# Schrödinger–Newton equation with spontaneous wave function collapse

Lajos Diósi<EMAIL_ADDRESS>Wigner Research Centre for Physics, H-1525
Budapest 114, P. O. Box 49, Hungary
and
Eötvös Loránd University, H-1117 Budapest, Pázmány Péter stny. 1/A

###### Abstract

Based on the assumption that the standard Schrödinger equation becomes
gravitationally modified for massive macroscopic objects, two independent
proposals has survived from the nineteen-eighties. The Schrödinger–Newton
equation (1984) provides well-localized solitons for free macro-objects but
lacks the mechanism how extended wave functions collapse on solitons. The
gravity-related stochastic Schrödinger equation (1989) provides the
spontaneous collapse but the resulting solitons undergo a tiny diffusion
leading to an inconvenient steady increase of the kinetic energy. We propose
the stochastic Schrödinger–Newton equation which contains the above two
gravity-related modifications together. Then the wave functions of free
macroscopic bodies will gradually and stochastically collapse to solitons
which perform inertial motion without the momentum diffusion: conservation of
momentum and energy is restored.

##  I Introduction

The conjectured validity of quantum theory in the macroscopic world is
famously problematic. This is shown usually on the elementary example: free
motion of an isolated macroscopic mass $M$. Its center of mass (c.o.m) wave
packet is widening eternally while we would expect a certain stationary
localization. The most spectacular paradox concerns the Schrödinger cat states
which are superpositions of two distant wave packets, legitimate in the
microworld but problematic for a macroscopic body.

A Newtonian semiclassical (mean-field) modification of the Schrödinger
equation to ensure stationary localization was proposed first Diósi (1984).
This Schrödinger–Newton equation (SNE) yields well-localized soliton wave
packets for the mass $M$. However, the Schrödinger cat states remain
legitimate solutions and an independent mechanism is required to destroy them
— as stated in ref. Diósi (1984). To this end, another Newton-gravity-related
(G-related) mechanism was proposed in two steps. The G-related master equation
(G-ME) decoheres the cat states Diósi (1987). Its unraveling, the G-related
stochastic Schrödinger equation (G-SSE) Diósi (1989) collapses them
spontaneously to one or the other wave packet, and drives this component into
a soliton — with one annoying effect though. The c.o.m. motion of the soliton
never becomes stationary, it remains subject of tiny stochastic fluctuations
generating kinetic energy at constant rate.

Here we propose the combination of the SNE and the G-SSE, obtaining the new
stochastic SNE (SSNE). We show that the momentum diffusion of solitons of
massive isolated objects get canceled by the interplay between the
stochasticity of the collapse mechanism and the semiclassical Newtonian self-
interaction. The remainder of the stochasticity is merely an extreme small
universal coordinate diffusion proportional to $\hbar/M$. Our SSNE is
partially realizing the concept of ref. Diósi (2019) where the diffusion
effects of a stereotypical collapse equation [our eq. (11) below] were
completely eliminated by imposing frame drag. Also a related vision of induced
gravity was put forward.

Secs. II and III recapitulate the basics of the SNE and the G-SSE, invoking
the results of Refs. Diósi (1984, 1987, 1989). The combination of the two
equations is introduced in sec. IV. Then sec. V proves that the soliton
momenta of SSNE are constant, their diffusion is canceled. Sec. VI contains
final remarks.

##  II Semiclassical Schrödinger–Newton Equation

Consider the standard Schrödinger equation
$d\left|\Psi\right>/dt=-(i/\hbar){\hat{H}}\left|\Psi\right>$ of a massive non-
relativistic many-body system, with Hamiltonian ${\hat{H}}$ not including the
Newtonian pair potential. Let ${\hat{\varrho}}(\mathbf{r})$ denote the
operator of mass density at location $\mathbf{r}$, normalized to the total
mass $M$. Instead of the Newton pair potential, one postulates that the
classical gravitational is sourced by the mean-field
$\left<{\hat{\varrho}}(\mathbf{s})\right>=\left<\Psi\right|{\hat{\varrho}}(\mathbf{r})\left|\Psi\right>$.
The semiclassical SNE reads

$\displaystyle\frac{d\left|\Psi\right>}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}{\hat{H}}\left|\Psi\right>+\frac{i}{\hbar}G\int\int\frac{{\hat{\varrho}}(\mathbf{r})\left<{\hat{\varrho}}(\mathbf{s})\right>d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}\left|\Psi\right>$
(1) $\displaystyle\equiv$
$\displaystyle-\frac{i}{\hbar}({\hat{H}}+{\hat{V}}_{\Psi})\left|\Psi\right>,$

where ${\hat{V}}_{\Psi}$ is the $\Psi$ dependent Newtonian semiclassical
(mean-field) interaction.

The SNE possesses soliton solutions for the c.o.m. wave function of isolated
bodies. Consider the motion of a single spherical symmetric rigid mass $M$,
the canonical position and momentum operators are
$\hat{\mathbf{x}},\hat{\mathbf{p}}$ respectively. The mass density operator
can be written as

${\hat{\varrho}}(\mathbf{r})=Mf(\mathbf{r}-\hat{\mathbf{x}})$ (2)

where $f$ is a normalized non-negative spherical symmetric function. Let us
introduce the frequency parameter $\omega_{G}$ as defined in ref. Diósi
(2007):

$\omega_{G}^{2}=\frac{4\pi}{3}GM\int f^{2}(\mathbf{r})d\mathbf{r}.$ (3)

This sets an effective strength of Newtonian self-attraction (of the G-related
spontaneous collapse, too, cf. secs. III-V) when the position uncertainty
$\Delta\mathrm{x}$ in state $\left|\Psi\right>$ is much smaller than the
characteristic length scale(s) of $f(\mathbf{r})$ not . Then, ignoring higher
(than 2nd) order terms in
$\hat{\mathbf{x}}_{c}=\hat{\mathbf{x}}-\left<\hat{\mathbf{x}}\right>$, we can
write the SNE (1) into the simple form (cf. Appendix A.):

$\frac{d\left|\Psi\right>}{dt}=-{\frac{i}{\hbar}}{\hat{H}}\left|\Psi\right>-\frac{i}{2\hbar}M\omega_{G}^{2}\hat{\mathbf{x}}_{c}^{2}\left|\Psi\right>.$
(4)

If ${\hat{H}}=\hat{\mathbf{p}}^{2}/2M$ and $\left<\hat{\mathbf{x}}\right>=0$
then the ground state coincides with the ground state of a central harmonic
oscillator of frequency $\omega_{G}$:

$\Psi_{0}(\mathbf{x})=\mathcal{N}\exp\left(-\frac{M\omega_{G}\mathbf{x}^{2}}{2\hbar}\right).$
(5)

This is a soliton standing at the origin. The soliton and its inertially
traveling versions are suitably representing the stationary localization of
free macro objects.

However, there is no mechanism that drives larger solitons and large
macroscopically extended wave functions toward the basic soliton states of
shape (5). Consider, e.g., a Schrödinger cat state which is superposition of
two solitons at a large distance $\ell$ from each other:

$\left|\mbox{Cat}\right>=\frac{\left|\mbox{left}\right>+\left|\mbox{right}\right>}{\sqrt{2}}.$
(6)

They attract each other via ${\hat{V}}_{\Psi}$, can even form a Kepler system,
but cannot collapse to a single soliton. Ref. Diósi (1984) emphasizes that
collapse needs an additional mechanism, not present in the SNE.

