THE CONES OF MAXIMUM VOLUME
Modeling an Ideal
The concept “the most efficient packaging of volume” can be modeled in geometry by any number of forms. Simply put, the idea is to package the most volume using the least surface area possible. Usually, the form of the sphere is associated with this concept since it is the supreme packaging form. You can’t do any better than the sphere. But given certain parameters, or constraints, we can glean from geometry other models of this concept. One of these models is a family of forms I have come to call The Cones of Maximum Volume. There are three such cones within this very special family.
The first of these cones is extracted from a range of cones all having a fixed edge length (defined as the distance from the tip of the cone to its base circumference). I liken this cone to the teepees of the American Indians to distinguish it from the other two, because we can imagine the cone’s fixed edge lengths to be analogous to the equal length poles delineating the teepee configuration. Now keeping the poles attached at the very top point of the cone (or teepee) we can vary its “height” at will. As the height lowers, its circular base grows larger; raise the height, and the cone’s circular base diminishes. This range of cone proportions varies from a very flat wide based cone with little or no volume (like a two dimensional circle), to one that’s tall and narrow, with little or no base area, and little or no volume. The two extremes of this range are an erect line approaching the cone’s edge length containing no volume within; and a circle having a diameter approaching twice the length of the cone’s edge. Somewhere in between there is one specific cone proportioning that captures the most volume possible.
To form this ideal cone from a flat circle, a pie-shaped wedge measuring (1 – (2/3)^1/2) of 360 degrees must be removed from the circle (66.06123… degrees). When the opposing radials of the circle are drawn together to form the cone, it is found that the cones circular base is 2/3 the area of the original flat circle, and its base circumference is (2/3)^1/2 the original circle’s circumference. The cross-section of this ideal volume maximizing cone and its originating circular proportioning is illustrated below. From this diagram we learn that this cone’s formative triangle (half the cross-section) has its three edges proportioned to the “square” roots of 1, 2, and 3.
The second Cone of Maximum Volume arises when it is stipulated that a cone be made from a given surface area such that the resulting cone captures the most volume possible. Again the range of possible cone shapes extends from the very low and wide to very tall and narrow. Somewhere in between all these cones of various aspect ratios and identical surface areas is the one specific proportioning that captures the most volume within. The cone that emerges has a formative cross-sectional triangle identical to the teepee cone above. It is simply rotated ninety degrees. A pie-shaped wedge measuring
2 / ((3)^1/2 + 3) of 360 degrees
(152.1539…degrees) removed from a circle will form this volume maximizing cone.
The fixed surface area Cone of Maximum Volume discussed above stipulates that all the surface area be incorporated into the cones side measure, ignoring its circular base area.
The third cone in this unique family considers this circular base area as well as the side measure when containing the most volume possible given a constant surface quantity. The cross-section of this cone reveals a base angle equal to twice the based angle of the teepee cone. Its pie-shaped wedge measures 2/3 of 360 degrees (240 degrees). This cone’s cross-section and circular derivation is illustrated in the diagram below.
These three particular forms are models of the concept most efficient, or economical packaging of volume. Each of these cones is uniquely shaped with respect to the others; yet they all share the same angular structuring. This is because they truly are models of the same “concept”. Thus far they have been discussed as pure forms, void of all scale or size. But when scaled to various aspects of “unity”, they reveal themselves to be deeply rooted members of the geometric hierarchy.
There is bountiful evidence that geometry embraces this “family” and incorporates it into the structural relationships of its most primal and formative organizations. For example, 1.0 surface-unit encapsulates the most volume possible only if it assumes the form of a spherical surface. Now if for some reason the parameters change, and it is stipulated that the surface unit perform the identical function (encapsulate volume in the most economical fashion) but be limited to the form of a cone, then the 1.0 surface-unit transforms from a spherical shaped surface into the surface of the third Cone of Maximum Volume presented above, where the circular base of the cone is a component of the 1.0 surface-unit.
Try to imagine this transformation in the laboratory of your mind. You should see a solitary sphere, and then see it to (magically?) change its surface into the shape of this particular Cone of Maximum Volume. Now because this cone is not a sphere, it can’t hold all of the sphere’s volume. But The Geometry of Form demands a specific and model-able accounting of surface and volume at the beginning and end of all transformations. So what the viewer will see happen along with the transformation of the sphere’s surface into the cone’s surface is the expulsion of the “excess volume” in the form of two tetrahedronal packages. Why two packages? Why in the form of two tetrahedrons? Well, I didn’t make this up. It’s just that’s how the geometry of this transformation is structured. Take a look at the drawing below.
