7 Rational Versions

Definition 45

✓

We define the two functions $\sin _{\mathbb {Q}}, \cos _{\mathbb {Q}}: \operatorname{\mathbb {R}}\to \operatorname{\mathbb {R}}$ by:

\begin{align*} \sin _{\mathbb {Q}}(x) & :=x-\frac{x^3}{3}+\frac{x^5}{5!}\mp \dots +\frac{x^{25}}{25!},\\ \cos _{\mathbb {Q}}(x) & :=1-\frac{x^2}{2}+\frac{x^4}{4!}\mp \dots +\frac{x^{24}}{24!}. \end{align*}

Further, by replacing $\sin ,\cos $ with $\sin _{\mathbb {Q}},\cos _{\mathbb {Q}}$ we define the functions

\[ R_{\operatorname{\mathbb {Q}}}(\alpha ), R'_{\operatorname{\mathbb {Q}}}(\alpha ), X_{\operatorname{\mathbb {Q}}}(\theta , \varphi ), M_{\operatorname{\mathbb {Q}}}(\theta , \varphi ), M_{\operatorname{\mathbb {Q}}}^{\theta }(\theta ,\varphi ),M_{\operatorname{\mathbb {Q}}}^{\varphi }(\theta ,\varphi ). \]

Lemma 46

✓

\[ |\sin _{\mathbb {Q}}(x)-\sin (x)|\leq \frac{|x|^{27}}{27!} \quad \text{and} \quad |\cos _{\mathbb {Q}}(x)-\cos (x)|\leq \frac{|x|^{26}}{26!}. \]

Proof ▶

Appeal to Taylor series bounds, using the fact that all absolute values of higher derivatives of sine and cosine never exceed 1.

Lemma 47

✓

For every $x\in [-4,4]$ it holds that

\[ |\sin _{\mathbb {Q}}(x)-\sin (x)| \leq \frac{\kappa }{7} \quad \text{and} \quad |\cos _{\mathbb {Q}}(x)-\cos (x)|\leq \frac{\kappa }{7}. \]

Proof ▶

Straightforward numerical calculation from Lemma 46.

Lemma 48

✓

Let $A = (a_{i,j})_{1 \leq i \leq m,\ 1 \leq j \leq n} \in \operatorname{\mathbb {R}}^{m \times n}$ and $\delta {\gt}0$. Assume that $|a_{i,j}| \leq \delta $. Then it holds that $\| A\| \leq \delta \sqrt{mn}.$

Proof ▶

For any $v\in \operatorname{\mathbb {R}}^n$ we have

\begin{align*} \| Av\| ^2 & =\sum _{i=1}^m \left(\sum _{j=1}^na_{i,j}v_j\right)^2 \leq \sum _{i=1}^m\left(\sum _{j=1}^n \delta |v_j|\right)^2 = \delta ^2 m\left(\sum _{j=1}^n |v_j|\right)^2 \leq \delta ^2 m n \| v\| ^2 \end{align*}

using the Cauchy-Schwarz inequality. Dividing by $\| v\| $ and taking the square root proves the claim.

Lemma 49

✓

Let $A(x,y)$ be an $m\times n$ matrix with $1 \leq m,n\leq 3$ such that every entry is of the form $a_1(x)\cdot a_2(y)$ where $a_i(z)\in \{ 0,1,-1,\pm \sin (z),\pm \cos (z)\} .$ Define $A_{\mathbb {Q}}(x,y)$ by replacing $\sin $ with $\sin _{\mathbb {Q}}$ and $\cos $ with $\cos _{\mathbb {Q}}$. Then for every $x,y\in [-4,4]$ it holds that $\| A(x,y)-A_{\mathbb {Q}}(x,y)\| \leq \kappa $.

Proof ▶

We’ve replaced the assumption $a_i(z)\in \{ 0,1,-1,\pm \sin (z),\pm \cos (z)\} $ in [ SY25 ] ’s Lemma 40 with $a_i(z)\in [-1,1]$.

