square format pictures of cardboard models}
[pic cube] [pic classic tesseract] [pic centrexed cube] [pic RD within cube] [pic RD]
{Ideally, we can write a Wolfram CDF that animates the centrexing process and embed it right here on the page}
{Ideally, we can write a Wolfram CDF that animates the centrexing process and embed it right here on the page}
A Rhombic Dodecahedron consists of twelve identical rhombi, each having opposing acute (narrower) angles of 70.53° and opposing obtuse (wider) angles of 109.47°. The rhombic faces meet each other at a 120° interior angle (Dihedral Angle) in a Rhombic Dodecahedron, so the edges of the rhombi have a bevel angle of half that - 60°.
When setting a cutting tool to cut a bevel angle other than 45°, it is important to know that the setting for the tool is arrived at by subtracting the bevel angle from 90° - thus the cutting tool must be set at 30° when cutting a bevel angle of 60°.
This situation is not encountered when beveling panels that will meet at a 90° angle (which is the case for most of simple carpentry), because a bevel of 45° is used - and since 45° is exactly half of 90°, the bevel angle and cutting tool setting for this case are the same.
When using a Rhombic Dodecahedron as a speaker enclosure, one of the corners is truncated and the baffle plate for the speaker driver is located at the plane of truncation.
There are two kinds of corners on a Rhombic Dodecahedron - four-line corners and three-line corners. The acute (narrower) angles of the rhombi come together in juxtaposition to form the four-line corners. The obtuse (wider) angles of the rhombi come together in juxtaposition to form the three-line corners. The four-line corners correspond to the faces of the original cube, so there are six of them. The three-line corners correspond to the corners of the original cube, so there are eight of them.
Recommended Components & Materials
Integrated co-axial drivers are strongly recommended.
2(?) sheets of ¼" gum plywood
8 Rubber feet with adhesive attachment
Sanding paper & sander
Stain
Sealant
1(?) sheet of grill cloth
grill cloth frame material
Making the pieces
Truncating a four-line corner provides a square baffle-plate and an enclosure having the smallest volume for a given woofer size.
 |
| Square truncation Hypercube Speaker |
The four rhombi surrounding the square baffle-plate are cut in half along their minor axis. Therefore when making a square-truncation Hypercube Speaker enclosure, you cut 10 rhombi with 60° bevels and then cut two of them in half along their minor axis, yielding 4 half-rhombi (a.k.a. isosceles triangles), having an apex angle of 70.53° and the base angles are 54.735°.
[Pic of two rhonbi with the minor axes indicated] [Pic of those rhombi after cutting]
The minor axis edges, which correspond with the bases of the isosceles triangles, will meet with the edges of the square baffle-plate at a dihedral angle of ??°, so the bases of the triangles and the edges of the square baffle-plate should have a ??° bevel. Set the cutting tool for ??° when beveling these edges.
Determining the rhombus size
Start with the size of the largest driver in your speaker system, which will be the woofer.
This will be mounted on the baffle plate. Decide how much clearance you want between the edge of the woofer and the edge of the baffle-plate. Multiply the desired clearance by two and add the product to the outside diameter of the woofer. This will give you the length of the side of the square baffle-plate.
If we drop a line bisecting the apex angle, we divide the isosceles triangle into two mirrored right triangles, whose angles are (A) 35.265°, (B) 54.735°, and (C) 90°, and that the Opposite leg of that right triangle is half the length of the base of the original isosceles triangle, so it is half of the length of the side of the baffle plate. The length of the hypotenuse of this right triangle will (finally!) be the length we are looking for, the length of the edge of the rhombus.
Now we have enough information that we can use trigonometry to get the rhombus edge-length we are looking for:
sin A = Opposite / Hypotenuse, therefore:
Hypotenuse = Opposite / sin A
Side of Rhombus = ½(side of baffle-plate) / sin 35.265° Thus:
Example: A woofer with an outside diameter of 10½" is being used. A clearance of ½" is desired. ½" x 2 = 1", 1" + 10½" = 11½", so the baffle plate will be a square with an edge that is 11½" long. A circular hole with a diameter equal to the inside diameter of the woofer will be cut in the exact center of this square baffle-plate.
To locate the center of the baffle-plate, use a pencil on the narrower (inner) side of the beveled baffle-plate to draw lines from each corner to its opposite corner, forming an "X" on the inside surface of the baffle-plate. The point of intersection will be the center of the circle for the woofer hole.
We divide the length of the baffle-plate side, 11½", by two, yielding 5¾". We then divide that by 0.5773, giving us a rhombus side length of 9.96". Since this is just four hundredths of an inch shy of 10", we can safely use a 10" rhombus in this particular case, given the precision of most rulers and cutting tools.