##  III Gravity-Related Wavefunction Collapse

Alternatively to the SNE (1), the G-SSE considers the following stochastic
modification of the standard Schrödinger equation:

$\frac{d\left|\Psi\right>}{dt}=-\frac{i}{\hbar}{\hat{H}}\left|\Psi\right>-\frac{G}{2\hbar}\int\int\frac{{\hat{\varrho}}_{c}(\mathbf{r}){\hat{\varrho}}_{c}(\mathbf{s})d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}\left|\Psi\right>+\frac{1}{\hbar}\int{\hat{\varrho}}_{c}(\mathbf{r})\Phi(\mathbf{r})d\mathbf{r}\left|\Psi\right>,$
(7)

where
${\hat{\varrho}}_{c}(\mathbf{r})={\hat{\varrho}}(\mathbf{r})-\left<{\hat{\varrho}}(\mathbf{r})\right>$
and $\Phi(\mathbf{r},t)$ is a white-noise field of spatial correlation

$\mathbf{M}\Phi(\mathbf{r},t)\Phi(\mathbf{s},\tau)=\frac{G\hbar}{|\mathbf{r}-\mathbf{s}|}\delta(t-\tau),$
(8)

$\mathbf{M}$ stands for the stochastic mean. The white-noise is to be taken in
terms of the Ito differential calculus. This G-SSE yields the spontaneous
collapse of massive macroscopic spatial superpositions, of the Schrödinger cat
states in particular. The average state, i.e., the density matrix
${\hat{\rho}}=\mathbf{M}\left|\Psi\right>\left<\Psi\right|$ satisfies the
G-ME:

$\displaystyle\frac{d{\hat{\rho}}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[{\hat{H}},{\hat{\rho}}]-\frac{G}{2\hbar}\int\int\frac{[{\hat{\varrho}}(\mathbf{r}),[{\hat{\varrho}}(\mathbf{s}),{\hat{\rho}}]]d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}$
(9) $\displaystyle\equiv$
$\displaystyle-\frac{i}{\hbar}[{\hat{H}},{\hat{\rho}}]+\mathcal{D}{\hat{\rho}}.$

This suppresses the coherence of macroscopically distinct superpositions of
the mass density, yielding their statistical mixture without the collapse. The
G-SSE (7) adds the collapse as well to the decoherence. The two distant
wavepackets of the Schrödinger cat state (6) collapse onto one of them
randomly at the rate

$\frac{\Delta E_{G}}{\hbar},$ (10)

where $\Delta E_{G}$ denotes how much the collapse reduces the gravitational
energy of the cat Penrose (1996, 1998, 2014).

The G-SSE (7) becomes simple for a single mass in the regime of small
$\Delta\mathrm{x}$ (cf. Appendices B,C):

$\frac{d\left|\Psi\right>}{dt}=-\frac{i}{\hbar}{\hat{H}}\left|\Psi\right>-\frac{1}{2\hbar}M\omega_{G}^{2}\hat{\mathbf{x}}_{c}^{2}\left|\Psi\right>+\sqrt{\frac{M}{\hbar}}\omega_{G}\hat{\mathbf{x}}_{c}\mathbf{w}\left|\Psi\right>,$
(11)

where $\mathbf{w}$ is the vector of three independent standard white-noises.
(Correspondence with notations in ref. Diósi (1989) is
$\Gamma=2M\omega_{G}^{2}/\hbar$ and
$d\xi=\sqrt{M/\hbar}\omega_{G}\mathbf{w}$.) Note, the Hermitian self-
attracting potential in the SNE (4) becomes anti-Hermitian here. For free
bodies (${\hat{H}}=\hat{\mathbf{p}}^{2}/2M$), the solutions converge to
solitons of the steady shape

$\Psi_{0}(\mathbf{x})=\mathcal{N}\exp\left(-(1-i)\frac{M\omega_{G}\mathbf{x}^{2}}{2\hbar}\right).$
(12)

This is slightly different from (5), by the new complex factor $1-i$ which
leaves the spread $\Delta\mathrm{x}=\sqrt{(\hbar/2M\omega_{G}})$ unchanged
just creating the correlation $\hbar/2$ between $\hat{\mathbf{x}}$ and
$\hat{\mathbf{p}}$. But the major difference is in the c.o.m. motion which is
plagued by a certain correlated diffusion of both the position and the
momentum:

$\displaystyle\frac{d}{dt}\left<\hat{\mathbf{x}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left<\hat{\mathbf{p}}\right>}{M}+\sqrt{\hbar/M}\mathbf{w},$
(13) $\displaystyle\frac{d}{dt}\left<\hat{\mathbf{p}}\right>$ $\displaystyle=$
$\displaystyle\sqrt{\hbar M}\omega_{G}\mathbf{w}.$ (14)

The momentum diffusion (14) is increasing the kinetic energy at rate
$\frac{1}{2}\hbar\omega_{G}^{2}$ which is probably an unphysical artifact of
the collapse model. We get rid of it below..

##  IV Schrödinger–Newton Equation with Wavefunction Collapse

We slightly alter the original 1989-version (7) of the G-SSE. We insert a
factor $\exp(-i\pi/4)=(1-i)/\sqrt{2}$ in front of the anti-Hermitian
stochastic potential:

$\frac{d\left|\Psi\right>}{dt}=-\frac{i}{\hbar}{\hat{H}}\left|\Psi\right>-\frac{G}{2\hbar}\int\int\frac{{\hat{\varrho}}_{c}(\mathbf{r}){\hat{\varrho}}_{c}(\mathbf{s})d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}\left|\Psi\right>+\frac{\mathrm{e}^{-i\pi/4}}{\hbar}\int{\hat{\varrho}}_{c}(\mathbf{r})\Phi(\mathbf{r})d\mathbf{r}\left|\Psi\right>.$
(15)

This yields the same G-ME (9) for the density matrix, and also encodes the
collapse of macroscopic superpositions. The novel feature is the appearance of
the Hermitian stochastic potential $\Phi(\mathbf{r})/\sqrt{2}$ which will
cancel the momentum diffusion (14) of the solitons — provided we include the
gravitational self-attraction contained in ${\hat{V}}_{\Psi}$ of the SNE.

Our new proposal is the following combination of the SNE (1) and the modified
G-SSE (15):

$\frac{d\left|\Psi\right>}{dt}=-\frac{i}{\hbar}({\hat{H}}\\!+\\!{\hat{V}}_{\Psi})\\!\left|\Psi\right>-\frac{G}{2\hbar}\int\int\frac{{\hat{\varrho}}_{c}(\mathbf{r}){\hat{\varrho}}_{c}(\mathbf{s})d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}\left|\Psi\right>+\frac{\mathrm{e}^{-i\pi/4}}{\hbar}\\!\\!\int\\!\\!{\hat{\varrho}}_{c}(\mathbf{r})\Phi(\mathbf{r})d\mathbf{r}\\!\left|\Psi\right>\\!.$
(16)

The ME of the average state reads:

$\frac{d{\hat{\rho}}}{dt}=-\frac{i}{\hbar}[{\hat{H}}+{\hat{V}}_{\Psi},{\hat{\rho}}]+\mathcal{D}{\hat{\rho}}.$
(17)

This is not a closed equation for ${\hat{\rho}}$ since ${\hat{V}}_{\Psi}$
depends on the pure state $\left|\Psi\right>$. The lack of a closed linear ME
is the signature of anomalies Gisin (1989) already troubling the SNE and
inherited by our SSNE (16). There is, however, a difference. The spontaneous
collapse might shadow the anomalies. One of them, the fake action-at-a-
distance is based on the attraction caused by ${\hat{V}}_{\Psi}$ between the
two halves $\left|\mbox{left}\right>,\left|\mbox{right}\right>$ of the
Schrödinger cat Diósi (2016). Unlike in case of the SNE, the cat now has the
finite lifetime $\hbar/\Delta E_{G}$ which may be too short to reach
detectable shifts of the left or the right wavepackets Großardt (2022).

##  V Solitons with Energy Conservation

We consider the soliton solutions of the new spontaneous collapse dynamics
SSNE in the small-$\Delta\mathrm{x}$ approximation. The single body special
case of the SSNE (16) takes this form:

$\frac{d\left|\Psi\right>}{dt}=-{\frac{i}{\hbar}}{\hat{H}}\left|\Psi\right>-\frac{1+i}{2\hbar}M\omega_{G}^{2}\hat{\mathbf{x}}_{c}^{2}\left|\Psi\right>+(1-i)\sqrt{\frac{M}{2\hbar}}\omega_{G}\hat{\mathbf{x}}_{c}\mathbf{w}\left|\Psi\right>.$
(18)

Note the important complex factors $(1+i)$ and $(1-i)$ compared to the
small-$\Delta\mathrm{x}$ approximation (11) of the G-SSE model. With these two
factors, as we show below, the soliton’s momentum diffusion (14) cancels.

For the time-dependent wave function of the soliton we take the following
Ansatz:

$\displaystyle\Psi_{t}(\mathbf{x})$ $\displaystyle=$
$\displaystyle\mathcal{N}\exp\left(-(1-i)\frac{|\mathbf{x}-\left<\hat{\mathbf{x}}\right>_{t}|^{2}}{4\Delta\mathrm{x}^{2}}+\frac{i}{\hbar}\left<\hat{\mathbf{p}}\right>\hat{\mathbf{x}}\right),$
$\displaystyle\Delta\mathrm{x}^{2}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{\sqrt{2}M\omega_{G}},$
$\displaystyle\frac{d}{dt}\left<\hat{\mathbf{x}}\right>_{t}$ $\displaystyle=$
$\displaystyle\frac{\mathbf{p}}{M}+\sqrt{\hbar/M}\mathbf{w}_{t}.$ (19)

These solutions correspond to inertial c.o.m. motion at constant momentum
$\left<\hat{\mathbf{p}}\right>$, apart from a minuscule diffusion of the
coordinate.