By assumption, for fixed $x,y$ every entry of $A(x,y)-A_{\mathbb {Q}}(x,y)$ is of the form $a b - \widetilde{a}\widetilde{b}$ for some $a,b\in [-1,1]$ and $|a-\widetilde{a}|,|b-\widetilde{b}|\leq \kappa /7$ by lemma 47. This implies that

\begin{align*} |ab-\widetilde{a}\widetilde{b}|& \leq |a b-a\widetilde{b}|+|a \widetilde{b}-\widetilde{a}\widetilde{b}| =|a|\cdot |b-\widetilde{b}|+|\widetilde{b}|\cdot |a-\widetilde{a}| \leq 1\cdot \kappa /7+(1+\kappa /7) \cdot \kappa /7 {\lt}\kappa /3. \end{align*}

So we can apply lemma 48 and obtain that $\| A(x,y)-A_{\operatorname{\mathbb {Q}}}(x,y)\| {\lt}\kappa /3\cdot \sqrt{3\cdot 3}=\kappa $.

Corollary 50

✓

Let $\alpha ,\theta ,\varphi \in [-4,4]$. Then it holds that

\begin{align*} \| R(\alpha )-R_{\operatorname{\mathbb {Q}}}(\alpha )\| , \| R’(\alpha )-R_{\operatorname{\mathbb {Q}}}’(\alpha )\| ,\| X(\theta ,\varphi )-X_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| , \| M(\theta , \varphi )-M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| , \\ \| M^\theta (\theta ,\varphi )-M_{\operatorname{\mathbb {Q}}}^\theta (\theta ,\varphi )\| , \| M^\varphi (\theta ,\varphi ) - M_{\operatorname{\mathbb {Q}}}^\varphi (\theta ,\varphi )\| \leq \kappa . \end{align*}

Moreover,

\[ \| R_{\operatorname{\mathbb {Q}}}(\alpha )\| , \| M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| \leq 1+\kappa \]

Proof ▶

The first statement is a direct application of lemma 49 and the second statement follows immediately after using lemma 12 and the triangle inequality.

Note: the original paper’s Corollary 41 makes a stronger claim that the norms of the derivatives of the rotation matrices are bounded by $1 + \kappa $ as well. This doesn’t directly follow from lemma 12 as currently stated. If we need these other bounds, we should strengthen lemma 12 to assert that the derivatives’ operator norms are at most 1.

Lemma 51

✓

For $1 \leq i \leq n$ let $(A_i,B_i)$ be pairs of real matrices, such that for each $i$ the dimensions of $A_i$ and $B_i$ are equal. Assume moreover that the products $A_1\cdots A_n$ and $B_1 \cdots B_n$ are well defined. Finally, assume that $\| A_i-B_i\| \leq \kappa $ and let $\delta _i\geq \max (\| A_i\| ,\| B_i\| ,1)$. Then it holds that $\| A_1\cdots A_n-B_1\cdots B_n\| \leq n\kappa \cdot \delta _1\cdots \delta _n$.

Proof ▶

See [ SY25 ] , Lemma 42.

Lemma 52

Let $\alpha , \theta , \varphi \in [-4,4]$, $P\in \operatorname{\mathbb {R}}^3$ with $\| P\| \leq 1$ and let $\widetilde{P}$ be a $\kappa $-rational approximation of $P$. Set $M = M(\theta , \varphi )$ and $M_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$, $M^\theta = M^\theta (\theta , \varphi )$, $M^\theta _{\operatorname{\mathbb {Q}}} = M^\theta _{\operatorname{\mathbb {Q}}}(\theta , \varphi )$, $M^\varphi = M^\varphi (\theta , \varphi )$, $M^\varphi _{\operatorname{\mathbb {Q}}} = M^\varphi _{\operatorname{\mathbb {Q}}}(\theta , \varphi )$ as well as $R = R(\alpha )$, $R_{\operatorname{\mathbb {Q}}} = R_{\operatorname{\mathbb {Q}}}(\alpha )$, $R' = R'(\alpha )$, $R'_{\operatorname{\mathbb {Q}}} = R'_{\operatorname{\mathbb {Q}}}(\alpha )$. Finally let $w \in \operatorname{\mathbb {R}}^2$ with $\| w\| = 1$. Then:

\begin{align} | \langle M P, w\rangle - \langle M_{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 3\kappa , \label{eq:boundskappa1} \\ | \langle M^\theta P, w\rangle - \langle M^\theta _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 3\kappa ,\\ | \langle M^\varphi P, w\rangle - \langle M^\varphi _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 3\kappa ,\\ | \langle R M P, w\rangle - \langle R_{\operatorname{\mathbb {Q}}} M_{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa ,\label{eq:boundskappa4} \\ | \langle R’ M P, w\rangle - \langle R’_{\operatorname{\mathbb {Q}}} M_{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa ,\\ | \langle R M^\theta P, w\rangle - \langle R_{\operatorname{\mathbb {Q}}} M^\theta _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa ,\\ | \langle R M^\varphi P, w\rangle - \langle R_{\operatorname{\mathbb {Q}}} M^\varphi _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa . \end{align}

Proof ▶

See [ SY25 ] , Lemma 44.

