Copyright ©1994, 1995, 1996, 1997, 1998, 1999 by Alternity, Inc. All rights reserved. The Projected Tesseract (Rhombic Dodecahedron) is a relatively obscure shape which has been found to be a superlative form for loudspeaker enclosures. An analysis of the behaviour of sound inside this unique geometry reveals that the boundary conditions are the reverse of those usually found in speaker enclosures; these boundary conditions account for the unprecedented improvements obtained by utilizing conventional electrodynamic drivers in PT cabinets. Sound waves are spatiotemporal phenomena. The 4 dimensions of 3space plus time intertwine in the phase of a sound wave. When a continuously emitting source is located near a plane (or on the face of a box), the reflected sound interacts with the emitted sound. At certain frequencies the interaction patterns form standing waves (stationary patterns of energy which throb at the characteristic frequencies). Mounted on an ordinary 3dimensional box, the standing waves interfere with the speaker's motion and induce distortion in the speaker's output. Mounted on an Inverted Projected Tesseract, however, the standing waves cooperate with the woofer's motion, adding very little distortion to the sound emitted by the speaker. A wave is an energy manifestation which travels from place to place. If you drop a penny n a bathtub full of water, the impact of the penny transfers some of its kinetic energy into surface of the water. This disturbance dissipates outwards as a circular ripple whose radius grows as its height decreases. As the ripple passes a point on the water's surface, the water level at that point rises and falls. Just as lights flashing in sequence appear to be a moving light, this rising and falling of the water appears to be a moving (growing) circular wavefront. Sound is a wave. If an object moves rapidly enough, it compresses the air in front of it, creating a highpressure region. This region appears to move, just as the raised ripple appears to move in the surface of the water in the tub. If an object vibrates back and forth, it emanates a continuous train of higherthenlower pressure waves. The way to express this mathematically is: Let x be the distance from the source A be an arbitrary scaling constant f be the frequency of the sound wave t be the elapsed time since the wave was emitted l be the wavelength p' be the elevated or lowered air pressure. Then the formula describing ordinary plane waves can be written as p' = Re[Ae^{i2p[(x/l)ft]}] or p' = Acos(2p[(x/l)ft]). If we freeze time, the pressure elevation is distributed spatially as a cosine curve. If we looked at a fixed position in space, we see a pressure which rises and falls as a cosine curve over time. The wave intensity at any point in space and time is determined by the quantity f = 2p (x/l ft) This quantity is called the phase of the wave. The phase determines the amount that the local air pressure is raised or lowered relative to the ambient air pressure. When two wave pass through the same point in space at the same time, they will either be "in phase" (both raising or lowering the air pressure at the same time) or "out of phase" (working against each other and cancelling each other out) to some degree. In phase waves add to produce a more intense raising or lowering of pressure; waves that are completely out of phase tend to cancel and result in no raising or lowering of air pressure at all. Sound waves can change their direction of motion by bouncing or reflecting off objects or barriers in their path. Usually this merely changes the direction the wave is travelling in. In cases in which the wave strikes a wall dead on, instead of at an angle, the wave merely reverses its direction of motion. When a source of sound waves is emitting a continuous train of waves, and is placed near a wall so that the waves reflect straight back to their source, at certain frequencies (determined by the distance between the source and the wall) the waves moving in opposite directions combine to form a standing wave, in which the spatial pattern appears to be motionless but the temporal pressure fluctuation is intensified. Standing wave patterns are common in musical instruments. In string instruments such as the guitar or the violin, a wire fixed and motionless at both ends creates a standing wave of backandforth or upanddown motion. When the wire is plucked or strummed, the travelling waves rebound from the fixed ends of the wire and pass through each other to create a standing wave at a frequency determined by the length, linear mass density, and tension of the wire. The fixed ends of the wire, where the wire cannot move, are called displacement nodes in the standing wave's spatial pattern. The regions where the wire vibrates backandforth the most are called displacement antinodes. Standing waves also occur in an organ pipe. The longer the pipe, the lower the characteristic frequency, because longer wavelengths (lower frequencies) can fit into a longer pipe. The occurrences in a pipe, however, are more complex than the simple upanddown vibrations of a wire. This is because the density of the guitar wire remains relatively constant (metal is fairly incompressible), but in a sound wave the density, pressure, and molecular displacement are all wave quantities. This means that in regions where the air is free to move, there is very little pressure change, but where the air is constrained or crowded, there is little movement but the pressure fluctuates. In other words, in a in a