Inflating the Platonic Solids While Preserving Distance

Inﬂating the Platonic Solids while Preserving Distance

Seth Arnold

Senior Project

Southern Illinois University Edwardsville

May 2014

In R2, it can be shown by the isoperimetric inequality that, of all simple closed curves with length L, the circle encloses the largest area. Similarly, in R3, we find that the sphere encloses the largest volume under a specified surface area. I am interested in increasing the volumes of the Platonic solids while preserving the distance between points on the surface of the solid. The process of increasing the volume of the solid will be referred to as inflating. Imagine that you are holding a hollow cube with a mylar surface. This cube has a known volume dependent on its side length. We will now pump air into the mylar cube. We ask ourselves how will this inflated solid look, and how does the volume of the inflated solid compare to the volume of the Platonic solid. In this paper, I will outline crucial definitions and theorems, demonstrate that an upper bound exists for the volumes of the inflated solids, provide a construction proof for increasing the volume, and present my constructed solids of increased volumes isometric to its respective Platonic solid. However, before we

1 jump head ﬁrst into the mathematics, we must ﬁrst pave our path and develop some essential tools and nomenclature that will assist us later.

1 Deﬁnitions and Theorems

The goal of this project is to inﬂate the Platonic solids while preserving the distance between points on the surface. Hence, we need some way to characterize curves on the surface of the Platonic solids. This is where we will yield to geodesics.

Deﬁnition 1.1. [Gal03] A geodesic on the surface of a polyhedron P is a curve γ of locally minimal length which does not pass through the vertices of the polyhedron. It means for that any two points A, B ∈ γ lying close to each other, the length of the part [A, B] of the curve γ with the endpoints A and B is less than the length of any other curve σ on the polyhedron surface with the same endpoints A, B

Since we do not want to increase or decrease the distance between points on the surface of the solid, we want to make sure that our inﬂations do not increase or decrease the length of the geodesic segments connecting points.

Definition 1.2. [MP77] An isometry from surface M to surface N is a one-to- one, onto, differentiable function f : M → N such that for any rectifiable curve γ :[c, d] → M, the length of γ equals the length of rectifiable curve (f ◦ γ). M and N are isometric if such an isometry exists.

Later in this paper, I will construct a solid of increased volume that is isometric to a Platonic solid. This construction utilizes a hypothesis characterized by the following deﬁnition.

Deﬁnition 1.3. A set C ⊆ Rn is called convex if for all x, y ∈ C, (1 − t)x + ty ∈ C for t ∈ [0, 1].

2 We will show that the inﬂated solid is isometric to a Platonic solid, but it would also be interesting to understand what other properties does this solid maintain.

Deﬁnition 1.4. [Gal03]The defect of a vertex v on a polyhedron is

X ∆v := 2π − αi i∈v

where the {αi} are all the plane angles of the polyhedral angle with vertex v.

Definition 1.5. [Gal03] The total curvature or defect of a polyhedron P is the sum of the vertices’ defects, X ∆P = ∆v. v∈P Russian mathematician Aleksandr Danilovich Aleksandrov was able to shed some light on the properties of the inflated polyhedra with the following theorem that is used to characterize our inflation. This theorem utilizes the concept of total curvature of polyhedra.

Theorem 1.6 (Alexandrov’s Uniqueness Theorem). [DAKS06]If the total curvature

of a convex polyhedron P1 is 4π, then every isometric mapping φ of P1 onto a convex polyhedron P2 can be realized by a motion or a motion and a reﬂection.

I will now elaborate on the important theorems that motivate our construction because without them, we are not guaranteed the existence of an isometric surface with greater volume. The most important of these is Theorem 1.11 proven by Pak. Later on in this paper, I will provide a proof of Theorem 1.11 for a restricted case. How- ever, we will need to review the following deﬁnitions that are helpful in understanding Pak’s construction.

Deﬁnition 1.7. [Pak06]We say that a surface S of a polyhedron P in R3 is simplicial if all of the faces of P are triangles.

3 The concept of the distance between two points is important when discussing isometry. From this point on, || · || will denote the standard Euclidean metric, and

| · |S will denote the metric on the surface of the solid. Namely, | · |S is the inﬁmum of the length of all curves between two points on the surface.

Deﬁnition 1.8. [Mun00] Let X and Y be topological spaces; let f : X → Y be a bijection. If both the function f and the inverse function f −1 : Y → X are continuous, then f is called a homeomorphism.

Deﬁnition 1.9. [Pak06] We say that a surface S0 ⊆ Rd is submetric to S, write S0 S, if there exists a homeomorphism φ : S → S0 which does not increase the

geodesic distance: |x, y|S ≥ |φ(x), φ(y)|S0 for all x, y ∈ S.

In Pak’s construction, he yields to a submetry. This means that the distance between the points on the surface of the inﬂated solids are less than or equal to the distance between their pre-images. This submetry is later corrected to become an isometry. We are guaranteed the existence of such an isometry by the following theorem.

3 Theorem 1.10. [Pak06] Let S1 be a surface submetric to surface S in R , and let

> 0 be any given constant. Then there exist a surface S2 isometric to S, such that

S2 is in -neighborhood of S1, that is, for all x ∈ S1 there exist y ∈ S2 such that ||x − y|| < .

Theorem 1.11. [Pak06] For every convex simplicial surface S in R3, there exists a volume increasing bending of S.

The volume increasing bending utilized by Pak is what we will call an inﬂation. This process of inﬂating a solid is a function. Moreover, it is a special type of function called a homotopy.

4 Deﬁnition 1.12. [Mun00] If f and f 0 are continuous maps of the space X into the space Y , we say that f is homotopic to f 0 if there is a continuous map F : X ×I → Y such that F (x, 0) = f(x) and F (x, 1) = f 0(x)

for each x. (Here I = [0, 1].) The map F is called a homotopy between f and f 0.

Theorem 1.11 implies that there exist a homotopy that takes a Platonic solid to its inﬂation. H is a homotopy such that H : I × S2 → R3 where I is an interval [0, a] and a is dependent upon the Platonic solid with which we are working. Theorem 1.11 is an important result because it is a restriction of our overarching goal since not every Platonic solid is simplicial. In Pak’s paper, he is able to extend this to a general case for convex polyhedra.

Theorem 1.13. [Pak06]For every convex surface S in R3, there exist a submetric convex surface S0 ∼ S of greater volume: vol(S0) > vol(S).

Notice that this theorem tell us that for every Platonic solid, we have a submetric convex solid of greater volume. We are not interested in a submetric solid, so will uti- lize Theorem 1.10 that states an isometry to the Platonic solid is in an -neighborhood of the convex submetry of greater volume. Now referring back to the Alexandrov’s Uniqueness Theorem, this inflated solid is isometric to its respective Platonic solid, but we will see that our bending is not a rigid motion, reflection, or composition of the two. Thus, we can conclude that our inflated solid is non-convex. We will come to find that our construction is an isometry in each face of the polyhedron. However, we will use the following theorem to guarantee that our construction is an isometry.

3 3 Theorem 1.14. If f1: U1 → R and f2: U2 → R are isometries with f1(x) = f2(x)

5 3 for all x ∈ U1 ∩ U2, then f:U1 ∪ U2 → R deﬁned by

  f1(x): x ∈ U1 f(x) =  f2(x): x ∈ U2

is an isometry.

Proof. In order to show that f is an isometry, it is suﬃcient to show that f is bijective

and diﬀerentiable. Let x, y ∈ U1 ∩ U2. Assume that f(x) = f(y). By assumption,

f1(U1 ∩ U2) = f2(U1 ∩ U2), so f1(x) = f2(x) = f1(y) = f2(y) where f1 and f2 are isometries. The cases where x, y ∈ U1 and x, y ∈ U2 follow from assumption, so f is injective. We will now show that f is surjective by analyzing two cases.

−1 Case 1 : Let x ∈ R(f1) and u ∈ U1 such that u = f1 (x) then f(u) = f1(u) =

−1 f1f1 (x) = x

−1 Case 2 : Let x ∈ R(f2) and v ∈ U2 such that v = f2 (x) then f(v) = f2(v) =

−1 f2f2 (x) = x

We now must show that f is differentiable. We know f1 is differentiable on U1 and f2 is differentiable on U2, then f1 and f2 are both differentiable on U1 ∩ U2 so f is differentiable on U1 ∪ U2. Hence, f is an isometry since f1 and f2 preserve the lengths of curves and we showed that f is a differentiable bijection.

Lastly, we need to guarantee that our inflations do in fact have larger volume. To assist with the volume calculations, we would like to subdivide the volume of our inflated solid into solids of easily calculable volumes. Hence, we will use the following definition.

Deﬁnition 1.15. [Mun00] A subset A of Rn is said to be star shaped if for some

point a0 of A, all the line segments joining a0 to other points of A lie in A.

6 This deﬁnition will allow for us to subdivide the solid into pyramids with the

faces of the inflation as bases and a point a0 as the origin. This concludes all of the necessary definitions and theorems that will be referenced in the construction of inflations for the Platonic solids, and we are now ready to begin our construction of isometric inflations of the Platonic solids, but in order to develop the upper bounds for the inflations we must include one more theorem. This theorem is related to the isoperimetric inequality in R2. According to Osserman, the isoperimetric inequality can be extended to Rn, but since we are only concerned with the Platonic solids we will only work with its extension to R3.

n n Theorem 1.16. [Oss78] Deﬁne V (A) to be the volume of set A, Br (a) = {x ∈ R :

n n |x − a| < r} and ωn = V (B1 (0)). If D is a connected open set in R , its volume V and surface area A are related by

n n n−1 A ≥ n ωnV

n with equality if and only if D = Br (a) for some r and a.