Theorem 53 Rational Global Theorem

Let $\operatorname{\mathbf{P}}$ be a pointsymmetric convex polyhedron with radius $\rho =1$ and $\widetilde{\operatorname{\mathbf{P}}}$ a $\kappa $-rational approximation. Let $\widetilde{S} \in \widetilde{\operatorname{\mathbf{P}}}$. Further let $\varepsilon {\gt}0$ and $\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {Q}}\cap [-4,4]$. Let $w\in \operatorname{\mathbb {Q}}^2$ be a unit vector. Denote $\overline{M_1}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1, \overline{\varphi }_1)$, $ \overline{M_2}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2, \overline{\varphi }_2)$ as well as $\overline{M_1}^{\theta } :=M_{\operatorname{\mathbb {Q}}}^\theta (\overline{\theta }_1, \overline{\varphi }_1)$, $\overline{M_1}^{\varphi } :=M_{\operatorname{\mathbb {Q}}}^\varphi (\overline{\theta }_1, \overline{\varphi }_1)$ and analogously for $\overline{M_2}^{\theta }, \overline{M_2}^{\varphi }$. Finally set

\begin{align*} G^{\operatorname{\mathbb {Q}}}& :=\langle R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}\widetilde{S},w \rangle - \varepsilon \cdot \big(|\langle R_{\operatorname{\mathbb {Q}}}’(\overline{\alpha }) \overline{M_1}\widetilde{S},w \rangle |+|\langle R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}^\theta \widetilde{S},w \rangle |+|\langle R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}^\varphi \widetilde{S},w \rangle |\big) \\ & \hspace{11cm}- 9\varepsilon ^2/2 - 4\kappa ( 1 + 3 \varepsilon ),\\ H^{\operatorname{\mathbb {Q}}}_P & :=\langle \overline{M_2}P,w \rangle + \varepsilon \cdot \big(|\langle \overline{M_2}^\theta P,w \rangle |+|\langle \overline{M_2}^\varphi P,w \rangle |\big) + 2\varepsilon ^2 + 3\kappa ( 1+2\varepsilon ). \end{align*}

If $G^{\operatorname{\mathbb {Q}}}{\gt}\max _{P\in \widetilde{\operatorname{\mathbf{P}}}} H^{\operatorname{\mathbb {Q}}}_P$ then there does not exist a solution to Rupert’s condition to $\operatorname{\mathbf{P}}$ with

\[ (\theta _1,\varphi _1,\theta _2,\varphi _2,\alpha ) \in [\overline{\theta }_1\pm \varepsilon ,\overline{\varphi }_1\pm \varepsilon ,\overline{\theta }_2\pm \varepsilon ,\overline{\varphi }_2\pm \varepsilon ,\overline{\alpha }\pm \varepsilon ]. \]

Proof ▶

Definition 54

✓

Let $\theta , \varphi \in \operatorname{\mathbb {Q}}\cap [-4,4]$ and $M_{\operatorname{\mathbb {Q}}} :=M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$. Three points $\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3 \in \operatorname{\mathbb {Q}}^3$ with $\| \widetilde{P}_1\| , \| \widetilde{P}_2\| , \| \widetilde{P}_3\| \leq 1+\kappa $ are called $\varepsilon $-$\kappa $-spanning for $(\theta , \varphi )$ if it holds that:

\begin{align*} \langle R(\pi /2) M_{\operatorname{\mathbb {Q}}} \widetilde{P}_1,M_{\operatorname{\mathbb {Q}}} \widetilde{P}_{2}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ) + 6\kappa ,\\ \langle R(\pi /2) M_{\operatorname{\mathbb {Q}}} \widetilde{P}_2,M_{\operatorname{\mathbb {Q}}} \widetilde{P}_{3}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ) + 6\kappa ,\\ \langle R(\pi /2) M_{\operatorname{\mathbb {Q}}} \widetilde{P}_3,M_{\operatorname{\mathbb {Q}}} \widetilde{P}_{1}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ) + 6\kappa . \end{align*}

Lemma 55

✓

Let $P_1, P_2, P_3 \in \operatorname{\mathbb {R}}^3$ with $\| P_i\| \leq 1$ and $\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3 \in \operatorname{\mathbb {Q}}^3$ be their $\kappa $-rational approximations. Assume that $\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3$ are $\varepsilon $-$\kappa $-spanning for some $\theta , \varphi \in \operatorname{\mathbb {Q}}\cap [-4,4]$, then $P_1, P_2, P_3$ are $\varepsilon $-spanning for $\theta , \varphi $.