The blue semi-circles at the bottom represent hemispheres of the 1.0 surface-unit sphere. The orange triangle is the cross-section of the same 1.0 surface-unit in the form of the ideal Cone of Maximum Volume. Now if this was any other cone form (other than the one created as an embodiment of the concept most economical packaging of volume), it’s circular base certainly would NOT be the same circle as the sphere’s cross-sectional circle, such as in the case of the sphere and cone in this diagram. Nor would any other cone’s height be equivalent not only to the two tetrahedronal packages of excess volume stacked atop one another, but also to the two sphere’s which result from another, slightly different transformation: the division of the 1.0 surface-unit sphere’s surface into two new spheres (which also requires the ejection of the same two tetrahedronal packages of excess volume). These two new spheres’ combined volume equals the volume of the cone. More proof of geometry’s affirmation that we have discovered its packaging preferences can be seen in how perfectly this cone nestles over the ½ surface-unit sphere. This sphere is the same height as the tetrahedronal excess volume package and is also the same volume as this tetra-form’s spun-into-cone (of Maximum Volume, less base-circle) counterpart.
When these three different cone models of the same volume maximizing concept are scaled to the same base circle, there is again abundant evidence that geometry has incorporated conceptual modeling into the very framework of its structural organization.
The next diagram shows these cones atop the same circular base, again the cross-section of the 1.0 surface-unit sphere. It can be calculated that the tip of the edge-constant cone (1) is the center of the ½ surface-unit sphere; the tip of the side-constant cone (2) is the point of tangency between the two ½ surface-unit spheres; and of course, the tip of the base-included cone (3) coincides with the combined height of the two ½ surface-unit spheres. The height of the side-constant cone is 2 times; and the base-area-included cone 4 times the height of the edge-constant cone. And since they all have the same base area, their volumes relate as 1, 2, and 4 times the volume of the edge-constant cone.
So far we’ve looked at the cones as they relate to the spherical surface-unit and found overwhelming confirmation for their inclusion into the primal geometric hierarchy. But what do these ideal cones look like when they are set to equal 1.0 volume-unit?
The diagram below depicts two slightly different arrangements of the ideal (base-included) Cone of Maximum Volume scaled to equal 1.0 volume-unit. Both depict its relation to its 1.0 volume-unit spherical counterpart and this sphere’s division of volume into two into ½ volume-unit spheres.
The arrangement on the left is showing that the two ½ volume-unit spheres are the same height as the cone when they are stacked atop one another. We already know that their combined volumes are the same as the cone; and so are their combined surface areas. As depicted, the cone sits upon the cross-section of the 1.0 volume-unit sphere. It has room to slide about within the cone, unlike the ½ volume-unit sphere above, which nestles perfectly within the confines of the cone, the sphere above and the base of the cone.
But when these ideal forms are arranged a bit differently, as depicted on the right side of the diagram, this apparent “sloppiness” is eliminated. Here the 1.0 volume-unit sphere is placed at the tip of the cone and another directly below and tangent to it. Now the 1.0 volume-unit sphere fits perfectly within the confines of the cone. All of the points of tangency revealed in the drawing have been confirmed (by my own calculations) to be mathematically commensurate. This drawing certainly reveals the powerful relationship between the concept the most efficient packaging of volume and the prime geometric forms for accomplishing such a task. For if two spheres is the next most efficient packaging choice after a single form, then this ideal cone is a true equivalent to the two spheres for it contains the same volume within the same surface. It should be noted (as we saw earlier), this is also the case when the cone and spheres considered are 1.0 and ½ surface-units respectfully.
When the side-area only version of the Cone of Maximum Volume is scaled to contain 1.0 volume-unit we find its height measure is equal to the diameter of the 1.0 volume-unit sphere.
We can see this in the following diagram:
And the edge-constant, or “teepee” version of this ideal cone likewise reveals its affinity to the spherical 1.0 volume-unit. In this case the cone’s height is the same height as a hemisphere containing the 1.0 volume-unit, as is illustrated below.
These direct relationships between “unity” and the ideal volume maximizing forms couldn’t be clearer. They are built into the very fabric of geometry.
The individual domain of a single sphere, when closest packed with other same size spheres, is defined by a rhombic dodecahedron. Packed together, these twelve-sided geometric solids fill all of space leaving no voids. It is the most efficient packager of volume with respect to all-space filling polyhedrons made from the same polygons.