We still owe to show that the wavefunciton $\Psi_{t}(\mathbf{x})$ satisfies
the eq. (18). It is sufficient if we prove it for the soliton initially at
rest at the origin, i.e., for $\left<\hat{\mathbf{x}}\right>_{0}=0$ and
$\left<\hat{\mathbf{p}}\right>=0$. First, we apply the eq. (18) to
$\Psi_{0}(\mathbf{x})$:

$\frac{d\Psi_{0}(\mathbf{x})}{dt}=\left(\\!i\hbar\frac{\nabla^{2}}{2M}\\!-\\!\frac{1+i}{2\hbar}M\omega_{G}^{2}\mathbf{x}^{2}\\!+\\!(1\\!-\\!i)\sqrt{\frac{M}{2\hbar}}\omega_{G}\mathbf{x}\mathbf{w}\\!\right)\\!\\!\Psi_{0}(\mathbf{x}).$

Second, we derive $d\Psi_{0}(\mathbf{x})/dt$ from the Ansatz (V):

$\frac{d\Psi_{0}(\mathbf{x})}{dt}=\left(-\sqrt{\hbar/M}\mathbf{w}\nabla+\frac{\hbar}{2M}\nabla^{2}\right)\Psi_{0}(\mathbf{x}),$
(20)

where the second term is comes from the Ito-correction
$(\mathbf{x}\mathbf{w}dt\nabla)^{2}=\mathbf{x}^{2}\nabla^{2}dt$. We substitute
the expression

$\nabla\Psi_{0}=-(1-i)\frac{\mathbf{x}}{2\Delta\mathrm{x}^{2}}\Psi_{0}=-(1-i)\frac{\mathbf{x}}{2}\frac{\sqrt{2}M\omega_{G}}{\hbar}\Psi_{0}$

and then observe that the stochastic term coincides with that of (V). We also
substitute the expression

$\nabla^{2}\Psi_{0}=\left(-i\frac{\mathbf{x}^{2}}{2\Delta\mathrm{x}^{4}}-(1-i)\frac{1}{2\Delta\mathrm{x}^{2}}\right)\Psi_{0}$

in both eq. (V) and eq. (20). Then, after elementary algebraic steps, their
deterministic parts will also coincide.

##  VI Final remarks

Penrose also proposed the spontaneous collapse rate (10) of the Schrödinger
cat as well as the SNE (1) to generate the stationary states after the
collapse Penrose (1996, 1998, 2014). He is treating the G-related spontaneous
collapse and the SNE together. However, he has been in a holding position
regarding any concrete dynamics (like the G-SSE or others) of the collapse.

With the goal of reaching a closed linear ME to avoid the anomalies of the
SNE, ref. Nimmrichter and Hornberger (2015) proposed a formal completion of
the SNE by stochastic terms. Ref. Tilloy and Diósi (2016) showed that this
goal can be achieved by combining the SNE with the G-SSE, of course
differently from the present proposal SSNE which, contrary to refs.
Nimmrichter and Hornberger (2015); Tilloy and Diósi (2016), sacrifices the
closed ME in favor of energy-momentum conservation — at least in the free
c.o.m. motion of the macroscopic body.

The advantage of the newly proposed SSNE (16) compared to the G-SSE (7) is
this: when the state of a free massive body is collapsing towards the
localized soliton, the spontaneous gain of kinetic energy is gradually
disappearing and in the soliton states the energy-momentum conservation
becomes restored. The non-conservations by the G-SSE (and by other collapse
models) are probably warnings of infancy of the models. Yet, their related
predictions are the only testable effects currently Bassi  _et al._ (2013),
since massive Schrödinger cat states are not available in the laboratory to
date. To test the G-SSE, ref. Helou  _et al._ (2017) searched for the
predicted c.o.m. momentum diffusion in the super-precise data of Lisa
Pathfinder experiment. Work Donadi  _et al._ (2021) was hunting the
spontaneous radiation predicted by the momentum diffusion of the nuclei inside
the detector in the super-low-background Gran Sasso laboratory. Both works put
an upper bound on the strength of the G-related spontaneous collapse. If we
adopt the new SSNE, a different interpretation of the Lisa Pathfinder data is
possible. But the interpretation of the Gran Sasso data can be retained
because the new model SSNE has not removed the momentum diffusion of the
microscopic constituents but of the c.o.m. of the macro-object.

Our proposal is the first dynamical model of spontaneous collapse with partial
restoration of energy-momentum conservation at the price of typical anomalies
of semiclassical theories which might become partially masked by the collapse
mechanism. Future works should aim at more complete restoration of energy-
momentum conservation and at better understanding whether the said anomalies
would become completely neutralized.

This research was funded by the Foundational Questions Institute and Fetzer
Franklin Fund, a donor advised fund of Silicon Valley Community Foundation
(Grant No. FQXi-RFP-CPW-2008, and a mini-grant), the National Research,
Development and Innovation Office for “Frontline” Research Excellence Program
(Grant No. KKP133827), research grant (Grant. No. K12435), and the John
Templeton Foundation (Grant 62099).

## A.

To derive ${\hat{V}}_{\Psi}$ of Eq. (1) for small $\Delta\mathrm{x}$, we start
from

${\hat{V}}_{\Psi}=-GM^{2}\int\int\frac{f(\mathbf{r}-\hat{\mathbf{x}})\left<f(\mathbf{s}-\hat{\mathbf{x}})\right>d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|},$
(21)

and consider the expansions

$f(\mathbf{r}-\hat{\mathbf{x}})=\left(1-(\hat{\mathbf{x}}_{c}\nabla)+\frac{1}{2}(\hat{\mathbf{x}}_{c}\nabla)^{2}\right)f(\mathbf{r}-\left<\hat{\mathbf{x}}\right>)\left<f(\mathbf{r}-\hat{\mathbf{x}})\right>=\left(1+\frac{1}{2}(\hat{\mathbf{x}}_{c}\nabla)^{2}\right)f(\mathbf{r}-\left<\hat{\mathbf{x}}\right>)$
(22)

omitting higher order terms in $\hat{\mathbf{x}}_{c}$. Translation invariance
of ${\hat{V}}_{\Psi}$ allows us to set $\left<\hat{\mathbf{x}}\right>=0$.

$f(\mathbf{r}-\hat{\mathbf{x}})\left<f(\mathbf{s}-\hat{\mathbf{x}})\right>=\left[1-\hat{\mathbf{x}}_{c}\nabla_{r}+\frac{1}{2}(\hat{\mathbf{x}}_{c}\nabla_{r})^{2}\right]f(\mathbf{r})\left[1+\frac{1}{2}(\hat{\mathbf{x}}_{c}\nabla_{s})^{2}\right]f(\mathbf{s}).$
(23)

Rotational invariance of ${\hat{V}}_{\Psi}$ cancels the linear term and yields
the identity
$(\hat{\mathbf{x}}_{c}\nabla)^{2}=(1/3)\hat{\mathbf{x}}_{c}^{2}\Delta$, hence

$f(\mathbf{r}-\hat{\mathbf{x}})\\!\left<f(\mathbf{s}-\hat{\mathbf{x}})\right>\\!=\\!f(\mathbf{r})f(\mathbf{s})\\!+\\!\frac{1}{6}\hat{\mathbf{x}}_{c}^{2}\left[f(\mathbf{s})\Delta
f(\mathbf{r})\\!+\\!f(\mathbf{r})\Delta f(\mathbf{s})\right],$ (24)

ignoring higher orders of $\hat{\mathbf{x}}_{c}$. Using this in Eq. (21):

$\displaystyle{\hat{V}}_{\Psi}$ $\displaystyle=$ $\displaystyle-
GM^{2}\int\int\frac{f(\mathbf{r})f(\mathbf{s})d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}+\frac{GM^{2}\hat{\mathbf{x}}_{c}^{2}}{6}\int\int\frac{f(\mathbf{s})\Delta
f(\mathbf{r})+f(\mathbf{r})\Delta
f(\mathbf{s})}{|\mathbf{r}-\mathbf{s}|}d\mathbf{r}d\mathbf{s}$ (25)
$\displaystyle=$ $\displaystyle 2E_{G}+\frac{4\pi
GM^{2}}{3}\hat{\mathbf{x}}_{c}^{2}\int
f^{2}(\mathbf{r})d\mathbf{r}=2E_{G}+\frac{1}{2}M\omega_{G}^{2}\hat{\mathbf{x}}_{c}^{2},$

where partial integrations and the indetity
$\Delta|\mathbf{r}-\mathbf{s}|^{-1}=-4\pi\delta(\mathbf{r}-\mathbf{s})$ have
been used. The constant $E_{G}$ stands for the gravitational self-energy.