standing sound wave regions that are displacement antinodes are pressure nodes and vice versa. Either the air moves back and forth, or the pressure increases and decreases, but not both. Where the air moves the most, the pressure changes the least. It is more difficult to picture sound waves in an organ pipe than on a guitar string, because the wire moves up and down as the waves travel down it, whereas the air in a pipe moves in the same direction that the sound wave moves. We say that the wire waves are transverse waves (vibrating perpendicular to the wave travel) whereas air movement in sound waves are longitudinal, meaning that they are parallel to the long axis of the pipe, and hence parallel to the direction of travel of the wave itself. Standing waves occur in pipes because sound tends to reflect back from the ends of the pipe. In a pipe closed at both ends, the ends are displacement nodes, like the fixed ends of a guitar string. At the closed ends of the pipe, the air cannot move back and forth, so the sealed ends are displacement nodes. The pressure at these sealed ends, however, can rise and fall as the air molecules crown into them like people crowding into an elevator. The sealed pipe ends become pressure antinodes. Conversely, in the middle regions of the pipe, where the air can move back and forth, there is no "crowding" of the air molecules, so the middle regions of the sealed pipe tend to be pressure nodes and displacement antinodes. When the ends of the pipe are open, sound waves travelling inside the pipe will still reflect back from the ends of the pipe, because of a pressure "overshoot" that occurs at the open pipe ends. When a high pressure wavefront reaches the open pipe end, it finds itself suddenly freed from the constriction of the pipe and expands outward from the end of the pipe. This sudden expansion of the high pressure wave pulls air molecules out of the pipe, causing the air pressure there to go lower than the normal atmospheric pressure. Thus, the exit of the highpressure wavefront from the open pipe end creates a lowpressure inside the pipe, which then travels back down the pipe in the opposite direction. The opposite occurs when a lowpressure wavefront reaches the open pipe end. In this case, the lowpressure region pulls air molecules into the pipe from outside, creating a highpressure reflection. It is important to remember that the pressure overshoot condition reverses the phase of the reflected sound wave. This is in contrast to the reflection from sealed pipe ends, in which only the wave's direction of travel reverses. Since reflection (of a sort) occurs at the open pipe ends, standing waves can again be created inside the pipe. In this case, however, the open pipe ends, where the air is free to move, turn out to be displacement antinodes and pressure nodes. Conversely, the middle of the pipe, where converging waves crowd or stretch the air, tends to be a pressure antinode and a displacement node. It can readily be seen from the preceding descriptions that the wave behaviour inside a simple resonator such as a cylindrical pipe is strongly influenced by the waves in which the sound waves must behave at the ends of the pipe. We call the rules imposed by the ends of the pipe the boundary conditions which must be satisfied by the equations describing the motion and behaviour of the sound waves inside the pipe. While we may change the material the pipe is made of, the gas inside the pipe, or the nature of the device used to generate the sound waves, the boundary conditions which obtain at the ends of the pipe are controlled mainly by the geometry of the pipe ends. Sealed ends give one set of boundary conditions; open ends generate an opposite set of boundary conditions. As we shall see, the ability of geometry to control the boundary conditions satisfied by the waves is a key to understanding and improving the design of loudspeakers. 100 years ago the electrodynamic loudspeaker was created by Tesla as a "telephone repeater" to let more than one person listen to a phone call. A current in the speaker's "voice coil" made the coil into a temporary magnet, attracted to or repelled by a permanent magnet on the back of the speaker. Moving toward or away from the magnet, the voice coil dragged a cardboard or paper cone back and forth, generating sound waves. Unfortunately, both the front and the back of the cone emit sound waves when the cone is dragged back and forth. These two emissions are out of phase. This is because the cone cannot move forwards and backwards at the same time. When the "frontwave" emitted by the front of the speaker is a highpressure wavefront, the "backwave" is the opposite: a lowpressure wavefront. They are out of phase. Because of the destructive interference (cancelling out) that would occur if both of these wavefronts reached a listener's ear, they cannot both be allowed to escape into a listening area. This is a problem made inevitable by the backandforth movement of the speaker cone. Two main strategies have been employed for decades to try to get around the problem of the speaker's backwave. The first is to mount the speaker on the front of a sealed box: only the frontwave can escape and travel to the listener's ear, so there will be no cancellation. This is known as the "Acoustic Suspension" design. Unfortunately, "hiding" the backwave this way creates new problems. First, throwing away half of the speaker's output energy reduces the efficiency of the device. Second, and more important, the backwave has nowhere to