In R2, the relationship between the length of a closed curve, L, and the area, A, that L encloses can be expressed as 4πA ≤ L2.

3 3 For Theorem 1.16, in R , ω3 = V (B1 (0)) represents volume of the unit sphere, so 4π ω3 = 3 . The isoperimetric inequality can then be expressed in terms of surface area, A, as

3 A 2 V ≤ √ . (1.1) 6 π

The isoperimetric inequality extension to R3 will serve as an upper bound to the volume of the Platonic solids after applying any bending function. We only have

7 equality in Equation 1.1 when the surface area is 4π. We can bound each unit platonic solid by calculating the surface area and using the isoperimetric inequality extension

to R3. I determined that the inflations of the unit tetrahedron, cube, octahedron, √ 2 √ √ 1 1 q 6 5 3(85+38 5) 4 dodecahedron, and icoshedron are bounded above by √ √ , , √π , √ , √ 2 4 3 π π 4 3 2 π 5 5 and √π respectively. Hence, we know that the inflation constructions have a supre- 2 4 3 mum. We will now use to following construction to find lower bounds for the volume inflations.

2 Construction of Isometric Inﬂations

Pak’s construction of inﬂations is more general then the following proof that restricts his construction to simple, convex, regular polyhedra. It should be noted that Pak’s construction requires that P only be simplicial and convex. Since the Platonic solids are regular, this proof is simpler than Pak’s because we can omit superﬂuous details that are not necessary due to their symmetry. I will remark on these details in the following subsection.

Restriction of Theorem 1.11 to the case of simple, convex, regular polyhedron. Let P

be a convex simple regular polyhedron in R3. This means each vertex v of P has de- gree 3. P consists of vertices v, faces F , and edges e. Let > 0 be a parameter. A vertex v in a face F is the intersection of two edges e and e0. For each vertex v and

0 0 edges e and e containing v, denote xv,F to be the point in the face F of e and e

0 such that xv,F is away from e and e . We know such a point exist because the edges

0 e and e intersect at v. A line segment from vertex v to xv,F should be created on the surface. This line will be used later in the construction. Let XF be the polygon consisting of the convex hull of xv,F in a face F . As of this moment in the construction, each face F consists of XF , two triangles per vertex, and one rectangle per edge

8 deﬁned by an edge and two xv,F . This subdivision of P can be seen in Figure 2.1.

Figure 2.1: Subdivision of the Surface of P [Pak06]

0 Let T be the union of two adjacent rectangles in F and F that share an edge e. Think of T as a creased rectangle. We wish to continuously deform T until it is coplanar. After deforming T, we see that the xv,F ’s are mapped to new vertices

3 since they comprise the vertices of T. In R , using the standard Euclidean metric, the distance, ||xv,F − xv,F 0 || ≤ 2. This follows from the triangle inequality. We know that there exist line segments that connect xv,F to edge e and xv,F 0 to edge e of length

. Say that these lines segments intersect e at e(t0). We will draw a line segment from xv,F to xv,F 0 . Hence we have

||xv,F − xv,F 0 || ≤ |xv,F − e(t0)|S + |xv,F 0 − e(t0)|S = 2.

Thus, we want to push XF in each face such that the images of xv,F ’s deﬁned by adjacent faces F and F 0 that share an edge e have a distance of 2.

9 Push each XF a distance δ > 0 in the direction of the normal to XF away from the center of P . The new vertices after pushing each xv,F will be denoted yv,F and similarly, we will denote the pushed polygonal faces as YF . Let Q be the convex hull

0 of {yv,F : v ⊂ F }. Q consists of YF , rectangles T, and polygonal faces determined by

0 vertices of yv,F for each F that shares a vertex v. We want the height of T to be less than or equal to the height of T. Thus we choose δ such that ||yv,F − yv,F 0 || = 2. δ is

function of . Consider two adjacent edges a and b of XF . We will consider a and b as vectors both sharing the same initial point. The lengths of a and b are a function of

epsilon since we required each xv,F be away from the two edges of F that intersect at v. The normal to F is the cross-product of vectors a and b since both a and b are elements of F . δ is scalar quantity. Hence we have δ = ||a × b|| = ||a|| · ||b||sinθ where θ is the angle between a and b. Thus, we have shown that δ is a function of . Since P is simple, we note that Q has a triangular surface deﬁned by three adjacent

yv,F which are derived from three xv,F sharing a vertex v, we will call this triangle

Uv. Consider the union of triangles around a vertex v. This is a triangular cone

shape and we want to transform this cone into a pyramid with Uv as its base. This transformation can be seen in Figure 2.2. The lines connecting v to xv,F will serve as

Figure 2.2: Transformation of the Cone [Pak06]

edges to the pyramid. Hence as we push XF , we see that triangular cone is stretched

over P . This construction provides a pyramid with Uv as its base. We have preserved

10 the distances between adjacent xv,F ’s through this transformation, but we have not

preserved the length of the line segment connecting v to the pushed xv,F . Since regular polyhedra have rotational symmetry and we are performing the same deformation in every face, we should be able to ﬁnd a vertex v0 such that the pyramid deﬁned by v0 and all of the yv,F ’s around an initial vertex v is isometric to the union of triangles around the vertex v on the surface of P . The new vertex v0 for the pyramid is located somewhere on the line connecting the initial vertex v to the center of the solid P .

Thus using length of the segments connecting v to xv,F , we are able to find a scalar that we can multiply to the vector defining v from the origin to define the new vertex v0 that characterizes and isometry. The inflated solids consist of three components, an pushed internal polygonal face, YF , from F , flattened rectangles, and pyramids. The distance between points on the pushed faces are preserved as are the distances between points on the rectangles. We have just shown that the pyramid with vertex

0 v and base Uv is isometric to the union of triangles around a vertex of P , so by Theorem 1.14, we can conclude that the constructed solid is isometric to the initial convex, simple, regular polyhedron.

2.1 Remark on Theorem 1.11

The theorem proven above is a restriction of the theorem proven by Pak to the case where P is a convex, simple, regular polyhedron. Pak’s generalization works for all convex simplicial polyhedra, but I imposed the extra condition of regularity since arguments made by Pak are superﬂuous for the inﬂations of the Platonic solids. However, I will discuss Pak’s construction starting with the development of the tri-

angular pyramid whose base is Uv. The preliminary construction of subdividing the surface and pushing the internal polygons is the same for the general case. A tri-

11 angular cone shape can be obtained from the union of triangles around a vertex v.

We wish to map this cone onto Uv. We will stretch the cone such that the vertices

of the pushed xv,F serve as vertices for a new triangular cone. From the construc-

tion, |xv,F − xv,F 0 |S = 2, however for Pak’s generalization, we will push each face of XF such that |yv,F − yv,F 0 | ≤ 2. We now will shrink each side of the cone from

|xv,F − xv,F 0 |S to |yv,F − yv,F 0 |. By shrinking the cone we obtain a triangular pyramid that can be seen in Figure 2.2. We are not guaranteed a submetry by shrinking the sides of a triangle, but we will later show that this map can be corrected to become a submetry.

If the vertex v of this pyramid projects inside of Uv, we have a submetry, however there is a possibility for v to project outside of Uv. If v does project outside of Uv, we know that at most two of the dihedral angles at the base of the pyramid are obtuse. I will discuss the case where only one dihedral angle is obtuse. Assume that the dihedral angle of edge e of Uv is obtuse. We can then ﬁnd a hyperplane He through e such that v projects onto the intersection of the hyperplane with the triangular pyramid. Project the remainder of the pyramid that is on the same side of He as v onto the intersection of He and the pyramid. This projection yields a new pyramid

0 called ∆v0 , with a vertex v that projects onto Uv. Hence, we have a submetry. I will not discuss the case when there are two obtuse dihedral angles.

Let Q be the union of all of the ∆v0 ’s with Q. We will now show that the construction can be corrected to become a submetry while increasing the volume.

0 Assume that Q is a submetry of S, the surface of P , and let P be the convex hull of all of the xv,F ’s. We then know that the volume of Q is bounded below by

0 X vol(Q) ≥ vol(P ) + δ()area(XF ) (2.1) F ⊂S

12 where δ is a function of because we can say δ ≤ c for some c. Now let us look at

XF . The length of each side of XF is a function of . Hence, we can say

area(XF ) = area(F ) − O().

We can now rewrite Equation 2.1 such that

0 vol(Q) = vol(P ) + δ()(area(S) − O()). (2.2)

We will now subdivide P into regions by using hyperplanes. These planes will cut through each Uv and each rectangle deﬁned by {xv,F , xv0,F , xv0,F 0 , xv,F 0 }. Thus we have regions of three types: the region P 0, a triangular pyramid for each vertex, and triangular prism for each edge. The length of every edge of the pyramid is a function

2 of , thus we can conclude the area of Uv is O( ). Also, the height of the pyramid is O(). Hence the volume of each pyramid is O(3). The area of the triangular base of the triangular prism is O(2) and each prism has height |e|, so the volume of the triangular prism is O(2). Thus, vol(P 0) = vol(P ) − O(2), and referring back to Equation 2.2, we conclude

2 vol(Q) = vol(P ) + δ()area(S) − O( ).