## B

To derive the double integral in Eq. (7) for small $\Delta\mathrm{x}$, we
start from

$\frac{GM^{2}}{2}\int\int\frac{f_{c}(\mathbf{r}-\hat{\mathbf{x}})f_{c}(\mathbf{s}-\hat{\mathbf{x}}_{c})d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|},$
(26)

and substitute the expansion

$f_{c}(\mathbf{r}-\hat{\mathbf{x}})=f(\mathbf{r}-\hat{\mathbf{x}})-\left<f(\mathbf{r}-\hat{\mathbf{x}})\right>=-(\hat{\mathbf{x}}_{c}\nabla)f(\mathbf{r}-\left<\hat{\mathbf{x}}\right>).$
(27)

Again, we can take $\left<\hat{\mathbf{x}}\right>=0$, yielding

$\frac{GM^{2}\hat{\mathbf{x}}_{c}^{2}}{6}\int\int\frac{\nabla_{r}f(\mathbf{r})\nabla_{s}f(\mathbf{s})d\mathbf{r}d\mathbf{s}}{|\mathbf{r}-\mathbf{s}|}=\frac{4\pi
GM^{2}\hat{\mathbf{x}}_{c}^{2}}{6}\int
f^{2}(\mathbf{r})d\mathbf{r}=\frac{1}{2}M\omega_{G}^{2}\hat{\mathbf{x}}_{c}^{2}.$
(28)

## C

To derive the stochastic term in Eq. (7) for small $\Delta\mathrm{x}$, we
write

$\displaystyle\frac{1}{\hbar}\int{\hat{\varrho}}_{c}(\mathbf{r})\Phi(\mathbf{r})d\mathbf{r}=\frac{M}{\hbar}\hat{\mathbf{x}}_{c}\int\nabla
f(\mathbf{r})\Phi(\mathbf{r})d\mathbf{r}$ (29)

and introduce the new stochastic variable, linear in the old stochastic field
$\Phi(\mathbf{r})$:

$\mathbf{w}=\sqrt{\frac{M}{\hbar}}\omega_{G}^{-1}\int\nabla
f(\mathbf{r}-\left<\hat{\mathbf{x}}\right>)\Phi(\mathbf{r})d\mathbf{r}.$ (30)

For the correlation function we obtain the following:

$\displaystyle\mathbf{M}\mathbf{w}_{t}\circ\mathbf{w}_{\tau}$ $\displaystyle=$
$\displaystyle\frac{M}{\hbar}\omega_{G}^{-2}\int\int(\nabla
f(\mathbf{r})\circ\nabla
f(\mathbf{s}))\mathbf{M}\Phi(\mathbf{r},t)\Phi(\mathbf{s},\tau)d\mathbf{r}d\mathbf{s}$
(31) $\displaystyle=$ $\displaystyle\frac{M}{\hbar}\omega_{G}^{-2}\int\int
f(\mathbf{r})f(\mathbf{s})\nabla_{r}\circ\nabla_{s})\frac{\hbar
G}{|\mathbf{r}-\mathbf{s}|}d\mathbf{r}d\mathbf{s}\delta(t-\tau)$
$\displaystyle=$ $\displaystyle\omega_{G}^{-2}\frac{4\pi}{3}I_{3\times
3}GM\int f^{2}(\mathbf{r})d\mathbf{r}\delta(t-\tau)=I_{3\times
3}\delta(t-\tau)$

where the $\mathbf{M}\Phi(\mathbf{r},t)\Phi(\mathbf{s},\tau)$ has been
substituted by the expression (8).

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# Peabodies of Constant Width

Isaac Arelio  ,  Luis Montejano  and  Deborah Oliveros

##  1\. Introduction

Bodies of constant width and their properties have been known for centuries.
Leonard Euler, for example, studied them in the eighteenth century under the
name of orbiforms. They have received considerable attention in popular
mathematics, in such contexts as videos, surveys, devices, and art, among
others. There is a broad, diverse body of knowledge on bodies of constant
width supported by an extensive and sophisticated theoretical framework. See,
for instance, the book _Bodies of Constant Width: An Introduction to Convex
Geometry With Applications_ by Birkhäuser, 2019 [5].

It is well known that there is a non-constructive procedure to complete a set
to a body of constant width of the same diameter, but besides the two Meissner
solids, the obvious constant width bodies of revolution, and the Meissner
polyhedra, there are only a few tangible examples in the literature of
constant width bodies that have a concrete finite procedure of construction.
The purpose of this paper is to describe a new $3$-dimensional family of
bodies of constant width that we have called peabodies, obtained from the
Reuleaux tetrahedron by replacing a small neighborhood of all six edges with
sections of an envelope of spheres. This family contains, in particular, the
two Meissner solids and a body with tetrahedral symmetry that we have called
Robert’s body, described by Patrick Roberts (see Figure 1) (we encourage the
reader to watch the animation presented at the beginning of [9] as well).

Figure 1. Robert’s body of constant width

It was a very pleasant surprise to discover that behind the construction of
this family lies the classical notion of confocal quadrics discussed, for
example, by Hilbert in his famous book _Geometry and Imagination_ [4]. This
notion has been used as early as Dupin in the nineteenth century to build
surfaces that are envelopes of spheres with surprisingly interesting
properties; see [3].

Figure 2. Dupin cyclide

In Section 2, we study confocal quadrics and prove that the distances of an
alternating sequence of four points in two confocal quadrics always satisfies
a simple equation; a result that is interesting in its own right. In Sections
3 and 4 we construct this new family of $3$-dimensional bodies and show that
they are bodies of constant width. Next, in Section 5 we analyze the
particular case of Robert’s body, showing that it has tetrahedral symmetry and
its boundary is smooth except for the $4$ vertices of the tetrahedron, in
which it has vertex singularities. Moreover, we observe that this body is not
the Minkowski sum of the two Meissner bodies of constant width by showing that
they differ in one of their sections. We also note that Robert’s body can not
be the extremum body for the Blaschke-Lebesgue conjecture about the minimum
volume among all 3-dimensional bodies of constant width. Furthermore, we show
that there is a continuous deformation along the collection of peabodies of
constant width from the most symmetric Robert’s body to the classic Meissner
bodies.

Finally, in Section 6 we point out that the construction of bodies of constant
width obtained from ball polyhedra whose singularities are self dual-graphs,
defined in [7] by Montejano and Roldan, can be adapted by replacing the
singularities of these ball polyhedra with sections of an envelope of spheres
to achieve constant width without changing the group of symmetries of the ball
polyhedra.

##  2\. Confocal quadrics

Let us consider two quadrics lying in orthogonal planes whose axes are the
same. See Figure 3. We say that these two conics are confocal if the focus of
each of them lies on the other. See Section 2.4 of the book _Geometry and the
Imagination_ by Hilbert and Cohen-Vossen [4].

Figure 3. Confocal quadrics

In the standard case, when one of the two confocal quadrics is the ellipse and
the other a hyperbola, Equation (1) gives us its planar equations.

(1) $\frac{z^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1\quad\quad\text{ and
}\quad\quad\frac{z^{2}}{a^{2}-b^{2}}-\frac{y^{2}}{b^{2}}=1.$

Similarly, in the case when the two confocal quadrics are parabolas and the
origen is the midpoint of their focus, the planar equations are given in (2).

(2) $z=\frac{x^{2}}{4a}-\frac{a}{2}\quad\quad\text{ and }\quad\quad
z=-\frac{y^{2}}{4a}+\frac{a}{2}.$

The following theorem is interesting in its own right and, as we will see in
Section 3, it is relevant to our construction of peabodies of constant width.
Throughout the paper we will abuse the notation by denoting by $ab$ both the
interval with extremes at the points $a$ and $b$, as well as its length.

######  Theorem 2.1.

Let $C_{1}$ and $C_{2}$ be two confocal quadrics. Suppose $a_{1}\in C_{1}$,
$a_{2}\in C_{2}$, $a_{3}\in C_{1}$ and $a_{4}\in C_{2}$.