Proof ▶

See [ SY25 ] , Lemma 46.

Lemma 56

Let $P,Q \in \operatorname{\mathbb {R}}^3$ with $\| P\| ,\| Q\| \leq 1$ and $\widetilde{P},\widetilde{Q}$ some respective $\kappa $-rational approximations. Moreover, let $\alpha , \theta , \varphi \in \operatorname{\mathbb {R}}\in [-4,4]$ and set $X = X(\theta , \varphi )$, $X_{\operatorname{\mathbb {Q}}} = X_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$ as well as $M = M(\theta , \varphi )$, $M_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$. Then

\begin{align} |\langle X, P \rangle - \langle X_{\operatorname{\mathbb {Q}}}, \widetilde{P} \rangle | & \leq 3 \kappa , \label{eq:boundskappa3.1}\\ |\langle MP, MQ \rangle - \langle M_{\operatorname{\mathbb {Q}}} \widetilde{P}, M_{\operatorname{\mathbb {Q}}}\widetilde{Q} \rangle | & \leq 5 \kappa , \label{eq:boundskappa3.3}\\ |\| M Q \| - \| M_{\operatorname{\mathbb {Q}}}\widetilde{Q} \| | & \leq 3 \kappa .\label{eq:boundskappa3.2} \end{align}

Proof ▶

See [ SY25 ] , Lemma 49.

Corollary 57

In the setting of lemma 56, let additionally $\overline{\theta }, \overline{\varphi }\in \operatorname{\mathbb {R}}\cap [-4,4]$ and set $\overline{M} = M(\overline{\theta }, \overline{\varphi })$, $\overline{M}_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\overline{\theta }, \overline{\varphi })$. Then

\[ |\| R(\alpha ) M P - \overline{M} Q\| - \| R_{\operatorname{\mathbb {Q}}}(\alpha ) M_{\operatorname{\mathbb {Q}}} \widetilde{P} - \overline{M}_{\operatorname{\mathbb {Q}}} \widetilde{Q}\| | \leq 6 \kappa \]

Proof ▶

See [ SY25 ] , Corollary 50.

Corollary 58

In the setting of lemma 56, let $\sqrt[+]{x}$ be an upper-$\operatorname{\mathbb {Q}}$-square-root function and set $\| x\| _{+} :=\sqrt[+]{\| x\| ^2}$. Set

\[ A = \frac{\langle M P, M(P-Q)\rangle - 2 \varepsilon \| P-Q\| \cdot (\sqrt{2}+\varepsilon )}{ \big(\| M P\| +\sqrt{2} \varepsilon \big) \cdot \big(\| M(P-Q)\| +2 \sqrt{2} \varepsilon \big)} \]

as well as

\[ A_{\operatorname{\mathbb {Q}}} = \frac{\langle M_{\operatorname{\mathbb {Q}}} \widetilde{P}, M_{\operatorname{\mathbb {Q}}} (\widetilde{P}-\widetilde{Q})\rangle - 10\kappa - 2 \varepsilon ( \| \widetilde{P}-\widetilde{Q}\| _{+} + 2 \kappa ) \cdot (\sqrt{2}+\varepsilon )}{ \big(\| M_{\operatorname{\mathbb {Q}}} \widetilde{P}\| _{+}+\sqrt{2} \varepsilon + 3\kappa \big) \cdot \big(\| M_{\operatorname{\mathbb {Q}}}(\widetilde{P}-\widetilde{Q})\| _{+}+2 \sqrt{2} \varepsilon + 6\kappa \big)}. \]

Then it holds that $A \geq A_{\operatorname{\mathbb {Q}}}$.

Proof ▶

See [ SY25 ] , Corollary 51.