Hence, the volume is increased for small enough , and we have shown that a volume increasing submetry of a solid exist. Later in Pak’s paper, he is able to remove the required condition of being simplicial for Theorem 1.11 and reduce it to a general case for convex polyhedra. I will not discuss this generalization.

13 3 Inﬂations of the Platonic Solids

In this section, we will consider the unit Platonic solids centered at the origin. We will use the construction above to create a non-convex solid that is isometric to the Platonic solid but with increased volume. We know that the inflated solid is isometric to the initial solid by Theorem 1.14. Also, these solids are non-convex by the Alexandrov Uniqueness Theorem (Thm. 1.6) since our inflation is not a motion, reflection, or composition of the two. Now, we must develop a function of that represents the volume of the solid. This will be done by dividing the solid into pyramids whose bases are the faces of the inflated Platonic solid. The origin will serve as a vertex for every pyramid as well. Since all of the inflated solids are star shaped with the origin as the star point, we are able calculate the volume of the solid by finding the volume of the pyramids with each of face types as bases. The face types are the pushed internal polygonal faces, the rectangles, and the triangular faces the of the pyramids from the construction. Since the Platonic solids are regular, we are able to count the number of pyramids of each type, and hence we can describe the volume of the inflated solid. Now that we have a method of calculating the volume of the solid, we must determine how we are going to find the volume of the pyramids.

1 The volume of a pyramid is 3 Bh where B represents the area of the base and h is the height of the pyramid. The height h is found by ﬁnding a unit vector normal to the face and taking the dot product of this unit normal with a point in the face serving as the base for our pyramid of interest.

3.1 Cube

The unit cube has a volume of 1 cubic unit. Refer to Table 1 for inflations of the cube with different values of . The inflation of the cube with the largest volume can

14 be seen in Figure 3.1.

Table 1: Inﬂations of the Cube

= 0 = 0.1 = 0.2

= 0.3 = 0.4 = 0.5

Now we will determine the volume of the inflated cube as a function of . This will be done by utilizing the star shaped property of the inflated solid, and subdividing the solid into pyramids. We see there are three types of pyramids in the subdivision of the inflated cube. We have 6 pyramids defined by the internal square face. There are 12 pyramids defined by the rectangles of length 1 − 2 and width 2. Lastly, each vertex has three pyramids with congruent triangular bases. Hence, there are 24 pyramids with triangular bases. The area of the triangular bases can be determined using Heron’s formula.

0 The distance between the vertices of the pushed XF ’s from faces F and F sharing and edge e needs to be 2. We pushed each face of XF some distance δ and the normal

15 Figure 3.1: Optimum Inﬂation of the Cube vectors for adjacent faces on the cube are orthogonal. Hence by the Pythagorean √ theorem, (2)2 = 2(δ + )2. Thus we arrive at the conclusion that δ = ( 2 − 1). By utilizing the Mathematica code for the cube similar to the one found in the appendix for the octahedron, we conclude that the volume of the inﬂated cube as a function of is √ ! √ √ 88 2 v() = 1 + 6( 2 − 1) + 24( 2 − 1)2 + − 32 3. 3

By plotting this function with the the volume of the cube (Figure 3.2), it is apparent that an interval exists where the volume of the inflated cube is greater than the volume of the cube, and the volume of the inflated cube is maximized at 1.18205 cubic units when ≈ 0.163036. Previously, we bounded the inflation

q 6 volume above by π ≈ 1.38198 cubic units. Hence there may exist diﬀerent inﬂation constructions for the cube that yield greater volume than the volume we obtained from our construction. However, we know that the volume is bounded above from

16 Figure 3.2: Volume of Inﬂated Cube vs.

Table 2: Inﬂations of the Tetrahedron

= 0 = 0.15 = √1 2 3

the isoperimetric inequality extension.

3.2 Tetrahedron

√ 3 The area of one face of a tetrahedron with unit length is 4 . Also the height of the q 2 tetrahedron is 3 . Hence we conclude the volume of the unit side length tetrahedron is √1 since the volume of a solid is 1 A h where A is the area of the base, and h is 6 2 3 B B the height. The optimum inflation of the tetrahedron can be seen in Figure 3.3, and the inflation of the tetrahedron for different values of can be seen in Table 2.

17 There are 4 pyramids with regular triangular bases, 6 pyramids with rectangular bases, and 12 with triangular bases. The volume function of the inflated tetrahedron can be seen in Figure 3.4. Using a numerical approximation, we find that the maximum volume for the inflated tetrahedron is around 0.156032 cubic units. This volume occurs when is approximately 0.0954153. We have bounded the volumes of

1 the inﬂated tetrahedrons above by by √ √ ≈ 0.214346, and so we conclude that 2 4 3 π other inﬂation constructions may exist that yield volumes larger than ours that is still bounded above by the isoperimetric inequality extension.

Figure 3.3: Inﬂated Tetrahedron

3.3 Octahedron

√ 2 The volume of the unit octahedron is 3 . If we cut the octahedron in half along the edges of unit length that define a square, we obtain two congruent square pyramids. √ 2 For each of these two pyramids, the area of the base is 1, and the height is 2 . Hence, √ 2 we conclude that the volume of the unit length octahedron is 3 . The inflations of the octahedron for different values of can be seen in Table 3. After optimally inflating

18 Figure 3.4: Volume of the Tetrahedron vs.

Table 3: Inﬂations of the Octahedron

= 0 = 0.15 = √1 2 3

the octahedron, we obtain the solid in Figure 3.5. Using a numerical approach, the volume as a function of is expressed in Figure 3.6. Clearly, there is an interval in which the volume of the inflated octahedron is greater than that of the octahedron. The maximum inflation occurs at 0.0952238. The volume at this point is 0.518244 cubic units. From the upper bound on the volume of the octahedron we see that the optimum inflation of the octahedron yields √ 2 a maximum volume between 0.518244 cubic units and √π ≈ 0.606261, yet again there 4 3 is still the possibility for inflation constructions yielding larger volumes bounded above

19 √ 2 by √π . 4 3

Figure 3.5: Inﬂated Octahedron

Figure 3.6: Volume of Inﬂated Octahedron vs.

3.4 Dodecahedron

Inﬂations of the dodecahedron can be found in Table 4. The volume as a function of can be seen graphically in Figure 3.7. It is also apparent that there is a local √ 15+7 5 extremum. The volume of the dodecahedron is 4 ≈ 7.66312 cubic units. This

20 Table 4: Inﬂations of the Dodecahedron

= 0 = 0.1 = 0.2

= 0.3 = 0.5 = 0.69

21 can be determined using the star shaped property of the dodecahedron. We have 12 √ √ 25+10 5 regular pentagonal faces. The area of each face is 4 , and we can determine the height by finding a unit normal on the face and taking the dot product of some point in the face with the unit normal. From the graph in Figure 3.7, we see that there does exist an such that the volume of the constructed inflation is greater than the volume of the dodecahedron. We know that the construction is isometric to the octahedron. Hence we know that we have achieved our result of inflating the octahedron while preserving distance.

Figure 3.7: Volume of Inﬂated Dodecahedron vs.

3.5 Icosahedron

The inflated icosahedrons can be seen in Table 5. By using the star shaped properties √ 3 of the icosahedron, we see that we have 20 equilateral triangular faces of area 4 and √ the height of the pyramids subdividing the volume of the icosahedron is 3+√ 5 . The 4 3 volume of the inflated icosahedron is expressed graphically in Figure 3.8. Similarly to the dodecahedron, we see that a local extremum exist for some . The volume of the √ 5(3+ 5) inflated icosahedron is greater than the volume of the icosahedron which is 12 for

22 Table 5: Inﬂations of the Icosahedron

= 0 = 0.15 = √1 2 3

some , and we know that our construction is an isometry. Hence, we have inflated the icosahedron while preserving distance. We also know that an upper bound on the volume of the inflations exist, and we conclude that an inflation exist that yields maximal volume which may or may not be different from our own construction.

Figure 3.8: Volume of Inﬂated Icosahedron vs.

23 Table 6: Inﬂation Data

Solid Volume Max Inﬂation % Inc. Upper bound % Inc. Needed 1 1 Tetrahedron √ 0.156032 32% √ √ 82% 6 2 2 4 3 π q Cube 1 1.18205 18% 6 38% √ π √ 2 Octahedron 2 0.518244 10% √π 29% 3 4 3 √ √ √ 1 5 3(85+38 5) 4 Dodecahedron 15+7 5 ≈ 8.00 ≈ 4% √ 15% 4 2√π √ 5 5(3+ 5) 5 Icosahedron ≈ 2.235 ≈ 2% √π 9% 12 2 4 3

4 Conclusion

In this paper, we have shown that non-convex polyhedra isometric to the Platonic solids exist. More importantly, we have shown that these polyhedra have volumes larger than the Platonic solids for some parameter of the inflation. Also, by Os- serman’s Isoperimetric Inequality extension, we are able to bound the volumes of isometric polyhedra conceived from different inflation constructions. A summary of important data obtained from our constructions can be seen in Tabel 6. I have in- cluded percentages by which our inflations have increased the volume of the solids, and the percent increases necessary to meet the volume of the upper bounds. The construction shown within this paper is but one of many other potential inflation constructions. This leaves opportunities for others to research different inflation constructions that may be better or worse than our construction. Also, a more challenging question would be to find the best inflation construction because we have guaranteed that one exist. Lastly, there is also potential research in determining if our outlined construction applies to non-convex solids provided self-intersection does not occur.