  1. a)

If $C_{1}$ is connected and $a_{2},a_{4}$ lie in the same component of
$C_{2}$, then

$a_{1}a_{2}+a_{3}a_{4}=a_{2}a_{3}+a_{1}a_{4}.$

  2. b)

If $C_{1}$ is connected and $a_{2}$ and $a_{4}$ lie in different components of
$C_{2}$, then

$a_{1}a_{2}+a_{1}a_{4}=a_{2}a_{3}+a_{3}a_{4}.$

###### Proof.

_The parabolic case_. Consider the most general case of two confocal parabolas
with focus at $\pm\frac{a}{2}$, parametrized by

$C_{1}\colon\Big{(}2at,0,at^{2}-\frac{a}{2}\Big{)}\quad\quad\text{and}\quad\quad
C_{2}\colon\Big{(}0,-2as,-as^{2}+\frac{a}{2}\Big{)},$

for $t,s\in\mathbb{R}$.

If we take any point $a_{1}=(2at_{1},0,at_{1}^{2}-\frac{a}{2})\in C_{1}$ and
any point $a_{2}=(0,-2as_{2},-as_{2}^{2}+\frac{a}{2})\in C_{2}$, then

$a_{1}a_{2}=|(2at_{1},2as_{2}),-a+as_{2}^{2}+at_{1}^{2})|=\sqrt{a^{2}(1+t_{1}^{2}+s_{2}^{2})^{2}}=a+at_{1}^{2}+as_{2}^{2}.$

If we now take any point $a_{3}=(2at_{3},0,at_{3}^{2}-\frac{a}{2})\in C_{1}$
and any point $a_{4}=(0,-2as_{4},-as_{4}^{2}+\frac{a}{2})\in C_{2}$, then

$\displaystyle a_{2}a_{3}$ $\displaystyle=a+as_{2}^{2}+at_{3}^{2},$
$\displaystyle a_{3}a_{4}$ $\displaystyle=a+at_{3}^{2}+as_{4}^{2},$
$\displaystyle a_{1}a_{4}$ $\displaystyle=a+at_{1}^{2}+as_{4}^{2}.$

Consequently,

$a_{1}a_{2}+a_{3}a_{4}=a_{2}a_{3}+a_{1}a_{4}$

as we wished.

_The elliptic-hyperbolic case a)_. Let $C_{1}=E$ be the ellipse in the
$xz$-plane parametrized by $(b\cos t,0,a\sin t)$, $0<t\leq\pi$, where $a$ is
the major axis and $b$ the minor axis. Let $C_{2}=H$ be the hyperbola in the
$yz$-plane parametrized by $(0,a\tan s,-\sqrt{a^{2}-b^{2}}\sec s)$,
$\dfrac{\pi}{2}<s<\dfrac{3\pi}{2}$. Note that the ellipse $E$ has a focus at
$(0,0,\pm\sqrt{a^{2}-b^{2}})$ and the hyperbola $H$ has a focus at $(0,0,\pm
a)$.

Let $a_{1}=(b\cos t_{1},0,a\sin t_{1})\in E$ and $a_{2}=(0,a\tan
s_{2},-\sqrt{a^{2}-b^{2}}\sec s_{2})\in H$. First, let us consider the square
of the distance from $a_{1}$ to $a_{2}$:

$\left|a_{1}-a_{2}\right|^{2}=b^{2}\cos^{2}t_{1}+(a^{2}-b^{2})\sec^{2}s_{2}+2a\sqrt{a^{2}-b^{2}}\sec
s_{2}\sin t_{1}+a^{2}\sin^{2}t_{1}+b^{2}\tan^{2}s_{2}.$

Using the identities $\cos^{2}t=1-\sin^{2}t$ and $\tan^{2}s=\sec^{2}s-1$, we
get

$\displaystyle|$ $\displaystyle
a_{1}-a_{2}|^{2}=a^{2}\sec^{2}s_{2}+2a\sqrt{a^{2}-b^{2}}\sec s_{2}\sin
t_{1}+(a^{2}-b^{2})\sin^{2}t_{1},$ $\displaystyle|$ $\displaystyle
a_{1}-a_{2}|^{2}=(a\sec s_{2}+\sqrt{a^{2}-b^{2}}\sin t_{1})^{2},$ (3)
$\displaystyle a_{1}a_{2}=|(a_{1}-a_{2}|=a\sec s_{2}+\sqrt{a^{2}-b^{2}}\sin
t_{1}.$

Now let $a_{3}=(b\cos t_{3},0,a\sin t_{3})\in E$ and $a_{4}=(0,a\tan
s_{4},-\sqrt{a^{2}-b^{2}}\sec s_{4})\in H$, where $a_{2},a_{4}$ lie in the
same component of $H$. By (3),

$a_{3}a_{4}=a\sec s_{4}+\sqrt{a^{2}-b^{2}}\sin t_{3}.$

Similarly, by (3),

$\displaystyle a_{3}a_{2}$ $\displaystyle=a\sec s_{2}+\sqrt{a^{2}-b^{2}}\sin
t_{3},\text{ and }$ $\displaystyle a_{1}a_{4}$ $\displaystyle=a\sec
s_{4}+\sqrt{a^{2}-b^{2}}\sin t_{1}.$

Consequently, $a_{1}a_{2}+a_{3}a_{4}=a_{2}a_{3}+a_{1}a_{4},$ as we wished.

_The elliptic-hyperbolic case b)_. This is the case in which $a_{4}$ is in a
different component of $a_{2}$. Let us temporarily fix $a_{1},a_{3}\in
E=C_{1}$ and let $H_{1}$ and $H_{2}$ be the two components of the hyperbola
$H=C_{2}$ in such a way that $a_{2}\in H_{1}$ and $a_{4}\in H_{2}$.

Note that by a),

$a_{1}a_{2}-a_{2}a_{3}=k,\quad\mbox{ for every }a_{2}\in H_{1},$

where $k$ is a constant, and similarly,

$a_{1}a_{4}-a_{4}a_{3}=\lambda,\quad\mbox{ for every }a_{4}\in H_{2},$

where $\lambda$ is a constant.

Let us prove that $k=-\lambda$. For this purpose, let $s_{i}\in H_{i}$ be the
foci of the ellipse $E$. By the above, $a_{1}s_{1}-s_{1}a_{3}=k$ and
$a_{1}s_{2}-s_{2}a_{3}=\lambda$. Since $s_{1},s_{2}$ are the foci of $E$ and
$a_{1},a_{3}\in E$, we have that $k=-\lambda$. But now
$a_{1}a_{2}-a_{2}a_{3}=-(a_{1}a_{4}-a_{4}a_{3})$, which implies that
$a_{1}a_{2}+a_{1}a_{4}=a_{2}a_{3}+a_{3}a_{4}$, as we wished. ∎

Theorem 2.1 gives geometric sense to the intuitive idea that for two confocal
quadrics, each of them is a focal curve of the other. Indeed, Theorem 2.1 b)
simultaneously summarizes both characteristic properties of ellipses and
hyperbolas.

##  3\. Confocal Pea Pod Devices

######  Definition 3.1.

A _frame of the pea pod_ $\Sigma$ in $\mathbb{R}^{3}$ consists of two circles
in a plane $H$ whose centers lie on an _axis_ line $L$ such that their
intersection contains a chord $I$ called the _longitudinal beam_ ,
perpendicular to $L$. Let $m=L\cap I$. We will say that a _pea pod frame_ is
an _elliptic frame_ when $m$ is not between the centers of the circles, is an
_hyperbolic frame_ when $m$ is between the centers of the circles, and is a
_parabolic frame_ when one of the circles is the line through $I$.

Next, we will construct a set of elements that will make up what we call a pea
pod device, which is crucial to the construction of peabodies and Theorem 4.5.

###  3.1. Elliptic Pea Pod Device

Consider an elliptic frame of a pea pod $\Sigma_{e}$. By definition, the point
$m$ is not between the centers of the circles. Call the circle whose center
$c_{e}$ is closer to $m$ the principal circle of the elliptic frame. The other
circle is called the secondary circle.

Consider the collection of all disks contained in the symmetric difference
between the principal and secondary circles and tangent to both of them. This
collection of disks is known as a Steiner chain (see for example [8, pp.
51-54]). Observe that their centers lie in an ellipse contained in $H$ with
the focus at the principal and secondary centers. We will call this curve _the
elliptic pea string_ of $\Sigma_{e}$ and denote it by $C_{e}$; see Figure 4.

$c_{e}$$C_{e}$$I$$m$$L$

Figure 4. Elliptic pea pod device $\Sigma_{e}$. For clarity the figure shows
only few spheres (“peas”) but the reader may imagine one sphere centered at
each point of the curve.

