2 The Noperthedron
2.1 Definition of the Noperthedron
We define three points \(C_1,C_2,C_3\in \mathbb {Q}^3\).
\[ C_1:=\frac{1}{259375205} \begin{pmatrix} {152024884}
\\ 0
\\ {210152163}
\end{pmatrix}, \qquad C_2:=\frac{1}{10^{10}} \begin{pmatrix} 6632738028
\\ 6106948881
\\ 3980949609
\end{pmatrix}, \]
\[ C_3:=\frac{1}{10^{10}} \begin{pmatrix} 8193990033
\\ 5298215096
\\ 1230614493
\end{pmatrix}. \]
Lemma
1
\(\| C_1 \| = 1\), \({98 \over 100} {\lt} \| C_2 \| {\lt} {99 \over 100}\), and \({98 \over 100} {\lt} \| C_3 \| {\lt} {99 \over 100}\).
Lemma
2
The radius of the Noperthedron is one.
Rotations about the \(x, y, z\) axes \(R_x,R_y,R_z:\) \(\mathbb {R}\to \mathbb {R}^{3\times 3}\) are defined in the usual way:
\[ R_x(\alpha ):=\begin{pmatrix} 1
& 0
& 0
\\ 0
& \cos \alpha
& -\sin \alpha
\\ 0
& \sin \alpha
& \cos \alpha
\end{pmatrix}, \hspace{1cm} R_y(\alpha ):=\begin{pmatrix} \cos \alpha
& 0
& -\sin \alpha
\\ 0
& 1
& 0
\\ \sin \alpha
& 0
& \cos \alpha
\end{pmatrix}, \]
\[ R_z(\alpha ):=\begin{pmatrix} \cos \alpha
& -\sin \alpha
& 0
\\ \sin \alpha
& \cos \alpha
& 0
\\ 0
& 0
& 1
\end{pmatrix}. \]
We define a 30-element set \(C_{30}\)
\[ \mathcal{C}_{30} :=\left\{ (-1)^\ell R_z\left(\frac{2\pi k}{15}\right) \colon k=0,\dots ,14; \ell =0,1\right\} . \]
of rotations.
We write \(\mathcal{C}_{30} \cdot P = \{ c P \, \text{ for } \, c \in \mathcal{C}_{30}\} \) for the orbit of \(P\) under the action of \(\mathcal{C}_{30}\).
Definition
3
The Noperthedron is polyhedron given by the vertex set
\[ \mathcal{C}_{30} \cdot C_1 \cup \mathcal{C}_{30} \cdot C_2 \cup \mathcal{C}_{30} \cdot C_3 \]
Lemma
4
The norm of any vertex in the Noperthedron is no more than 1.
Proof
▶
Evident from definitions.
Definition
5
A set \(S \subseteq \operatorname{\mathbb {R}}^3\) is point-symmetric if \(x \in S\) implies \(-x \in S\).
Lemma
6
The noperthedron is point-symmetric.
2.2 Refined Rupert’s property for the Noperthedron
Lemma
7
Let \(\operatorname{\mathbf{P}}= \operatorname{\mathbf{NOP}}\), then for all \(\theta , \varphi , \alpha \in \operatorname{\mathbb {R}}\), the following three identities hold (as sets):
\begin{align*} M({\theta +2\pi /15,\varphi })\cdot \operatorname{\mathbf{P}}& =M(\theta , \varphi ) \cdot \operatorname{\mathbf{P}},\\ R(\alpha +\pi )M(\theta , \varphi ) \cdot \operatorname{\mathbf{P}}& =R(\alpha )M(\theta , \varphi ) \cdot \operatorname{\mathbf{P}},\\ \begin{pmatrix} 1
& 0
\\ 0
& -1
\end{pmatrix} M(\theta , \varphi ) \cdot \operatorname{\mathbf{P}}& = M({\theta +\pi /15,\pi -\varphi }) \cdot \operatorname{\mathbf{P}}. \end{align*}
Corollary
8
If the noperthedron is Rupert, then there exists a solution with
\begin{align*} \theta _1,\theta _2& \in [0,2\pi /15] \subset [0,0.42], \\ \varphi _1& \in [0,\pi ] \subset [0,3.15],\\ \varphi _2& \in [0,\pi /2] \subset [0,1.58],\\ \alpha & \in [-\pi /2,\pi /2] \subset [-1.58,1.58]. \end{align*}