24 5 Appendix: Octahedron Mathematica Construc-

tion

I have developed Mathematica code that inflates each one of the Platonic solids by our construction. I will provide the code for the octahedron inflation and the dodecahedron inflation, but for the sake of brevity, I will not include the Mathematica code for all five Platonic solids.

5.1 Octahedron Inﬂation

25 In[21]:= octverts = PolyhedronData@"Octahedron", "VertexCoordinates"D; H*Utilizes built- in data for the vertices of the unit octahedron centered at the origin*L

midpoints = 8octverts@@1DD + H1 2L Hoctverts@@2DD - octverts@@1DDL, octverts@@1DD + H1 2L Hoctverts@@3DD - octverts@@1DDL, octverts@@1DD + H1 2L Hoctverts@@4DD - octverts@@1DDL, octverts@@1DD + H1 2L Hoctverts@@5DD - octverts@@1DDL, octverts@@6DD + H1 2L Hoctverts@@2DD - octverts@@6DDL, octverts@@6DD + H1 2L Hoctverts@@3DD - octverts@@6DDL, octverts@@6DD + H1 2L Hoctverts@@4DD - octverts@@6DDL, octverts@@6DD + H1 2L Hoctverts@@5DD - octverts@@6DDL, octverts@@3DD + H1 2L Hoctverts@@2DD - octverts@@3DDL, octverts@@4DD + H1 2L Hoctverts@@2DD - octverts@@4DDL, octverts@@3DD + H1 2L Hoctverts@@5DD - octverts@@3DDL, octverts@@4DD + H1 2L Hoctverts@@5DD - octverts@@4DDL<; verts = Join@octverts, midpointsD; H*Calculates the midpoints of each edge of the octahedron. These midpoints are joined with the vertex coordinates in "verts"*L

In[24]:= Clear@ΕD; Ε; everts = 8verts@@1DD + 2 Ε Hverts@@15DD - verts@@1DDL Norm@verts@@15DD - verts@@1DDD, verts@@2DD + 2 Ε Hverts@@8DD - verts@@2DDL Norm@verts@@8DD - verts@@2DDD, verts@@3DD + 2 Ε Hverts@@7DD - verts@@3DDL Norm@verts@@7DD - verts@@3DDD, verts@@1DD + 2 Ε Hverts@@16DD - verts@@1DDL Norm@verts@@16DD - verts@@1DDD, verts@@4DD + 2 Ε Hverts@@7DD - verts@@4DDL Norm@verts@@7DD - verts@@4DDD, verts@@2DD + 2 Ε Hverts@@9DD - verts@@2DDL Norm@verts@@9DD - verts@@2DDD, verts@@1DD + 2 Ε Hverts@@18DD - verts@@1DDL Norm@verts@@18DD - verts@@1DDD, verts@@5DD + 2 Ε Hverts@@9DD - verts@@5DDL Norm@verts@@9DD - verts@@5DDD, verts@@4DD + 2 Ε Hverts@@10DD - verts@@4DDL Norm@verts@@10DD - verts@@4DDD, verts@@1DD + 2 Ε Hverts@@17DD - verts@@1DDL Norm@verts@@17DD - verts@@1DDD, verts@@3DD + 2 Ε Hverts@@10DD - verts@@3DDL Norm@verts@@10DD - verts@@3DDD, verts@@5DD + 2 Ε Hverts@@8DD - verts@@5DDL Norm@verts@@8DD - verts@@5DDD, verts@@6DD + 2 Ε Hverts@@17DD - verts@@6DDL Norm@verts@@17DD - verts@@6DDD, verts@@5DD + 2 Ε Hverts@@12DD - verts@@5DDL Norm@verts@@12DD - verts@@5DDD, verts@@3DD + 2 Ε Hverts@@14DD - verts@@3DDL Norm@verts@@14DD - verts@@3DDD, verts@@6DD + 2 Ε Hverts@@18DD - verts@@6DDL Norm@verts@@18DD - verts@@6DDD, verts@@4DD + 2 Ε Hverts@@14DD - verts@@4DDL Norm@verts@@14DD - verts@@4DDD, verts@@5DD + 2 Ε Hverts@@13DD - verts@@5DDL Norm@verts@@13DD - verts@@5DDD, verts@@6DD + 2 Ε Hverts@@16DD - verts@@6DDL Norm@verts@@16DD - verts@@6DDD, verts@@2DD + 2 Ε Hverts@@13DD - verts@@2DDL Norm@verts@@13DD - verts@@2DDD, verts@@4DD + 2 Ε Hverts@@11DD - verts@@4DDL Norm@verts@@11DD - verts@@4DDD, verts@@6DD + 2 Ε Hverts@@15DD - verts@@6DDL Norm@verts@@15DD - verts@@6DDD, verts@@3DD + 2 Ε Hverts@@11DD - verts@@3DDL Norm@verts@@11DD - verts@@3DDD, verts@@2DD + 2 Ε Hverts@@12DD - verts@@2DDL Norm@verts@@12DD - verts@@2DDD<; H*Calculates the internal polygonal vertices x_8v,F< as a function of Ε*L

Printed by Wolfram Mathematica Student Edition 2 octahedron.inflation.nb

In[27]:= Clear@∆D; ∆; push@list_D := Module@8pushvec = 80, 0, 0<, newlist = 8<<, For@i = 1, i <= Length@listD, i = i + 3, pushvec = ∆ HCross@list@@i + 1DD - list@@iDD, list@@i + 2DD - list@@iDDDL Simplify@ Norm@Cross@list@@i + 1DD - list@@iDD, list@@i + 2DD - list@@iDDDD, Ε Î RealsD; AppendTo@newlist, list@@iDD + pushvecD; AppendTo@newlist, list@@i + 1DD + pushvecD; AppendTo@newlist, list@@i + 2DD + pushvecD; D; newlist D H*This is the pushing parameter ∆ determined by the cross product to two edge vectors of X_F*L

intpoly = Join@octverts, push@evertsDD; H*This is the compilation of the pushed faces of X_F and the vertices of the octahedron*L

flapverts = 8intpoly@@7DD + .5 Hintpoly@@16DD - intpoly@@7DDL, intpoly@@9DD + .5 Hintpoly@@17DD - intpoly@@9DDL, intpoly@@16DD + .5 Hintpoly@@13DD - intpoly@@16DDL, intpoly@@18DD + .5 Hintpoly@@14DD - intpoly@@18DDL, intpoly@@13DD + .5 Hintpoly@@10DD - intpoly@@13DDL, intpoly@@15DD + .5 Hintpoly@@11DD - intpoly@@15DDL, intpoly@@7DD + .5 Hintpoly@@10DD - intpoly@@7DDL, intpoly@@8DD + .5 Hintpoly@@12DD - intpoly@@8DDL, intpoly@@8DD + .5 Hintpoly@@30DD - intpoly@@8DDL, intpoly@@9DD + .5 Hintpoly@@29DD - intpoly@@9DDL, intpoly@@17DD + .5 Hintpoly@@21DD - intpoly@@17DDL, intpoly@@18DD + .5 Hintpoly@@20DD - intpoly@@18DDL, intpoly@@14DD + .5 Hintpoly@@24DD - intpoly@@14DDL, intpoly@@15DD + .5 Hintpoly@@23DD - intpoly@@15DDL, intpoly@@11DD + .5 Hintpoly@@27DD - intpoly@@11DDL, intpoly@@12DD + .5 Hintpoly@@26DD - intpoly@@12DDL, intpoly@@26DD + .5 Hintpoly@@30DD - intpoly@@26DDL, intpoly@@25DD + .5 Hintpoly@@28DD - intpoly@@25DDL, intpoly@@29DD + .5 Hintpoly@@21DD - intpoly@@29DDL, intpoly@@28DD + .5 Hintpoly@@19DD - intpoly@@28DDL, intpoly@@20DD + .5 Hintpoly@@24DD - intpoly@@20DDL, intpoly@@19DD + .5 Hintpoly@@22DD - intpoly@@19DDL, intpoly@@23DD + .5 Hintpoly@@27DD - intpoly@@23DDL, intpoly@@22DD + .5 Hintpoly@@25DD - intpoly@@22DDL<; H*These points serve only aestheic purposes. They are the midpoints of line segment connecting adjacent pushed x_8v,F<'s *L