Next, for every one of the disks in the Steiner chain and contained in the
interior of the principal circle, consider the corresponding $3$-dimensional
ball centered at the center of the disk and with the same radius. The
collection of all these balls is called the _elliptic pea pod_ devise of
$\Sigma_{e}$. The ball of the pea pod whose center lies in $L$ is called the
_bulb_.

###  3.2. Hyperbolic Pea Pod

Now consider a hyperbolic frame $\Sigma_{h}$, and recall that in this case the
point $m$ is between the centers of the circles. Here we will call the smaller
circle of the device with center at $c_{h}$ the principal circle of the frame.
Consider the collection of disks contained in the intersection between the two
circles of the frame such that they are tangent to both of them. It is easy to
see that their centers lie along an hyperbola with its focus at the centers of
the circles. We will call this curve _the hyperbolic pea string_ of
$\Sigma_{h}$ and denote it by $C_{h}$. Finally, as before, for every one of
these disks, we consider $3$-dimensional balls with the same center and the
same radius. The collection of all these balls is called the _hyperbolic pea
pod device_ of $\Sigma_{h}$. As before, the ball of the pea pod whose center
lies in $L$ is called the bulb. See Figure 5.

$c_{h}$$C_{h}$$I$$m$$L$

Figure 5. Hyperbolic pea pod device $\Sigma_{h}$.

###  3.3. Parabolic Pea Pod

In this case, $\Sigma_{p}$ consists of a frame where one of the circles is a
line through $I$ and the other, called the principal circle of the frame, has
its center at $c_{p}$. Consider the collection of disks contained in the
interior of the principal circle tangent to both the principal circle and the
longitudinal beam. Consider only those disks that are in the half plane
determined by $I$ opposite to $c_{p}$. It is easy to observe that in this case
their centers lie along a parabola contained in $H$ with focus at $c_{p}$. Let
us call this curve _the parabolic pea string_ of $\Sigma_{p}$ and denote it by
$C_{p}$. Finally, consider the $3$-dimensional balls centered at the center of
all these disks and with the same radius. The collection of all these balls is
called the _pea pod_ of $\Sigma_{p}$. The ball of the pea pod whose center
lies in $L$ is called the bulb. See Figure 6.

$c_{p}$$C_{p}$$I$$m$$L$

Figure 6. Parabolic pea pod device $\Sigma_{p}$.

For all three cases and with a slight abuse of notation, $\Sigma_{i}$ will
denote both the frame and the pea pod device.

###  3.4. Confocal Pea Pods

######  Definition 3.2.

Two pea pod devices $\Sigma_{i}$ and $\Sigma_{j}$ in $\mathbb{R}^{3}$,
$i,j\in\\{e,h,p\\}$ are _confocal_ if they satisfy:

  1. 1)

$\Sigma_{i}$ and $\Sigma_{j}$ share the same axis $L$,

  2. 2)

their corresponding planes are orthogonal, and

  3. 3)

their corresponding pea strings $C_{i}$ and $C_{j}$ are confocal quadrics.

Let $\Sigma_{i}$ and $\Sigma_{j}$ be two confocal pea pod devices,
$i,j\in\\{e,h,p\\}$. Clearly, if $i=e$ then $j=h$ (or vice versa) or if $i=p$
then $j=p$.

Let $\Sigma_{i}$ be a pea pod device $i\in\\{e,h,p\\}$ with principal circle
at center $c_{i}$ and radius $r_{i}$. Denote by $E(\Sigma_{i})$ the closed
sub-arc of its quadric pea string consisting of those points that are centers
of disks of the pea pod. For every $x\in E(\Sigma_{i})$, let us denote the
$3$-dimensional ball of the pea pod centered at $x$ by $B(x)$ and its radius
by $R(x)$.

The following theorem is our main result about confocal pea pods.

######  Theorem 3.3.

Let $\Sigma_{i}$ and $\Sigma_{j}$ be two confocal pea pod devices,
$i,j\in\\{e,h,p\\}$. Then for every $x\in E(\Sigma_{i})$ and $y\in
E(\Sigma_{j})$,

$xy+R(x)+R(y)=r_{i}+r_{j}-c_{i}c_{j}.$

###### Proof.

First, recall that $E(\Sigma_{i})$ and $E(\Sigma_{j})$, are subarcs of the
confocal pea quadric strings. Note that by construction $R(x)=r_{i}-c_{i}x$
and $R(y)=r_{j}-c_{j}y.$

Therefore,

$xy+R(x)+R(y)=\big{(}xy-c_{i}x-c_{j}y\big{)}+r_{1}+r_{2}.$

Since $\\{c_{i},x,y,c_{j}\\}$ are four alternate points in a pair of confocal
quadrics and $x$ and $c_{j}\in E(\Sigma_{i})$ and $y$ and $c_{i}\in
E(\Sigma_{j})$, then each pair lies in the same component of the corresponding
quadric. By Theorem 2.1 a), we have that

$xy+R(x)+R(y)=r_{i}+r_{j}-c_{i}c_{j},$

which is a constant independent of $x$ and $y$. ∎

######  Corollary 3.4.

Let $\Sigma_{i}$ and $\Sigma_{j}$ be two confocal pea pod devices,
$i,j\in\\{e,h,p\\}$ with $I=ab$ the longitudinal beam of $\Sigma_{i}$ and
$J=cd$ the longitudinal beam of $\Sigma_{j}$. Then

$ac=ad=bc=bd=r_{i}+r_{j}-c_{i}c_{j}.$

######  Definition 3.5.

Let $\Sigma_{i}$ and $\Sigma_{2}$ be two confocal pea pod devices,
$i,j\in\\{e,h,p\\}$. If in addition, the center $c_{i}$ of the principal
circle of one pea pod device is the center of the bulb of the other, then we
will say that they are _convex_ confocal pea pod devices. The notion of
convexity for confocal pea pod devices and the following observation will be
relevant for the proof of Lemma 4.2.

Observe that if $\Sigma_{i}$ and $\Sigma_{j}$ are convex confocal,
$i,j\in\\{e,j,p\\}$, with $I=ab$ the longitudinal beam of $\Sigma_{i}$ and
$J=cd$ the longitudinal beam of $\Sigma_{j}$, then the tetrahedron $abcd$
satisfies that $E(\Sigma_{i})\cap abcd=\\{a,b\\}$, $E(\Sigma_{j})\cap
abcd=\\{c,d\\}$, and the orthogonal projection along $L$ of $E(\Sigma_{i})$
and $I$ coincide and the orthogonal projection along $L$ of $E(\Sigma_{j})$
and $J$ coincide. (See Figure 7.)

$a$$b$$c$$d$$E(\Sigma_{e})$$E(\Sigma_{h})$$a$$b$$c$$d$$E(\Sigma_{i})$$E(\Sigma_{j})$

Figure 7. Convex confocal pea pod devices

###  3.5. The Wedge-Pod Surfaces of the Device

######  Definition 3.6.

Given a pea pod device $\Sigma$, the _realization_ of the pea pod of $\Sigma$,
denoted by $|\Sigma|$, is defined as

$|\Sigma|=\bigcup_{x\in E(\Sigma)}B(x).$

As a consequence of Theorem 3.3 and Corollary 3.4, we are able to measure the
diameter between the realizations of two confocal pea pods. Recall that for
two subsets $S,T$ of $\mathbb{R}^{3}$, the _diameter_ between $S$ and $T$ is
defined by

$d(S,T)=\operatorname{Sup}\\{ab\mid a\in S,b\in T\\}.$

######  Lemma 3.7.

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two confocal pea pod devices. Then

$d(\left|\Sigma_{1}\right|,\left|\Sigma_{2}\right|)\leq
r_{1}+r_{2}-c_{1}c_{2}.$

###### Proof.

Let $w_{i}\in|\Sigma_{i}|$, $i=1,2.$ Then, $w_{i}\in B(x_{i}),$ for some point
$x_{i}$ in the subarc $E(\Sigma_{i})$ of its pea quadric string, $i=1,2.$
Therefore, by the triangle inequality, $w_{1}w_{2}\leq
x_{1}x_{2}+R(x_{1})+R(x_{2}).$ By Theorem 3.3, $w_{1}w_{2}\leq
r_{1}+r_{2}-c_{1}c_{2}$.∎

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two confocal pea pod devices and as
before denote by $I_{1}=ab$ the longitudinal beam of $\Sigma_{1}$ and by
$I_{2}=cd$ the longitudinal beam of $\Sigma_{2}$. For every $x$ in the sub-arc
$E(\Sigma_{1})$ and $y$ in the sub-arc $E(\Sigma_{2})$, let $L(x,y)$ be the
line through $x$ and $y$. (See Figure 8.)