go, so its energy and momentum must reflect back to the speaker cone and interfere with the movement of the cone. This tugofwar on the speaker cone prevents it from accurately following its input signal, causing distortion in the speaker's sound output. Moreover, at certain frequencies determined by the box dimensions the air motions inside the box create standing waves. The result is an uneven response of the speaker to the musical program being reproduced. Some frequencies become resonant and as a result are louder than they should be. At other than resonant frequencies, partial selfcancellation reduces the speaker's output. The upshot of all of this is that the speaker "booms" at certain notes, is unnaturally quiet at other notes, adds significant distortion to the signal, and is inefficient. The second main strategy employed by speaker designers is to cut a hole in the box, so that some of the backwave can escape and (hopefully) reinforce the frontwave. This is known as the "vented box" design. this design, also, has serious flaws. Allowing the air to leak in and out of the box lets the speaker move more freely. This tends to make the speaker more efficient, but also adds distortion to the output because the speaker's motion is not as tightly controlled. The "vent" or "bass port" introduces other problems, too. The vent output cannot be completely in phase with the speaker's frontwave, except at certain frequencies determined by wavelength/pathlength relationships. At all other frequencies, the vent waves are at least partly out of phase with the frontwave, causing varying amounts of cancellation at these other frequencies. Despite years of changes and millions of research dollars, no speaker of this type has corrected the design flaws. The result have been distortion and uneven response. We see, therefore, that mainstream loudspeaker designers are left with a choice between two evils. Either they can opt for more control and less efficiency (sealed box), or less control and more efficiency (vented box). Either way they get uneven response and distorted output. it appears that what is needed to resolve these problems is a loudspeaker enclosure which combines the advantages of both designs without their drawbacks. In other words, what is needed is design that has the efficiency of the vented box and the control of the sealed box, and that produces less distortion than either design. When we look at both box designs we see a common element: the box. The rectangular box used in both designs is rigid and unyielding, requiring displacement nodes on all of its inner walls. But the speaker mounted on one of these walls must function as a displacement antinode in order to perform its function. this sets up builtin problems in both types of box design. Sealed box designer try to alleviate this problem by stuffing the box with a soft, absorbent padding to try to prevent backwave reflection and standing wave phenomena. Vented box designers try to solve the problem by cutting a hole somewhere on the box. This merely creates another displacement antinode (remember that holes in contact with the atmosphere must be pressure nodes and displacement antinodes). What is needed is a speaker cabinet shape which does not place the speaker's displacement antinode in the muffle of a wall displacement node. The speaker must be mounted somewhere on the cabinet where we would already expect a displacement antinode, so that the speaker's motion can blend right in. This is difficult to do; it would seem inevitable that any container made with rigid walls must have displacement nodes on its walls, since the walls cannot move. Fortunately, a cabinet shape does exist which allows the speaker to be mounted in the middle of a displacement antinode, eliminating the tugofwar of reflected backwave distortion. Unbelievable as it sounds, this shape has been known for over two thousand years and turns out to possess acoustical advantages no audio engineer or mathematician ever anticipated. This shape is found in nature in the crystals of gold, silver, copper, lodestone, and lapis lazuli.^{1}. The shape is known as the Projected Tesseract, known to mathematicians and crystallography experts as the Rhombic Dodecahedron. This shape, when hollow, creates a unique inner resonant cavity which has cubic symmetry and permits an antinodal displacement boundary. Cubeshaped standing waves can exist within the PT cavity; each face of the cube is a displacement antinode into which the speaker's vibration can blend perfectly. Remarkably, resonance becomes an advantage, not a problem. The tapered pyramidal corners of the PT allow each frequency to form its own antinodal surface at a unique crosssectional position. this means that no frequency is favored over the others, so no "booming" overemphasizes any note over the others: they all boom evenly. If the PT is made from thin enough materials, the cubical displacement antinode forces the walls of the PT to flex inwards and outwards, radiating sound out in all directions. This converts the dipole (twosided) speaker into a source omnidirectional lowdistortion sound. Imagine a pipe that has ends that taper down to points. The propagation of sound waves down the pipe is controlled by the relationship between the wavelength of the sound wave and the cross section of the pipe. Long wavelengths (low frequencies) need a wide pipe to travel well; shorter wavelengths (higher frequencies) can penetrate narrower sections of pipe. A sound wave travelling down a narrowing pipe will