Printed by Wolfram Mathematica Student Edition octahedron.inflation.nb 3

finverts = Join@intpoly, flapvertsD; H*Compiles all important vertices into one variable*L ratio = HNorm@Hfinverts@@22DD + finverts@@25DD + finverts@@28DD + finverts@@19DDL 4D + Sqrt@2D ΕL Norm@finverts@@6DDD; H*This scalar quantity will adjust the initial vertices to maintain an isometry with the initial solid*L gfinverts = Join@Join@ratio * octverts, push@evertsDD, flapvertsD; H*Defines the vertex set used for the remainder of the notebook*L ivfin = 887, 8, 9<, 810, 11, 12<, 813, 14, 15<, 816, 17, 18<, 819, 20, 21<, 822, 23, 24<, 825, 26, 27<, 828, 29, 30<, 87, 31, 32, 9<, 89, 40, 39, 8<, 88, 38, 37, 7<, 837, 7, 1<, 87, 31, 1<, 89, 32, 3<, 89, 40, 3<, 88, 39, 2<, 88, 38, 2<, 810, 37, 38, 12<, 812, 46, 45, 11<, 811, 36, 35, 10<, 810, 35, 1<, 810, 1, 37<, 812, 38, 2<, 812, 46, 2<, 811, 36, 4<, 811, 45, 4<, 813, 35, 36, 15<, 815, 44, 43, 14<, 814, 34, 33, 13<, 813, 35, 1<, 813, 33, 1<, 815, 4, 36<, 815, 4, 44<, 814, 5, 34<, 814, 5, 43<, 818, 42, 41, 17<, 816, 33, 34, 18<, 817, 32, 31, 16<, 816, 31, 1<, 816, 1, 33<, 818, 34, 5<, 818, 5, 42<, 817, 3, 32<, 817, 3, 41<, 828, 48, 47, 30<, 830, 39, 40, 29<, 829, 49, 50, 28<, 828, 50, 6<, 828, 6, 48<, 830, 47, 2<, 830, 39, 2<, 829, 49, 3<, 829, 40, 3<, 819, 50, 49, 21<, 821, 41, 42, 20<, 820, 51, 52, 19<, 819, 52, 6<, 819, 6, 50<, 821, 49, 3<, 821, 41, 3<, 820, 42, 5<, 820, 51, 5<, 822, 52, 51, 24<, 824, 43, 44, 23<, 823, 53, 54, 22<, 822, 54, 6<, 822, 52, 6<, 824, 51, 5<, 824, 5, 43<, 823, 44, 4<, 823, 4, 53<, 825, 54, 53, 27<, 827, 45, 46, 26<, 826, 47, 48, 25<, 825, 48, 6<, 825, 6, 54<, 827, 53, 4<, 827, 45, 4<, 826, 46, 2<, 826, 47, 2<<; H*Describes the relationships between vertices to develop the faces of the inflated solid*L

Show@Graphics3D@8Opacity@1D, Yellow, GraphicsComplex@gfinverts, Polygon@ivfinDD

Printed by Wolfram Mathematica Student Edition 4 octahedron.inflation.nb

In[43]:= v@e_D := ModuleB8d<, d = ∆ . HSolve@Norm@gfinverts@@13DD - gfinverts@@10DDD 2 Ε . Ε ® e, ∆D@@2DDL; 1 IfBe >= - .001, .47, H8 * v1 + 12 * v2 + 24 * v3L . 8Ε ® e, ∆ ® d< F 2 3 F H*Numerical definition of the volume function*L

1 PlotB8v@eD, Sqrt@2D 3<, :e, 0.0001, - .01>, AxesLabel ® 8Ε, Volume

Maximize@v@eD, e D; H*Numerically determines the maximum of the Volume Function*L

Printed by Wolfram Mathematica Student Edition 5.2 Dodecahedron Inﬂation

30 2 dodecV = ::- 1 + , 0, RootA1 - 20 ð12 + 80 ð14 &, 3E>, 5

2 : 1 + , 0, RootA1 - 20 ð12 + 80 ð14 &, 2E>, 5 1 :RootA1 - 20 ð12 + 80 ð14 &, 1E, J-3 - 5 N, RootA1 - 20 ð12 + 80 ð14 &, 3E>, 4 1 :RootA1 - 20 ð12 + 80 ð14 &, 1E, J3 + 5 N, RootA1 - 20 ð12 + 80 ð14 &, 3E>, 4

5 11 1 : + , J-1 - 5 N, RootA1 - 20 ð12 + 80 ð14 &, 3E>, 8 8 5 4

5 11 1 : + , J1 + 5 N, RootA1 - 20 ð12 + 80 ð14 &, 3E>, 8 8 5 4

1 5 11 :RootA1 - 20 ð12 + 80 ð14 &, 2E, J-1 - 5 N, + >, 4 8 8 5

1 5 11 :RootA1 - 20 ð12 + 80 ð14 &, 2E, J1 + 5 N, + >, 4 8 8 5

1 2 1 :- 1 + , - , RootA1 - 100 ð12 + 80 ð14 &, 1E>, 2 5 2

1 2 1 1 1 1 5 11 :- 1 + , , RootA1 - 100 ð12 + 80 ð14 &, 1E>, : + , - , + >, 2 5 2 4 2 5 2 8 8 5

1 1 1 5 11 1 : + , , + >, : J5 + 5 N , 0, RootA1 - 100 ð12 + 80 ð14 &, 1E>, 4 2 5 2 8 8 5 10 1 :RootA1 - 100 ð12 + 80 ð14 &, 1E, J-1 - 5 N, RootA1 - 20 ð12 + 80 ð14 &, 2E>, 4 1 :RootA1 - 100 ð12 + 80 ð14 &, 1E, J1 + 5 N, RootA1 - 20 ð12 + 80 ð14 &, 2E>, 4

5 11 :RootA1 - 5 ð12 + 5 ð14 &, 1E, 0, + >, 8 8 5 1 :RootA1 - 20 ð12 + 80 ð14 &, 3E, J-1 - 5 N, RootA1 - 100 ð12 + 80 ð14 &, 1E>, 4 1 :RootA1 - 20 ð12 + 80 ð14 &, 3E, J1 + 5 N, RootA1 - 100 ð12 + 80 ð14 &, 1E>, 4 31 , Printed by Wolfram Mathematica Student Edition 2 dodecahedron.inflation.nb

1 1 1 : + , J-3 - 5 N, RootA1 - 20 ð12 + 80 ð14 &, 2E>, 8 8 5 4

1 1 1 : + , J3 + 5 N, RootA1 - 20 ð12 + 80 ð14 &, 2E>>; 8 8 5 4 H*Utilizes built in Mathematica Data for the vertices of the Dodecahedron of unit length centered at the origin*L

H*Determines the midpoints of each edge used to find vertices of internal polygon*L midpoint = 8dodecV@@11DD + H1 2L * HdodecV@@5DD - dodecV@@11DDL Norm@dodecV@@5DD - dodecV@@11DDD, dodecV@@2DD + H1 2L * HdodecV@@5DD - dodecV@@2DDL Norm@dodecV@@5DD - dodecV@@2DDD, dodecV@@6DD + H1 2L * HdodecV@@2DD - dodecV@@6DDL Norm@dodecV@@2DD - dodecV@@6DDD, dodecV@@12DD + H1 2L * HdodecV@@6DD - dodecV@@12DDL Norm@dodecV@@6DD - dodecV@@12DDD, dodecV@@11DD + H1 2L * HdodecV@@12DD - dodecV@@11DDL Norm@dodecV@@12DD - dodecV@@11DDD, dodecV@@13DD + H1 2L * HdodecV@@2DD - dodecV@@13DDL Norm@dodecV@@2DD - dodecV@@13DDD, dodecV@@18DD + H1 2L * HdodecV@@13DD - dodecV@@18DDL Norm@dodecV@@13DD - dodecV@@18DDD, dodecV@@20DD + H1 2L * HdodecV@@18DD - dodecV@@20DDL Norm@dodecV@@18DD - dodecV@@20DDD, dodecV@@6DD + H1 2L * HdodecV@@20DD - dodecV@@6DDL Norm@dodecV@@20DD - dodecV@@6DDD, dodecV@@4DD + H1 2L * HdodecV@@20DD - dodecV@@4DDL Norm@dodecV@@20DD - dodecV@@4DDD, dodecV@@4DD + H1 2L * HdodecV@@8DD - dodecV@@4DDL Norm@dodecV@@8DD - dodecV@@4DDD, dodecV@@12DD + H1 2L * HdodecV@@8DD - dodecV@@12DDL Norm@dodecV@@8DD - dodecV@@12DDD, dodecV@@18DD + H1 2L * HdodecV@@10DD - dodecV@@18DDL Norm@dodecV@@10DD - dodecV@@18DDD, dodecV@@10DD + H1 2L * HdodecV@@15DD - dodecV@@10DDL Norm@dodecV@@15DD - dodecV@@10DDD, dodecV@@15DD + H1 2L * HdodecV@@4DD - dodecV@@15DDL Norm@dodecV@@4DD - dodecV@@15DDD, dodecV@@13DD + H1 2L * HdodecV@@17DD - dodecV@@13DDL Norm@dodecV@@17DD - dodecV@@13DDD, dodecV@@9DD + H1 2L * HdodecV@@17DD - dodecV@@9DDL Norm@dodecV@@17DD - dodecV@@9DDD, dodecV@@9DD + H1 2L * HdodecV@@10DD - dodecV@@9DDL Norm@dodecV@@10DD - dodecV@@9DDD, dodecV@@9DD + H1 2L * HdodecV@@14DD - dodecV@@9DDL Norm@dodecV@@14DD - dodecV@@9DDD, dodecV@@1DD + H1 2L * HdodecV@@14DD - dodecV@@1DDL Norm@dodecV@@14DD - dodecV@@1DDD, dodecV@@1DD + H1 2L * HdodecV@@15DD - dodecV@@1DDL Norm@dodecV@@15DD - dodecV@@1DDD, dodecV@@19DD + H1 2L * HdodecV@@17DD - dodecV@@19DDL Norm@dodecV@@17DD - dodecV@@19DDD, dodecV@@19DD + H1 2L * HdodecV@@3DD - dodecV@@19DDL Norm@dodecV@@3DD - dodecV@@19DDD, dodecV@@3DD + H1 2L * HdodecV@@14DD - dodecV@@3DDL Norm@dodecV@@14DD - dodecV@@3DDD, dodecV@@5DD + H1 2L * HdodecV@@19DD - dodecV@@5DDL Norm@dodecV@@19DD - dodecV@@5DDD, dodecV@@3DD + H1 2L * HdodecV@@7DD - dodecV@@3DDL Norm@dodecV@@7DD - dodecV@@3DDD, dodecV@@7DD + H1 2L * HdodecV@@11DD - dodecV@@7DDL Norm@dodecV@@11DD - dodecV@@7DDD, dodecV@@7DD + H1 2L * HdodecV@@16DD - dodecV@@7DDL Norm@dodecV@@16DD - dodecV@@7DDD, dodecV@@8DD + H1 2L * HdodecV@@16DD - dodecV@@8DDL , 32