$L(x,y)$$\phi_{2}(x,y)$$y$$x$$\phi_{1}(x,y)$$a$$d$$c$$b$

Figure 8. The line $L(x,y)$

Denote by $\phi_{1}(x,y)$ the point in $L(x,y)$ at a distance $R(x)$ from $x$,
with $x$ between $\phi_{1}(x,y)$ and $y$ and let $\phi_{2}(x,y)$ be the point
in $L(x,y)$ at a distance $R(y)$ from $y$, with $y$ between $x$ and
$\phi_{2}(x,y)$. Note that $\phi_{1}(x,y)\in|\Sigma_{1}|$,
$\phi_{2}(x,y)\in|\Sigma_{2}|$ and
$\phi_{1}(x,y)\phi_{2}(x,y)=r_{1}+r_{2}-c_{1}c_{2}$. Therefore,
$d(|\Sigma_{1}|,|\Sigma_{2}|)=r_{1}+r_{2}-c_{1}c_{2}.$

Moreover, the interval $\phi_{1}(x,y)\phi_{2}(x,y)$ is a binormal of the
convex hull $\operatorname{conv}(|\Sigma_{1}|\cup|\Sigma_{2}|)$. Indeed, if
$x\not=\\{a,b\\}$, then there is a unique support plane of
$\operatorname{conv}(|\Sigma_{1}|\cup|\Sigma_{2}|)$ at $\phi_{1}(x,y)$
orthogonal to $\phi_{1}(x,y)\phi_{2}(x,y)$ because this is the unique tangent
plane to $B(x)$ at $\phi_{1}(x,y)$. Similarly, if $y\not=\\{c,d\\}$, then
there is a unique support plane of
$\operatorname{conv}(|\Sigma_{1}|\cup|\Sigma_{2}|)$ at $\phi_{2}(x,y)$
orthogonal to $\phi_{1}(x,y)\phi_{2}(x,y)$.

Note that $\phi_{1}(a,y)=a$ and $\phi_{1}(b,y)=b$ for every $y\in
E(\Sigma_{2})$, and similarly, $\phi_{2}(x,c)=c$ and $\phi_{2}(x,d)=d$ for
every $x\in E(\Sigma_{1}).$ In any other case,

$\phi_{i}\colon E(\Sigma_{1})\times E(\Sigma_{2})\to\mathbb{R}^{3}\text{ is an
smooth embedding.}$

Denote by $(AB)$ the image of $\phi_{1}\big{(}E(\Sigma_{1})\times
E(\Sigma_{2})\big{)}$ and by $(CD)$ the image of
$\phi_{2}\big{(}E(\Sigma_{1})\times E(\Sigma_{2})\big{)}$ and note that by the
above, $(AB)$ and $(CD)$ are smooth surfaces with boundary, smoothly embedded
in $\mathbb{R}^{3}$ and contained in the boundary of $|\Sigma_{1}|$ and
$|\Sigma_{2}|$, respectively.

On the other hand, observe that $\tilde{C}_{AB}=\\{\phi_{1}(x,c)\mid x\in
E(\Sigma_{1})\\}$ is a curve contained in the sphere of radius
$r_{1}+r_{2}-c_{1}c_{2}$ with center at $c$, connecting $a$ with $b$ (see
Corollary 3.4). Similarly, $\tilde{D}_{AB}=\\{\phi_{1}(x,d)\mid x\in
E(\Sigma_{1})$} is a curve contained in the sphere of radius
$r_{1}+r_{2}-c_{1}c_{2}$ with center at $d$, connecting $a$ with $b$. Thus,
the boundary of the surface $(AB)$ is

$\partial(AB)=\tilde{C}_{AB}\cup\tilde{D}_{AB}.$

Likewise, $\tilde{A}_{CD}=\\{\phi_{2}(a,y)\mid y\in E(\Sigma_{2})\\}$ is a
curve contained in the sphere of radius $r_{1}+r_{2}-c_{1}c_{2}$ and center at
$a$, connecting $c$ with $d$. Similarly, $\tilde{B}_{CD}=\\{\phi_{2}(b,y)\mid
y\in E(\Sigma_{2})\\}$ is a curve contained in the sphere of radius
$r_{1}+r_{2}-c_{1}c_{2}$ with center at $b$, connecting $c$ with $b$. Thus,
the boundary of the surface $(CD)$ is

$\partial(CD)=\tilde{A}_{CD}\cup\tilde{B}_{CD}.$

The two surfaces with boundaries $(AB)$ and $(CD)$ are called the _wedge-pod
surfaces_ of the confocal pair of pea pod devices $\Sigma_{1}$ and
$\Sigma_{2}$. Note that if we fix an $x=x_{0}$ while $y$ varies in
$E(\Sigma_{2})$, then $\phi_{1}(x_{0},y)$ is an arc of circle contained in
$(AB)$, while if we fix $y=y_{0}$ while $x$ varies in $E(\Sigma_{1})$, then
$\phi_{2}(x,y_{0})$ is an arc of circle contained in $(CD)$.

In summary, we have the following lemma.

######  Lemma 3.8.

Let $(AB)$ and $(CD)$ be the wedge-pod surfaces of the confocal pair of pea
pod devices $\Sigma_{1}$ and $\Sigma_{2}$. Then
$d\big{(}(AB),(CD)\big{)}=r_{1}+r_{2}-c_{1}c_{2}$. Moreover, for every point
$u\in(AB)$, there is a point $v\in(CD)$ such that $uv=r_{1}+r_{2}-c_{1}c_{2}$,
and there is a plane tangent to $(AB)$ at $u$ and a plane tangent to $(CD)$ at
$v$ orthogonal to $uv$. Furthermore, if $u\in(AB)\setminus\\{a,b\\}$, then the
tangent plane of $(AB)$ at $u$ is unique, and similarly if
$v\in(CD)\setminus\\{c,d\\}$, then the tangent plane of $(CD)$ at $v$ is
unique.

Our strategy is now to construct the boundary of a body of constant width by
assembling wedge-pod surfaces together with sphere caps. See Figures 1 and 9.

##  4\.  Assembling a Peabody of Constant Width

Throughout the rest of the paper, we will assume that $\Sigma_{1}$ and
$\Sigma_{2}$ are convex confocal pea pod devices with longitudinal beams
$I_{1}$ and $I_{2}$. Moreover, $I_{1}=ab$ and $I_{2}=cd$ are opposite sides of
a regular tetrahedron $abcd$ of length, say $2$, and consequently by Corollary
3.4, $r_{1}+r_{2}-c_{1}c_{2}=2$. Furthermore, by the convexity of the confocal
pea pod devices, $E(\Sigma_{i})\cap abcd=\\{a,b\\}$, $E(\Sigma_{j})\cap
abcd=\\{c,d\\}$, the orthogonal projections along $L$ of $E(\Sigma_{1})$ and
$ab$ coincide and the orthogonal projections along $L$ of $E(\Sigma_{2})$ and
$cd$ coincide.

Next, for every other pair of opposite sides of the regular tetrahedron
$abcd$, we will choose confocal pea pod devices, having them as longitudinal
beams, and as in the previous section we construct six wedge-pod surfaces
$(AB),(CD),(AC),(BD),(AD),(BC)$. Each of these surfaces has $\partial(AB)$,
$\partial(CD)$, $\partial(AC)$, $\partial(BD)$, $\partial(AD)$ and
$\partial(BC)$ as a boundary, with its corresponding pair of curves defined in
the previous section.

Denote by $\mathbb{S}(x,2)$ the sphere with center at $x$ and radius $2$.
Observe that

$\tilde{A}_{BC}\cup\tilde{A}_{CD}\cup\tilde{A}_{DB}\subset\mathbb{S}(a,2)$

is a simple closed curve. Let $(A)$ be the spherical cap of $\mathbb{S}(a,2)$
bounded by the curve $\tilde{A}_{BC}\cup\tilde{A}_{CD}\cup\tilde{A}_{DB}$ in
such a way that $(A)$ is contained in the boundary of the Reuleaux tetrahedron
with vertices $\\{a,b,c,d\\}$. For the definition of the Reuleaux tetrahedron,
see the paragraph preceding Lemma 4.3 and [5, Section 8.2]).