When the proportions of this form are calculated, one finds that the triangle based on the “square-roots of 1, 2, and 3” define its twelve rhombic faces. This is the same triangle found at the heart of the ideal volume maximizing cones. In the Cone of Maximum Volume where the surface area of the cone without its base is set to equal 1.0 surface-unit, its volume (0.11667…) can be modeled by a cube with edge-lengths of 0.4886…, which is the edge-length of the Transit-tetrahedron. And when a system of seven (the Decahedron) transforms into a system of eight (the Star-tetrahedron), and the system of eight is likewise modeled by five tetrahedrons, 0.11667… of one tetra-volume is ejected by the Decahedron in the transformation as a Transit-tetrahedron of excess volume.
The Cylinder of Maximum Volume
Ideal Volume Maximizing Cones
Another model illustrating the principle of maximizing volume is the Cylinder of Maximum Volume. Any given surface area can be formed into a cylinder of any proportions, but there is only one cylinder proportion which will maximize volume given that surface area. In this model the circular top and bottom must be included in the given surface area since the alternative (surface area forming side only) will create tubes approaching infinite length and little volume to a circular strand approaching infinite circumference and great volume.
The Cylinder of Maximum Volume has its diameter and height in a 1:1 proportion. Its rectangular cross-section is a perfect square. The cylinder’s side is 2/3 of its total surface area. A sphere with its diameter equal to the cylinder’s diameter has a surface area equal to the cylinder’s side area, and a volume 2/3 that enclosed by the cylinder.
The constant-edged, or teepee form of the Cone of Maximum Volume can be made by removing a pie-shaped wedge from a circle of given area and then drawing together the edges either side of the missing section. If an area equivalent to the cone’s total originating circle of given area be fashioned into a cylinder of maximum volume, then the diameter of this resulting cylinder is equal to the radius of the cone’s base as in the diagram below.
When a surface unit in the form of the volume maximizing cylinder is superimposed with the same surface unit in the form of a sphere’s surface the “teepee” form of the Cone of Maximum Volume again emerges. The
cross-sections of this cylinder and sphere are arranged in the diagram below. The congruent points where the two forms intersect form the circular bases of two of these cones; and the center point of the intersecting sphere and cylinder is the common pointy of tangency for the tips of the two cones. The volume of one of these two internal cone structures compares to the volume of the cylinder in the same proportion as the volume of a one-surface-unit tetrahedron compares to a one-volume-unit tetrahedron.
Furthermore, either of these two cones of maximum volume formed by the congruent cross-sections of the cylinder and sphere can be derived from original circles of area equal to the sphere’s cross-section. The imprint of another of the ideal cones, side area only, is seen between the two “teepees” in blue triangles in the diagram below.
The importance of these diagrams is their illustrating how the “concept” the most economical packaging of volume translates into the actual structuring of the geometry of form.
The Cylinder of Maximum Volume
Volume Maximizing Number Pattern 1837
When the Cylinder of Maximum Volume is quanta-sized as being 1.0 volume-unit, its surface-area is 5.53581… This same surface in the form of a sphere’s surface captures a 1.8371173…^(1/3) volume exposing geometry’s prime volume-maximizing number to be at the heart of apportioning these two geometric qualities.
This volume-maximizing apportioning at essence is the rationing between Unity, or 1.0, and 1.8371173… and appears in the form of one power or another whenever the concept of maximizing volume is geometrically modeled. For example, a sphere with a 1.8371173… volume has a surface area of 7.253963… This same surface area, as an original circle of area giving birth to a side-area only cone of maximum volume, captures precisely 1.0 volume-unit within the confines of the surface area remaining as the cone. It was previously pointed out that the “teepee cone” apportions its original circle in a 293.93876…/360 degrees . . . and, 1.8371173…degrees times one-hundred-sixty is 293.93876… degrees.
The figure above shows this “teepee cone” version of the cones of maximum volume scaled to unity, (1.0), as the radius of its circular base. The apothem, or measure from the cone’s tip to its base edge, then becomes 1.224744…, which in a sense is disguising the deeper conceptual structuring reflecting the above volume maximizing ideal number patterning. But 1.224744… can also be expressed as 1.8371173…^(1/3) revealing this number pattern; and if we express the radius as 1.0^(1/3) then this ideal cone form emerges from the precise arrangement of two different cubes of these very specific volumes.
So entwined are these two quantities and the concept of volume maximizing that the radius length of the “teepee cone” equal to 1.8371173… in volume appears as the height of the same ideal cone form when its volume equals 1.0.
Furthermore, the volume of the cone of maximum volume in the previous diagram is .74048049… and is the number expression describing the maximum density that the closest packing of identical spheres may approach to fill any given volume of space.
This volume quantity can be equally expressed another way; as a ratio between two quantities:
and leads directly to another volume-maximizing model, The Maximum Potential Of A Sphere (see next chapter).