Printed by Wolfram Mathematica Student Edition dodecahedron.inflation.nb 3

Norm@dodecV@@16DD - dodecV@@8DDD, dodecV@@1DD + H1 2L * HdodecV@@16DD - dodecV@@1DDL Norm@dodecV@@16DD - dodecV@@1DDD<; H*Finds the midpoints of each edge of the dodecahedron*L extvert = Join@dodecV, midpointD; H*Joins the midpoints and vertices into a single variable*L Show@ Graphics3D@8Opacity@1D, Yellow, GraphicsComplex@extvert, Polygon@ivbDD

1 inverts = :extvert@@1DD + Ε J1 + 5 N * 4 Hextvert@@38DD - extvert@@1DDL Norm@Hextvert@@38DD - extvert@@1DDLD, 1 extvert@@14DD + Ε J1 + 5 N * Hextvert@@34DD - extvert@@14DDL 4 1 Norm@Hextvert@@34DD - extvert@@14DDLD, extvert@@9DD + Ε J1 + 5 N * 4 Hextvert@@41DD - extvert@@9DDL Norm@Hextvert@@41DD - extvert@@9DDLD, 1 extvert@@10DD + Ε J1 + 5 N * Hextvert@@40DD - extvert@@10DDL 4 1 Norm@Hextvert@@40DD - extvert@@10DDLD, extvert@@15DD + Ε J1 + 5 N * 4 Hextvert@@39DD - extvert@@15DDL Norm@Hextvert@@39DD - extvert@@15DDLD, 1 extvert@@15DD + Ε J1 + 5 N * Hextvert@@49DD - extvert@@15DDL 4 1 Norm@Hextvert@@49DD - extvert@@15DDLD, extvert@@4DD + Ε J1 + 5 N * 4 Hextvert@@50DD - extvert@@4DDL Norm@Hextvert@@50DD - extvert@@4DDLD, 1 extvert@@8DD + Ε J1 + 5 N * Hextvert@@41DD - extvert@@8DDL 4 1 Norm@Hextvert@@41DD - extvert@@8DDLD, extvert@@16DD + Ε J1 + 5 N * 4 Hextvert@@35DD - extvert@@16DDL Norm@Hextvert@@35DD - extvert@@16DDLD, 1 extvert@@1DD + Ε J1 + 5 N * Hextvert@@31DD - extvert@@1DDL 4 1 Norm@Hextvert@@31DD - extvert@@1DDLD, extvert@@1DD + Ε J1 + 5 N * 4 Hextvert@@46DD - extvert@@1DDL Norm@Hextvert@@46DD - extvert@@1DDLD, 1 extvert@@16DD + Ε J1 + 5 N * Hextvert@@44DD - extvert@@16DDL 4 1 Norm@Hextvert@@44DD - extvert@@16DDLD, extvert@@7DD + Ε J1 + 5 N * 4 Hextvert@@40DD - extvert@@7DDL Norm@Hextvert@@40DD - extvert@@7DDLD, 1 extvert@@3DD + Ε J1 + 5 N * Hextvert@@50DD - extvert@@3DDL 4 33 ,

Printed by Wolfram Mathematica Student Edition 4 dodecahedron.inflation.nb

1 Norm@Hextvert@@50DD - extvert@@3DDLD, extvert@@14DD + Ε J1 + 5 N * 4 Hextvert@@48DD - extvert@@14DDL Norm@Hextvert@@48DD - extvert@@14DDLD, 1 extvert@@14DD + Ε J1 + 5 N * Hextvert@@42DD - extvert@@14DDL 4 1 Norm@Hextvert@@42DD - extvert@@14DDLD, extvert@@3DD + Ε J1 + 5 N * 4 Hextvert@@37DD - extvert@@3DDL Norm@Hextvert@@37DD - extvert@@3DDLD, 1 extvert@@19DD + Ε J1 + 5 N * Hextvert@@39DD - extvert@@19DDL 4 1 Norm@Hextvert@@39DD - extvert@@19DDLD, extvert@@17DD + Ε J1 + 5 N * 4 Hextvert@@44DD - extvert@@17DDL Norm@Hextvert@@44DD - extvert@@17DDLD, 1 extvert@@9DD + Ε J1 + 5 N * Hextvert@@43DD - extvert@@9DDL 4 1 Norm@Hextvert@@43DD - extvert@@9DDLD, extvert@@16DD + Ε J1 + 5 N * 4 Hextvert@@25DD - extvert@@16DDL Norm@Hextvert@@25DD - extvert@@16DDLD, 1 extvert@@8DD + Ε J1 + 5 N * Hextvert@@47DD - extvert@@8DDL 4 1 Norm@Hextvert@@47DD - extvert@@8DDLD, extvert@@12DD + Ε J1 + 5 N * 4 Hextvert@@48DD - extvert@@12DDL Norm@Hextvert@@48DD - extvert@@12DDLD, 1 extvert@@11DD + Ε J1 + 5 N * Hextvert@@49DD - extvert@@11DDL 4 1 Norm@Hextvert@@49DD - extvert@@11DDLD, extvert@@7DD + Ε J1 + 5 N * 4 Hextvert@@32DD - extvert@@7DDL Norm@Hextvert@@32DD - extvert@@7DDLD, 1 extvert@@5DD + Ε J1 + 5 N * Hextvert@@46DD - extvert@@5DDL 4 1 Norm@Hextvert@@46DD - extvert@@5DDLD, extvert@@19DD + Ε J1 + 5 N * 4 Hextvert@@47DD - extvert@@19DDL Norm@Hextvert@@47DD - extvert@@19DDLD, 1 extvert@@3DD + Ε J1 + 5 N * Hextvert@@21DD - extvert@@3DDL 4 1 Norm@Hextvert@@21DD - extvert@@3DDLD, extvert@@7DD + Ε J1 + 5 N * 4 Hextvert@@45DD - extvert@@7DDL Norm@Hextvert@@45DD - extvert@@7DDLD, 1 extvert@@11DD + Ε J1 + 5 N * Hextvert@@43DD - extvert@@11DDL 4 1 Norm@Hextvert@@43DD - extvert@@11DDLD, extvert@@11DD + Ε J1 + 5 N * 4 Hextvert@@23DD - extvert@@11DDL Norm@Hextvert@@23DD - extvert@@11DDLD, 1 extvert@@12DD + Ε J1 + 5 N * Hextvert@@22DD - extvert@@12DDL 4 1 Norm@Hextvert@@22DD - extvert@@12DDLD, extvert@@6DD + Ε J1 + 5 N * 4 34 ,