reflect back when it reacher a section of pipe too narrow for it.Furthermore, this reflection is more like a reflection from an open end of pipe than it is like a reflection from a rigid wall, because the wave "turns around" before it reaches the end of the tapered pipe, so that at the point of reflection the air molecules can move. In other words, the point of reflection in the tapering pipe is a displacement antinode. If the pipe tapers down to a point, all frequencies must reflect back. Each does so at a different cross section, since each frequency has a distinct wavelength and can therefore penetrate the narrowing pipe to a distinct distance. If both ends taper down to points, standing waves can be formed at practically all audible frequencies. The PT shape is generated by connecting the corners of a cube to its center (which slices the cube into six inwardpointing pyramids) and then turning the structure insideout. This creates a cube of emptiness, from whose open faces pyramids now point outwards along the positive and negative x,y,z axes. Along each Cartesian axis the structure resembles the aforementioned pipe tapering to points at both ends. Each pair of opposing pyramids creates a pair of potential standing wave displacement antinodes. Subtracting or truncating one of these pyramids leaves in its place a square on one face of the inner cube; placing the woofer on this square puts it right where the displacement antinode of the missing pyramid would occur (where the air "wants" to vibrate the most), eliminating the tugofwar and the distortion it normally causes. In this placement, the displacement inside the cabinet can actually cooperate with the speaker's motion, increasing its efficiency without distorting its output. We have illustrated the pros and cons of traditional designs and have introduced Alternity's finding that the symmetrical antinodal displacement boundary of the PT harmonizes the necessary motion of air molecules in a speaker cabinet with the relative stillness the speaker cone needs in order to function with greatest fidelity. In order to give a more detailed description, the following sections attempt to describe through mathematical analysis the PT's boundary conditions that are the complete reversal of those usually found in speaker enclosures. The formulation of physical laws is facilitated by a notation developed by Grassman and Cartan, known as Exterior Calculus. It works well in any number of dimensions and in any coordinate system, and avoids the jumble of indices found in tensor notation^{2}. The principal entities in exterior calculus are the functions ("zeroforms"), Kforms, and the operators which act upon them. Functions are called zeroforms because they posses zerodimensional point values for each position in spacetime. Examples relevant to acoustics include pressure and density. Three important operators in exterior calculus are the differential operator d, "wedge" operator L, and the star operator "_{*}". The differential operator operates on kdimensional forms to create k+1 dimensional forms. For example, operating on the zeroform f(x), we obtain a oneform consisting of the product of the derivative of f with respect to x and the differential dx: d [f(x)] = f '(x)dx . (1) To give a higherdimensional example, we must first discuss "wedge" multiplication. Wedge multiplication is a way of expressing bilinear or multilinear functions. Given a coordinate space V containing two vectors v_{1} and v_{2}, and the two elements w^{i} and w^{j} of the "dual" space V^{x}, we can construct an alternating bilinear function by:^{3} [w^{i}Lw^{j}](v_{1},v_{2}) = w^{i}(v_{1})w^{j}(v_{2})  w^{i}(v_{2})w^{j}(v_{1}). (2) From this it is obvious that (aLb) =  (bLa) . Wedge products can be conveniently expressed as determinants; this facilitates extending the discussion to products involving more than two factors. Given a oneform fdg, where f and g are functions, we can now operate with the differential operator to obtain a twoform:^{4} d(fdg) = df L dg. (3) The Star operator is a rotationinvariant operator which can best be understood by considering examples of its action:^{5} *dx = dy Ldz *dy = dz Ldx *dz = dx Ldy *[f(x,y,z) + g(x,y,z)] = f dy Ldz + g dz Ldx (4) The existence of a scalar product creates a correspondence between a vector field and a differential form. For example, if we use the Cartesian vector field: > > > > A = A_{x}e_{x} + A_{y}e_{y} + A_{z}e_{z} , the associated oneform is:^{6} A = A_{x}dx + A_{y}dy + A_{z}dz (5) This allows us to express the familiar vector space operators in the form they take in exterior calculus notation:^{7} > grad f = df > > curl A = * dA > div A = * d *A . (6) The fundamental equation of motion for a volume element of fluid is (written in exterior calculus notation):^{8} (7) If we rewrite this in terms of derivatives at a fixed point, we obtain: (8) From this we obtain the fundamental differential equations for sound waves: (9) The particular nature of the functions for V and p' will be determined by the nature of the boundary conditions imposed upon the functions by the geometry of the enclosure. For example, in the tradition rectangular box enclosure, the rigidity of the walls located perpendicular to the x,y,z axes of symmetry requires that the functions for V must contain displacement nodes at the walls. This spatial periodicity, in turn, restricts the allowed frequencies to those whose wavelengths fit correctly into the boundary spacing. If we center the origin of our x,y,z coordinates on any corner