Theorem 59 Rational Local Theorem

Let $\operatorname{\mathbf{P}}$ be a polyhedron with radius $\rho =1$ and $\widetilde{P}_i$ be a $\kappa $-rational approximation of $P_i \in \operatorname{\mathbf{P}}$. Set $\widetilde{\operatorname{\mathbf{P}}} = \{ \widetilde{P}_i \text{ for } P_i \in \operatorname{\mathbf{P}}\} $. Let $P_1, P_2, P_3, Q_1, Q_2, Q_3 \in \operatorname{\mathbf{P}}$ be not necessarily distinct and assume that $P_1, P_2, P_3$ and $Q_1, Q_2, Q_3$ are congruent. Let $\varepsilon {\gt}0$ and $\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {Q}}\cap [-4,4]$. Set $\overline{X_1}:=X_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1,\overline{\varphi }_1), \overline{X_2}:=X_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2,\overline{\varphi }_2)$ as well as $\overline{M_1}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1,\overline{\varphi }_1), \overline{M_2}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2,\overline{\varphi }_2)$. Assume that there exist $\sigma _P, \sigma _Q \in \{ 0,1\} $ such that

\[ (-1)^{\sigma _P} \langle \overline{X_1},\widetilde{P}_i\rangle {\gt}\sqrt{2}\varepsilon + 3\kappa \quad \text{and} \quad (-1)^{\sigma _Q} \langle \overline{X_2}, \widetilde{Q}_i\rangle {\gt}\sqrt{2}\varepsilon + 3\kappa , \tag {A$^{\operatorname{\mathbb {Q}}}_\varepsilon $} \]

for all $i=1,2,3$. Moreover, assume that $\widetilde{P}_1,\widetilde{P}_2,\widetilde{P}_3$ are $\varepsilon $-$\kappa $-spanning for $(\overline{\theta }_1,\overline{\varphi }_1)$ and that $\widetilde{Q}_1,\widetilde{Q}_2,\widetilde{Q}_3$ are $\varepsilon $-$\kappa $-spanning for $(\overline{\theta }_2,\overline{\varphi }_2)$. Let $\sqrt[+]{x}$ and $\sqrt[-]{x}$ be upper- and lower-$\operatorname{\mathbb {Q}}$-square-root functions, then set $\| Z\| _{+} :=\sqrt[+]{\| Z\| ^2}$ and $\| Z\| _{-} :=\sqrt[-]{\| Z\| ^2}$ for $Z \in \operatorname{\mathbb {Q}}^n$. Finally, assume that for all $i = 1,2,3$ and any $\widetilde{Q}_j \in \widetilde{\operatorname{\mathbf{P}}} \setminus \widetilde{Q}_i$ it holds that

\[ \frac{\langle \overline{M_2}\widetilde{Q}_i,\overline{M_2}(\widetilde{Q}_i-\widetilde{Q}_j)\rangle - 10\kappa - 2 \varepsilon ( \| \widetilde{Q}_i-\widetilde{Q}_j\| _{+} + 2 \kappa ) \cdot (\sqrt{2}+\varepsilon )}{ \big(\| \overline{M_2}\widetilde{Q}_i\| _{+}+\sqrt{2} \varepsilon + 3\kappa \big) \cdot \big(\| \overline{M_2}(\widetilde{Q}_i-\widetilde{Q}_j)\| _{+}+2 \sqrt{2} \varepsilon + 6\kappa \big)} {\gt} \frac{\sqrt{5} \varepsilon + \delta }{r}, \tag {B$^{\operatorname{\mathbb {Q}}}_\varepsilon $} \]

for some $r {\gt}0$ such that $\min _{i=1,2,3}\| \overline{M_2}\widetilde{Q}_i \| _{-} {\gt} r + \sqrt{2} \varepsilon + 3\kappa $ and for some $\delta \in \operatorname{\mathbb {R}}$ with

\[ \delta = \max _{i=1,2,3}\left\| R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}\widetilde{P}_i-\overline{M_2}\widetilde{Q}_i\right\| _{+}/2 + 3\kappa . \]

Then there exists no solution to Rupert’s problem $R(\alpha ) M(\theta _1,\varphi _1)\operatorname{\mathbf{P}}\subset M(\theta _2,\varphi _2)\operatorname{\mathbf{P}}^\circ $ with

\[ (\theta _1, \varphi _1, \theta _2, \varphi _2, \alpha ) \in [\overline{\theta }_1\pm \varepsilon ,\overline{\varphi }_1\pm \varepsilon ,\overline{\theta }_2\pm \varepsilon ,\overline{\varphi }_2\pm \varepsilon ,\overline{\alpha }\pm \varepsilon ] \subseteq \operatorname{\mathbb {R}}^5. \]

Proof ▶