Printed by Wolfram Mathematica Student Edition dodecahedron.inflation.nb 5

Hextvert@@21DD - extvert@@6DDL Norm@Hextvert@@21DD - extvert@@6DDLD, 1 extvert@@2DD + Ε J1 + 5 N * Hextvert@@25DD - extvert@@2DDL 4 1 Norm@Hextvert@@25DD - extvert@@2DDLD, extvert@@5DD + Ε J1 + 5 N * 4 Hextvert@@24DD - extvert@@5DDL Norm@Hextvert@@24DD - extvert@@5DDLD, 1 extvert@@5DD + Ε J1 + 5 N * Hextvert@@36DD - extvert@@5DDL 4 1 Norm@Hextvert@@36DD - extvert@@5DDLD, extvert@@2DD + Ε J1 + 5 N * 4 Hextvert@@42DD - extvert@@2DDL Norm@Hextvert@@42DD - extvert@@2DDLD, 1 extvert@@13DD + Ε J1 + 5 N * Hextvert@@45DD - extvert@@13DDL 4 1 Norm@Hextvert@@45DD - extvert@@13DDLD, extvert@@17DD + Ε J1 + 5 N * 4 Hextvert@@22DD - extvert@@17DDL Norm@Hextvert@@22DD - extvert@@17DDLD, 1 extvert@@19DD + Ε J1 + 5 N * Hextvert@@26DD - extvert@@19DDL 4 1 Norm@Hextvert@@26DD - extvert@@19DDLD, extvert@@2DD + Ε J1 + 5 N * 4 Hextvert@@28DD - extvert@@2DDL Norm@Hextvert@@28DD - extvert@@2DDLD, 1 extvert@@6DD + Ε J1 + 5 N * Hextvert@@27DD - extvert@@6DDL 4 1 Norm@Hextvert@@27DD - extvert@@6DDLD, extvert@@20DD + Ε J1 + 5 N * 4 Hextvert@@26DD - extvert@@20DDL Norm@Hextvert@@26DD - extvert@@20DDLD, 1 extvert@@18DD + Ε J1 + 5 N * Hextvert@@23DD - extvert@@18DDL 4 1 Norm@Hextvert@@23DD - extvert@@18DDLD, extvert@@13DD + Ε J1 + 5 N * 4 Hextvert@@29DD - extvert@@13DDL Norm@Hextvert@@29DD - extvert@@13DDLD, 1 extvert@@10DD + Ε J1 + 5 N * Hextvert@@36DD - extvert@@10DDL 4 1 Norm@Hextvert@@36DD - extvert@@10DDLD, extvert@@9DD + Ε J1 + 5 N * 4 Hextvert@@27DD - extvert@@9DDL Norm@Hextvert@@27DD - extvert@@9DDLD, 1 extvert@@17DD + Ε J1 + 5 N * Hextvert@@33DD - extvert@@17DDL 4 1 Norm@Hextvert@@33DD - extvert@@17DDLD, extvert@@13DD + Ε J1 + 5 N * 4 Hextvert@@38DD - extvert@@13DDL Norm@Hextvert@@38DD - extvert@@13DDLD, 1 extvert@@18DD + Ε J1 + 5 N * Hextvert@@37DD - extvert@@18DDL 4 1 Norm@Hextvert@@37DD - extvert@@18DDLD, extvert@@4DD + Ε J1 + 5 N * 4 Hextvert@@33DD - extvert@@4DDL Norm@Hextvert@@33DD - extvert@@4DDLD,

Printed by Wolfram Mathematica Student Edition 6 dodecahedron.inflation.nb

1 extvert@@15DD + Ε J1 + 5 N * Hextvert@@28DD - extvert@@15DDL 4 1 Norm@Hextvert@@28DD - extvert@@15DDLD, extvert@@10DD + Ε J1 + 5 N * 4 Hextvert@@30DD - extvert@@10DDL Norm@Hextvert@@30DD - extvert@@10DDLD, 1 extvert@@18DD + Ε J1 + 5 N * Hextvert@@35DD - extvert@@18DDL 4 1 Norm@Hextvert@@35DD - extvert@@18DDLD, extvert@@20DD + Ε J1 + 5 N * 4 Hextvert@@34DD - extvert@@20DDL Norm@Hextvert@@34DD - extvert@@20DDLD, 1 extvert@@20DD + Ε J1 + 5 N * Hextvert@@32DD - extvert@@20DDL 4 1 Norm@Hextvert@@32DD - extvert@@20DDLD, extvert@@6DD + Ε J1 + 5 N * 4 Hextvert@@31DD - extvert@@6DDL Norm@Hextvert@@31DD - extvert@@6DDLD, 1 extvert@@12DD + Ε J1 + 5 N * Hextvert@@30DD - extvert@@12DDL 4 1 Norm@Hextvert@@30DD - extvert@@12DDLD, extvert@@8DD + Ε J1 + 5 N * 4 Hextvert@@29DD - extvert@@8DDL Norm@Hextvert@@29DD - extvert@@8DDLD, 1 extvert@@4DD + Ε J1 + 5 N * Hextvert@@24DD - extvert@@4DDL 4 Norm@Hextvert@@24DD - extvert@@4DDLD>;H*Finds the vertices of X_F*L

Clear@∆D; ∆; push@list_D := Module@8pushvec = 80, 0, 0<, newlist = 8<<, For@i = 1, i <= Length@listD, i = i + 5, pushvec = -∆ HCross@list@@i + 1DD - list@@iDD, list@@i + 2DD - list@@iDDDL Norm@Cross@list@@i + 1DD - list@@iDD, list@@i + 2DD - list@@iDDDD; AppendTo@newlist, list@@iDD + pushvecD; AppendTo@newlist, list@@i + 1DD + pushvecD; AppendTo@newlist, list@@i + 2DD + pushvecD; AppendTo@newlist, list@@i + 3DD + pushvecD; AppendTo@newlist, list@@i + 4DD + pushvecD; D; newlist DH*This function is responsible for determining the ∆ parameter for pushing each face of X_F*L

In[13]:= penvert = push@N@invertsDD; H*Set of push vertices appoximated numerically*L

flapverts = 8penvert@@1DD + H1 2L Hpenvert@@11DD - penvert@@1DDL, penvert@@1DD + H1 2L Hpenvert@@10DD - penvert@@1DDL, penvert@@2DD + H1 2L Hpenvert@@15DD - penvert@@2DDL, penvert@@5DD + H1 2L Hpenvert@@6DD - penvert@@5DDL, penvert@@2DD + H1 2L Hpenvert@@16DD - penvert@@2DDL, penvert@@3DD + H1 2L Hpenvert@@20DD - penvert@@3DDL, penvert@@3DD + H1 2L Hpenvert@@47DD - penvert@@3DDL, penvert@@4DD + H1 2L Hpenvert@@46DD - penvert@@4DDL, penvert@@4DD + H1 2L Hpenvert@@53DD - penvert@@4DDL, , 36

Printed by Wolfram Mathematica Student Edition In[13]:=

dodecahedron.inflation.nb 7

penvert@@5DD + H1 2L Hpenvert@@52DD - penvert@@5DDL, penvert@@6DD + H1 2L Hpenvert@@52DD - penvert@@6DDL, penvert@@7DD + H1 2L Hpenvert@@51DD - penvert@@7DDL, penvert@@7DD + H1 2L Hpenvert@@60DD - penvert@@7DDL, penvert@@8DD + H1 2L Hpenvert@@59DD - penvert@@8DDL, penvert@@9DD + H1 2L Hpenvert@@21DD - penvert@@9DDL, penvert@@8DD + H1 2L Hpenvert@@22DD - penvert@@8DDL, penvert@@9DD + H1 2L Hpenvert@@12DD - penvert@@9DDL, penvert@@10DD + H1 2L Hpenvert@@11DD - penvert@@10DDL, penvert@@12DD + H1 2L Hpenvert@@21DD - penvert@@12DDL, penvert@@13DD + H1 2L Hpenvert@@25DD - penvert@@13DDL, penvert@@13DD + H1 2L Hpenvert@@29DD - penvert@@13DDL, penvert@@14DD + H1 2L Hpenvert@@28DD - penvert@@14DDL, penvert@@14DD + H1 2L Hpenvert@@17DD - penvert@@14DDL, penvert@@15DD + H1 2L Hpenvert@@16DD - penvert@@15DDL, penvert@@17DD + H1 2L Hpenvert@@28DD - penvert@@17DDL, penvert@@18DD + H1 2L Hpenvert@@27DD - penvert@@18DDL, penvert@@18DD + H1 2L Hpenvert@@40DD - penvert@@18DDL, penvert@@19DD + H1 2L Hpenvert@@39DD - penvert@@19DDL, penvert@@19DD + H1 2L Hpenvert@@48DD - penvert@@19DDL, penvert@@20DD + H1 2L Hpenvert@@47DD - penvert@@20DDL, penvert@@22DD + H1 2L Hpenvert@@59DD - penvert@@22DDL, penvert@@23DD + H1 2L Hpenvert@@58DD - penvert@@23DDL, penvert@@23DD + H1 2L Hpenvert@@32DD - penvert@@23DDL, penvert@@24DD + H1 2L Hpenvert@@31DD - penvert@@24DDL, penvert@@24DD + H1 2L Hpenvert@@30DD - penvert@@24DDL, penvert@@25DD + H1 2L Hpenvert@@29DD - penvert@@25DDL, penvert@@26DD + H1 2L Hpenvert@@36DD - penvert@@26DDL, penvert@@27DD + H1 2L Hpenvert@@40DD - penvert@@27DDL, penvert@@26DD + H1 2L Hpenvert@@35DD - penvert@@26DDL, penvert@@30DD + H1 2L Hpenvert@@31DD - penvert@@30DDL, penvert@@32DD + H1 2L Hpenvert@@58DD - penvert@@32DDL, penvert@@33DD + H1 2L Hpenvert@@57DD - penvert@@33DDL, penvert@@33DD + H1 2L Hpenvert@@42DD - penvert@@33DDL, penvert@@34DD + H1 2L Hpenvert@@41DD - penvert@@34DDL, penvert@@34DD + H1 2L Hpenvert@@37DD - penvert@@34DDL, penvert@@35DD + H1 2L Hpenvert@@36DD - penvert@@35DDL, penvert@@37DD + H1 2L Hpenvert@@41DD - penvert@@37DDL, penvert@@38DD + H1 2L Hpenvert@@45DD - penvert@@38DDL, penvert@@38DD + H1 2L Hpenvert@@49DD - penvert@@38DDL, penvert@@39DD + H1 2L Hpenvert@@48DD - penvert@@39DDL, penvert@@50DD + H1 2L Hpenvert@@54DD - penvert@@50DDL, penvert@@46DD + H1 2L Hpenvert@@53DD - penvert@@46DDL, penvert@@55DD + H1 2L Hpenvert@@56DD - penvert@@55DDL, penvert@@51DD + H1 2L Hpenvert@@60DD - penvert@@51DDL, penvert@@45DD + H1 2L Hpenvert@@49DD - penvert@@45DDL, penvert@@44DD + H1 2L Hpenvert@@50DD - penvert@@44DDL, penvert@@44DD + H1 2L Hpenvert@@54DD - penvert@@44DDL, penvert@@43DD + H1 2L Hpenvert@@55DD - penvert@@43DDL, penvert@@42DD + H1 2L Hpenvert@@57DD - penvert@@42DDL, penvert@@43DD + H1 2L Hpenvert@@56DD - penvert@@43DDL<; H*Finds the midpoints of the rectangular faces*L

finverts = Join@Join@penvert, dodecVD, flapvertsD; H*Compiles all vertices relevant to inflated dodecahedron*L 37