of the box, the pressure elevation can be written as (10) Where k_{x}x = np k_{y}y = np k_{z}z = np For x = 0, L_{x} (L_{x} = length of box along xaxis ) y = 0, L_{y} z = 0, L_{z} guarantees a pressure antinode (V node) on the boundary surface. This requires that the frequency is given by L_{x} = nl/2 and so on for the y and z axes, giving three distinct resonant spectra determined by the length, width, and depth of the box. The discrete resonant spectra of the parallelepiped guarantee a bumpy response curve for the box loudspeaker. At resonant frequencies, the speaker will be moving either parallel or antiparallel to its reflected backwave. When parallel, the speaker's cone movement will be increased, boosting output amplitude. When antiparallel, the cone movement will be attenuated by opposing forces, diminishing output amplitude, Also, since the cone movement is restricted, more energy will remain in the system, resulting in greater thermal waste. An ideal loudspeaker would resonate evenly across the frequency spectrum, fitting each wavelength properly to form standing waves in the fundamental mode shape in which the reflected backwave is in phase with the speaker's frontwave. Cone movement would be maximized at frequencies, resulting in efficient power transduction and reduced thermal waste. Such an ideal loudspeaker would be difficult to design with boundary displacement nodes at all frequencies. It would probably have, instead, a boundary consisting of pressure nodes (displacement antinodes). Consider, however, the PT. Like the parallelepiped, the PT possesses Cartesian symmetry. But unlike the parallelepiped, the boundary surface of the PT does not contain rigid planes perpendicular to the x,y,z axes of symmetry. Instead, the outwardlypointing pyramidal concavities of the PT's interior allow displacement antinodes (pressure nodes) perpendicular to the x,y,z axes of symmetry. Consider a sound wave reflecting back and forth along the xaxis. Its "standing wave" pattern contains pressure nodes where p' = 0. Locating our origin of coordinates as before, we are led to try the following function for p': (11) Let us examine one pyramid; from symmetry, we will then know the behavior in the other pyramids. The p' function then determines (via 9a) the nature of the allowed V_{x} function: (12) Integrating over time, we obtain the expression for the velocity oneform (recall that this is associated with the velocity vector): (13) As a consistency check, we can use (9b) to calculate the pressure function associated with the velocity function: (14) Integrating this expression over time to obtain p', we get: (15a) And since (15b) We have: (15c) which was the pressure function (11) we began with. The motion of air inside an PTenclosed loudspeaker is a remarkable thing! The planar pressure nodes (displacement antinodes) present in each pyramidal subsection of the PT come together to form the faces of the inner cube, which cannot move left, right, up, down, forwards, or backwards. Pressed in on all six sides by inwardmoving incoming waves, the cube of air in the middle of the PT can only become smaller and larger as it remains in place. Another way to express this is to say that the cubeshaped volume of air in the center of the PTshaped cabinet oscillates only in the 4th direction, which means that its apparent intersection with our space becomes smaller and larger with respect to an equal volume of air in our 3space, in the same way that the inner cube of a projected tesseract looks smaller than the "outer" cube, because the inner cube is "farther away" along the 4th direction. That this state of motion can exist at even one frequency is remarkable. But we know from the tapering of the pyramidal corners of the PT that this state of motion can exist at any frequency within the operating range of the cabinet, because each frequency will penetrate the pyramids to a unique distance before the narrowing cross section forces it to reflect back upon itself. Not only this, but the general motion must be described as a superposition of all of the allowed motions (frequencies), including sub and superharmonics of the cabinet's operating range: virtually the entire audible spectrum can participate to some degree. Traditional speaker cabinet construction shuns thin materials because of the likelihood of creating "boomy" resonances that overemphasize particular frequencies and elevate impedance. This is due to the discrete resonance spectra of the parallelepiped mentioned above, which singles out certain frequencies for "preferential treatment". To avoid these resonances, traditional speaker cabinets are made from thick, heavy materials and are padded and braced internally, which significantly increases both the cost and the weight of the speaker. In the PT, however, the ability of the enclosed air volume to respond adroitly to a wide range of frequencies avoids any "preferential treatment". The cooperative reflected backwave of the PTenclosed speaker eliminates the need for internal padding of any kind; its ability to redirect and reradiate the backwave as omnidirectional supplementary frontwave means that the PT loudspeaker is more effective and more efficient if it is made from thinner, more resonant materials, like a violin or an acoustic guitar, than if it is made from the thicker woods of more traditional loudspeaker "furniture". In PT loudspeakers, resonance is not an isolated problem of a few unique frequencies, but a uniform advantage which enhances performance at all frequencies.