Make 2 square baffle-plates (beveled at ??º) with the woofer-holes cut in them and 20 rhombi (beveled at 60º). Cut 4 of the rhombi in half along their minor axis and bevel that minor-axis edge at ??º.
You should now have 2 beveled baffle plates, 16 beveled rhombi, and 8 beveled isosceles triangles (half-rhombi).
1] Put three beveled rhombi together , wider (outer) side up, with their obtuse (wider) angles adjacent to each other.Tape them together ( Leave them flat for now, don't tape the remaining edge yet) with Scotch Blue painter's Tape. Duct tape can also be used, but that will leave adhesive residue on the faces of the rhombi that will result in a lot more sanding during the finishing phase.
2] Turn the flat taped tri-panel over, so the narrower (inner) side is up and you see V-shaped grooves where the adjacent beveled rhombus edges meet. Using a water-filled spray bottle with a fine mist nozzle, apply water mist to the bevel edges of the rhombi. This will activate the glue used in the next step.
3] Apply (sparingly!) a polyurethane glue like Gorilla Glue to all of the bevel edges.
4] Bring the remaining two glued rhombic edges together. The tri-panel will assume a three-dimensional tetrahedral configuration Apply the tape across this edge tightly. Put it aside to dry.
5] Repeat steps [1] through [4], three more times, assembling a total of four tri-panels.
6] Once the glue in both tri-panels is dry, put the two tri-panels together with four acute (narrower) angles adjacent to each other so they form a four-line corner. Fix this alignment firmly in your mind, with attention to the tri-panel edges that come together.
7] Mist and apply (sparingly!) glue to the edges that were observed to meet in the previous step. Tape the two tri-panels together and put the assembly aside for the glue to dry.
8] Repeat steps [6] and [7], creating another tri-panel concatenation assembly, for a total of two.
9] Once the assembled tri-panel concatenations have dried, mist and glue the relevant edges and glue in the remaining four rhombi, taping across the glued edges. Set both of the resulting rhombic assemblies aside for the glue to dry.
10] Once both rhombic assemblies have dried, mist and glue the relevant edges and put in the isosceles triangles, aligning them so their base edges (the shorter edges) form the sides of a square hole and their longer edges meet with the remaining exposed rhombic edges. Tape them across the glued edges and set them aside to dry.
This will be mounted on the baffle plate. Decide how much clearance you want between the edge of the woofer and the edge of the baffle-plate. Multiply the desired clearance by two and add the product to the outside diameter of the woofer. This will give you the length of the side of the square baffle-plate.
The derivation of The Rule
(skip this part if math bores you to tears)
As noted above, the edge of the square baffle-plate will correspond to the minor axis of the adjacent half-rhombus (isosceles triangle). We know the base of this isosceles triangle is therefore equal to the length of the side of the baffle-plate. We know the base angles of this isosceles triangle are 54.735°. We know the apex angle of this triangle is 70.53°. If we drop a line bisecting the apex angle, we divide the isosceles triangle into two mirrored right triangles, whose angles are (A) 35.265°, (B) 54.735°, and (C) 90°, and that the Opposite leg of that right triangle is half the length of the base of the original isosceles triangle, so it is half of the length of the side of the baffle plate. The length of the hypotenuse of this right triangle will (finally!) be the length we are looking for, the length of the edge of the rhombus.
Now we have enough information that we can use trigonometry to get the rhombus edge-length we are looking for:
sin A = Opposite / Hypotenuse, therefore:
Hypotenuse = Opposite / sin A
Side of Rhombus = ½(side of baffle-plate) / sin 35.265° Thus:
The Rule:
Side of Rhombus = ½(side of baffle-plate) / 0.5773
Example: A woofer with an outside diameter of 10½" is being used. A clearance of ½" is desired. ½" x 2 = 1", 1" + 10½" = 11½", so the baffle plate will be a square with an edge that is 11½" long. A circular hole with a diameter equal to the inside diameter of the woofer will be cut in the exact center of this square baffle-plate.
To locate the center of the baffle-plate, use a pencil on the narrower (inner) side of the beveled baffle-plate to draw lines from each corner to its opposite corner, forming an "X" on the inside surface of the baffle-plate. The point of intersection will be the center of the circle for the woofer hole.
[Pic of square baffle-plate with "X"] [Pic of square baffle-plate with hole]
We divide the length of the baffle-plate side, 11½", by two, yielding 5¾". We then divide that by 0.5773, giving us a rhombus side length of 9.96". Since this is just four hundredths of an inch shy of 10", we can safely use a 10" rhombus in this particular case, given the precision of most rulers and cutting tools.
Make 2 square baffle-plates (beveled at ??º) with the woofer-holes cut in them and 20 rhombi (beveled at 60º). Cut 4 of the rhombi in half along their minor axis and bevel that minor-axis edge at ??º.