Printed by Wolfram Mathematica Student Edition 8 dodecahedron.inflation.nb

p1 = Sec@36 * Π 180D Ε; p2 = Tan@36 * Π 180D Ε; p3 = Sqrt@3D Ε; H*p1,p2,and p3 determine the side lengths for the triangle that divides the volume of the cap vertex pyramid along an edge leading towards the apex, the median of a triangular face, and the base of the pyramid*L s = Hp1 + p2 + p3L 2;H*Determines the semiperimeter*L area = FullSimplify@Sqrt@s * Hs - p1L * Hs - p2L * Hs - p3LD, Ε Î RealsD; H*Heron's Formula to determine area*L Solve@H1 2L Sqrt@3D Ε * H area, HD;H*Determines the height to adjust the vertex of the cap pyramid to maintain an isometry*L J-3 + 5 N Ε height = - ;H*Declaration of height variable*L 3 ratio = HNorm@Hfinverts@@25DD + finverts@@29DD + finverts@@13DDL 3D + heightL Norm@finverts@@67DDD;H*The scalar quantity that adjusts the cap vertices of the pyramid*L

gfinverts = Join@Join@penvert, ratio * dodecVD, flapvertsD; H*Compliation of all the vertices necessary to define the inflated dodecahedron*L ivd = 881, 2, 3, 4, 5<, 86, 7, 8, 9, 10<, 811, 12, 13, 14, 15<, 816, 17, 18, 19, 20<, 821, 22, 23, 24, 25<, 826, 27, 28, 29, 30<, 831, 32, 33, 34, 35<, 836, 37, 38, 39, 40<, 841, 42, 43, 44, 45<, 846, 47, 48, 49, 50<, 851, 52, 53, 54, 55<, 856, 57, 58, 59, 60<, 81, 2, 83, 81<, 82, 3, 86, 85<, 83, 87, 88, 4<, 84, 89, 90, 5<, 85, 84, 82, 1<, 81, 61, 81<, 81, 61, 82<, 82, 83, 74<, 82, 74, 85<, 83, 86, 69<, 83, 87, 69<, 84, 88, 70<, 84, 70, 89<, 85, 75, 90<, 85, 75, 84<, 86, 10, 82, 84<, 86, 91, 92, 7<, 87, 93, 94, 8<, 88, 96, 95, 9<, 89, 97, 98, 10<, 86, 84, 75<, 86, 91, 75<, 87, 64, 92<, 87, 64, 93<, 88, 94, 68<, 88, 68, 96<, 89, 95, 76<, 89, 76, 97<, 810, 61, 82<, 810, 98, 61<, 811, 98, 97, 12<, 812, 99, 100, 13<, 813, 101, 102, 14<, 814, 103, 104, 15<, 815, 83, 81, 11<, 811, 81, 61<, 811, 98, 61<, 812, 97, 76<, 812, 99, 76<, 813, 100, 67<, 813, 67, 101<, 814, 102, 63<, 814, 63, 103<, 815, 74, 83<, 815, 104, 74<, 816, 104, 103, 17<, 817, 105, 106, 18<, 818, 107, 108, 19<, 819, 109, 110, 20<, 820, 86, 85, 16<, 816, 85, 74<, 816, 74, 104<, 817, 63, 103<, 817, 63, 105<, 818, 79, 106<, 818, 79, 107<, 819, 77, 108<, 819, 77, 109<, 820, 69, 110<, 820, 69, 86<, 821, 95, 96, 22<, 822, 111, 112, 23<, 823, 113, 114, 24<, 824, 115, 116, 25<, 825, 100, 99, 21<, 821, 76, 99<, 821, 76, 95<, 822, 68, 96<, 822, 68, 111<, 823, 72, 112<, 823, 72, 113<, 824, 71, 114<, 824, 71, 115<, 825, 67, 116<, 825, 67, 100<, 826, 117, 118, 27<, 827, 106, 105, 28<, 828, 102, 101, 29<, 829, 116, 115, 30<, 830, 120, 119, 26<, 830, 71, 115<, 830, 71, 120<, 826, 65, 119<, 826, 65, 117<, 827, 79, 118<, 827, 79, 106<, 828, 63, 105<, 828, 63, 102<, 829, 67, 101<, 829, 67, 116<, 841, 124, 123, 42<, 842, 139, 140, 43<, 843, 138, 137, 44<, 844, 136, 135, 45<, 845, 128, 127, 41<, 841, 62, 124<, 841, 62, 127<, 842, 66, 123<, 842, 66, 139<, 843, 80, 140<, 843, 80, 138<, 844, 78, 137<, 844, 78, 136<, 845, 73, 135<, 845, 73, 128<, 851, 92, 91, 52<, 852, 90, 89, 53<, 853, 132, 131, 54<, 854, 137, 138, 55<, 855, 133, 134, 51<, 851, 64, 92<, 851, 64, 134<, 852, 75, 91<, 852, 75, 90<, 853, 70, 89<, 8132, 53, 70<, 854, 78, 131<, 854, 78, 137<, 855, 80, 138<, 855, 80, 133<, 850, 131, 132, 46<, 846, 88, 87, 47<, 847, 110, 109, 48<, 848, 130, 129, 49<, 849, 135, 136, 50<, 846, 70, 132<, 846, 70, 88<, 847, 69, 87<, 847, 69, 110<, 848, 77, 109<, 848, 77, 130<, 849, 73, 135<, 849, 73, 129<, 850, 78, 136<, 850, 78, 131<, 836, 126, 125, 37<, 837, 127, 128, 38<, 838, 129, 130, 39<, 839, 108, 107, 40<, 840, 118, 117, 36<, 836, 65, 117<, 836, 65, 126<, 837, 62, 125<, 837, 62, 127<, 838, 73, 128<, 838, 73, 129<, 839, 77, 130<, 839, 77, 108<, , , , , 38

Printed by Wolfram Mathematica Student Edition dodecahedron.inflation.nb 9

840, 79, 107<, 840, 79, 118<, 831, 114, 113, 32<, 832, 121, 122, 33<, 833, 123, 124, 34<, 834, 125, 126, 35<, 835, 119, 120, 31<, 831, 71, 120<, 831, 71, 114<, 832, 72, 113<, 832, 72, 121<, 833, 66, 122<, 833, 66, 123<, 834, 62, 124<, 834, 62, 125<, 835, 65, 126<, 835, 65, 119<, 856, 140, 139, 57<, 857, 122, 121, 58<, 858, 112, 111, 59<, 859, 94, 93, 60<, 860, 134, 133, 56<, 856, 80, 133<, 856, 80, 140<, 857, 66, 139<, 857, 66, 122<, 858, 72, 121<, 858, 72, 112<, 859, 68, 111<, 859, 68, 94<, 860, 64, 93<, 860, 64, 134<<; H*Information about how the vertices related to each other to develop faces for the inflated dodecahedron*L Show@ Graphics3D@8Opacity@1D, Yellow, GraphicsComplex@gfinverts, Polygon@ivdDD .7, .4, HH12 * v1 + 30 * v2 + 60 * v3LL . 8Ε ® e, ∆ ® d< D DH*Numberically defines the volume as a function of Ε*L

Plot@8v@eD, H1 4L H15 + 7 Sqrt@5DL<, 8e, 0.00001, .6<, AxesLabel ® 8Ε, Volume

Printed by Wolfram Mathematica Student Edition te

References

[DAKS06] N.S. Dairbekov, A.D. Alexandrov, Sem¨enSamsonovich Kutateladze, and A.B. Sossinsky, Convex polyhedra, Springer, 2006.

[Gal03] G. Galperin, Convex polyhedra without simple closed geodesics, Regular and Chaotic Dynamics Volume 8 (2003), no. 1.

[MP77] R. S. Millman and G. D. Parker, Elements of diﬀerential geometry, Prentice-Hall, 1977.

[Mun00] James R. Munkres, Topology, Second Edition, Prentice-Hall, 2000.

[Oss78] Robert Osserman, The isoperimetric inequality, Bulletin of the American Mathematical Society 84 (1978), no. 6, 1182–1238.

[Pak06] Igor Pak, Inﬂating polyhedral surfaces, 2006. http://citeseerx.ist.psu.edu/ viewdoc/download?doi=10.1.1.71.9361&rep=rep1&type=pdf.