You should now have 2 beveled baffle plates, 16 beveled rhombi, and 8 beveled isosceles triangles (half-rhombi).
[Pic of all pieces laid out]
Assembling the Pieces
[Pic of tri-panel flat and taped layout]
2] Turn the flat taped tri-panel over, so the narrower (inner) side is up and you see V-shaped grooves where the adjacent beveled rhombus edges meet. Using a water-filled spray bottle with a fine mist nozzle, apply water mist to the bevel edges of the rhombi. This will activate the glue used in the next step.
[Pic of tri-panel misting operation]
3] Apply (sparingly!) a polyurethane glue like Gorilla Glue to all of the bevel edges.
[Pic of tri-panel glue application]
[Pic of assembled tri-panel ]
5] Repeat steps [1] through [4], three more times, assembling a total of four tri-panels.
[Pic of four assembled tri-panels]
6] Once the glue in both tri-panels is dry, put the two tri-panels together with four acute (narrower) angles adjacent to each other so they form a four-line corner. Fix this alignment firmly in your mind, with attention to the tri-panel edges that come together.
[Pic of tri-panel alignment]
7] Mist and apply (sparingly!) glue to the edges that were observed to meet in the previous step. Tape the two tri-panels together and put the assembly aside for the glue to dry.
[pic of tri-panel concatenation assembly]
8] Repeat steps [6] and [7], creating another tri-panel concatenation assembly, for a total of two.
[pic of both tri-panel concatenation assemblies]
9] Once the assembled tri-panel concatenations have dried, mist and glue the relevant edges and glue in the remaining four rhombi, taping across the glued edges. Set both of the resulting rhombic assemblies aside for the glue to dry.
[pic gluing rhombus into a tri-panel concatenation assembly]] [pic of completed rhombic assemblies]
[pic of gluing triangle into rhombic assembly] [pic of completed non-baffle assemblies]
11] Once both non-baffle assemblies have dried, mist and glue the relevant edges and put in the bafle plates. Be sure to put tape tightly across all four edges of the baffle-plates, two pieces of tape per edge. Put both assembled enclosures aside to dry.
[pic of gluing baffle-plate into non-baffle assembly] [pic of completed enclosures]
12] Once the enclosures have dried, remove all tape and sand their outer surfaces to smoothness
[pic of sanding operation]
13] Once the enclosure surfaces are smooth and all sanding dust has been completely removed (a canister vacuum is useful here), apply the stain of your choice. Follow the directions that came with your chosen stain carefully. Put the stained enclosures aside to dry completely.
[pic of staining operation]
14] Once the stained enclosures have completely dried, apply the final clear glossy polyurethane finish - this will seal the wood and bring up the wood-grain in the stained wood, providing a beautiful glossy durable finish. Put the finished enclosures aside to dry completely.
[pic of sealing/finish application operation]
[pic of back-plate installation] [pic of both enclosures with installed back-plates]
16] Install the crossovers.
[pic of crossover installation]
[pic of silicon rubber application]
18] Install the drivers. Put the assembled Hypercube Speakers aside for the silicon rubber to cure, approximately 24 hours.
[pics of driver installation]
19] At this point, the Hypercube Speakers are ready for operation, and you can stop here, if you wish. However, it is emphatically recommended that you protect the driver(s) from inadvertent harm with protective grills. What sort of grills is a matter of personal preference - however you can make pyramidal grill cloths that complete the lines of the Hypercube for maximal protective and aesthetic effect.
[pic completed Hypercube Speakers] [pic of Hypercube Speakers with pyramidal grills]
Making the Pyramidal Protective Grills
20]
Well done. This is a good start. I hope the world is listening. --MRK
ReplyDeleteThank you so much for posting this. The dihedral/bevel angles for the half-rhombi joining the baffle show up as question marks (at least with firefox and internet explorer) so I'm guessing they're 135/67.5 degrees respectively? Also I cannot see the pics. Anyway, I am planning to construct these enclosures using CAD software provided by eMachineShop. With 26 pieces the cost definitely adds up especially with plywood. However, polycarbonate seems to be a much more affordable option, so I'm hoping that it's a suitable material. I'm thinking of attempting a small build first (4" full-range driver) and a bigger version later if everything works out. I imagine there are many people who would like the appearance of a flush-mounted (counter-sunk) driver which could be done for a number of drivers if 3/8" material is used. That thickness would still be about half the wavelength of the highest treble frequencies so I'm assuming this is ok? OK last thing - might there be any parameters that would indicate that a particular driver is better suited for this enclosure than another (driver)? For example, would it be wasteful to use a driver with a large excusion cabability? Thanks again for this info and any response would be greatly appreciated. --Greg
ReplyDelete