In a similar way we define the sphere cap $(B)$ of $\mathbb{S}(b,2)$ with
boundary $\tilde{B}_{CD}\cup\tilde{B}_{DA}\cup\tilde{B}_{AC}$, the sphere cap
$(C)$ of $\mathbb{S}(c,2)$ with boundary
$\tilde{C}_{AB}\cup\tilde{C}_{BD}\cup\tilde{C}_{DA}$, and the sphere cap $(D)$
of $\mathbb{S}(d,2)$ with boundary
$\tilde{D}_{AB}\cup\tilde{D}_{BC}\cup\tilde{D}_{CA}.$

Note that if $u\in\tilde{A}_{(BC)}\setminus\\{b,c\\}$, then the line through
$au$ is a unique line normal to both surfaces, $(A)$ and $(BC)$. This allows
us to glue together all these surface pieces; four sphere caps, one for each
vertex, and six wedge-shaped surfaces, called wedge-pod surfaces, one for each
side of the tetrahedron. In this way, we obtain a smooth submersion $\xi$ from
the sphere $\mathbb{S}^{2}$ into $\mathbb{R}^{3}$,

$\xi\colon\mathbb{S}^{2}\to\mathbb{R}^{3}$

with image $\xi(\mathbb{S}^{2})=\Im$. (See Figure 9.)

$d$$c$$(DC)$$(A)$$(B)$$(AC)$$(BC)$$\tilde{B}(DC)$$\tilde{A}(DC)$$\tilde{B}(AC)$$\tilde{A}(BC)$

Figure 9. Assembling a peabody

A consequence of Lemma 3.8, the following lemma emphasizes the fundamental
property of the smooth submersion, $\xi\colon\mathbb{S}^{2}\to\mathbb{R}^{3}$.

######  Lemma 4.1.

For every point $x\in\mathbb{S}^{2}$, there is a unique point
$y\in\mathbb{S}^{2}$ such that the length of $\xi(x)\xi(y)$ is $2$ and the
planes orthogonal to $\xi(x)\xi(y)$ at $\xi(x)$ and $\xi(y)$ are tangent
planes of the smooth submersion $\Im$. Furthermore, if $\xi(x)$ is different
from $\\{a,b,c,d\\}$, the tangent plane of the submersion $\Im$ at $\xi(x)$ is
unique.

###  4.1. The Diameter of the Smooth Submersion $\Im$

In this section we would like to prove that $\Im$ is the boundary of a body of
constant width. For this purpose, it is essential to prove that the diameter
of $\Im$ is $2$.

We begin by considering the following set of lines $L(x,y)$, corresponding to
the opposite sides $ab$ and $cd$ of the tetrahedron $abcd$,

$\mathcal{L}_{ab,cd}=\\{L(x,y)\mid x\in E(\Sigma_{1})\setminus\\{a,b\\}\text{
and }y\in E(\Sigma_{2})\setminus\\{c,d\\}\\}.$

Define $\mathcal{L}_{ac,bd}$ and $\mathcal{L}_{ad,bc}$ similarly, and let
$\mathcal{L}=\mathcal{L}_{ab,cd}\cup\mathcal{L}_{ac,bd}\cup\mathcal{L}_{ad,bc}$.

######  Lemma 4.2.

The three families $\mathcal{L}_{AB,CD},\mathcal{L}_{AC,BD},$ and
$\mathcal{L}_{AD,BC}$ are pairwise disjoint

###### Proof.

Let $\mho$ be the space of all lines in $\mathbb{R}^{3}$ except for those
lines parallel to a face of the tetrahedron $abcd$. We shall prove that
$\mathcal{L}_{ab,cd}$, $\mathcal{L}_{ac,bd}$ and $\mathcal{L}_{ad,bc}$ lie in
different components of $\mho$. The space $\mho$ consists precisely of those
lines with the property that the orthogonal projection along them maps the
vertices of the tetrahedron $abcd$ to four points in general position.
Therefore, $\mho$ has seven connected components and three of them correspond
to those lines with the property that the orthogonal projection along them
maps the vertices of the tetrahedron $abcd$ to four points in convex position.
Indeed, if $L_{1}$ is the line through the midpoints of $ab$ and $cd$, $L_{2}$
is the line through the midpoints of $ac$ and $bd$ and $L_{3}$ is the line
through the midpoints of $ad$ and $bd$, then one of $L_{1}$, $L_{2}$ and
$L_{3}$ lies in each of these three connected components of $\mho$.

We will next prove that $\mathcal{L}_{ab,cd}\subset\mho$. By symmetry, it is
enough to verify that $L(x,y)\in\mathcal{L}_{ab,cd}$ is not parallel to the
face $abc$ of the tetrahedron. Suppose $L(x,y)\subset\Gamma$, where $\Gamma$
is a plane parallel to the face $abc$. If the plane through $abc$ separates
$\Gamma$ from $d$, then $\Gamma$ does not intersect $E(\Sigma_{2})$, but if
the plane through $abc$ does not separate $\Gamma$ from $d$, then $\Gamma$
does not intersect $E(\Sigma_{1})$. This is so because of the convexity of the
confocal pea pod devices $\Sigma_{1}$ and $\Sigma_{2}$. If $\Gamma$ is the
plane through $abc$, then $L(x,y)\notin\mathcal{L}_{ab,cd}$, because $y=c$ and
$x$ is equal to either $a$ or to $b$. Similarly,
$\mathcal{L}_{ac,bd}\subset\mho$ and $\mathcal{L}_{ad,bc}\subset\mho$.

The lemma follows now from the fact that these three sets
$\mathcal{L}_{ab,cd}$, $\mathcal{L}_{ac,bd}$ and $\mathcal{L}_{ad,bc}$ are
connected subsets of $\mho$, and each of $L_{1}\in\mathcal{L}_{ab,cd}$,
$L_{2}\in\mathcal{L}_{ac,bd}$ and $L_{3}\in\mathcal{L}_{ac,bd}$ lies in a
different connected component of $\mho$.∎

Consider the tetrahedron $abcd$ with sides $2$ and having, at each of its
vertices, a ball of radius $2$ centered at the vertex and containing the other
three vertices on its boundary. It is well known that the intersection of
these four balls of radius $2$ is the Reuleaux tetrahedron $T$. It contains
the tetrahedron $abcd$. and the singularities of the boundary of $T$ are an
embedded copy of the complete graph $K_{4}$, where the six edges are arcs of a
circle and the vertices are $\\{a,b,c,d\\}$. Furthermore, the smooth pieces of
the boundary of $T$ consist of four spherical caps of radius $2$. See Figure
10. For more about the Reuleaux tetrahedron $T$, see [5, Section 8.2].

$a$$b$$c$$d$

Figure 10. The Reuleaux tetrahedron

######  Lemma 4.3.

$\Im$ is contained in the Reuleaux tetrahedron $T$.

###### Proof.

By symmetry, it will be enough to show that $(AB)$ is contained in the
Reuleaux tetrahedron $T$. By the proof of Lemma 3.7, it is obvious that $(AB)$
is contained in the interior of $\mathbb{S}(c,2)$ and the interior of
$\mathbb{S}(d,2)$. Then it is left to show that $(AB)$ is contained in the
interior of $\mathbb{S}(a,2)$ and in the interior of $\mathbb{S}(b,2)$ as
well. Let $w$ be the point in $(AB)$ with the property that it is the furthest
point away from $a$. Suppose $w\in B(x)$ for some $x\in E(\Sigma_{1})$. Then
the unique line normal to $(AB)$ at $w$ must contain $a$ and $x$, the center
of $B(x)$, but then $B(x)$ is the ball of radius zero centered at $b$ and
hence $w=b$. This implies that $(AB)$ is contained in the interior of
$\mathbb{S}(a,2)$. Simillarily, $(AB)$ is contained in the interior of
$\mathbb{S}(b,2)$. ∎

######  Lemma 4.4.

The diameter of $\Im$ is $2$.

###### Proof.

Let $xy$ be a diameter of $\Im$ and let $L$ be the line through $x$ and $y$.
If $x$ lies in some of the sphere caps, say $x\in(A)$, then the vertex $a$
lies in $L$. Therefore, $xa$ is a chord of the Reauleaux tetrahedron, but by
Lemma 4.3, the Reauleaux tetrahedron contains $\Im$ and consequently $xa$ is a
chord of $\Im$. This implies that $y=a$ and therefore, in this case, the
diameter of $\Im$ is $2$.

Suppose now that $x$ lies in the relative interior of one of the wedge-pod
surfaces, say, $(AB)$, and hence $L=L(x_{0},y_{0})$, for some $x_{0}\in
E(\Sigma_{1})\setminus\\{a,b\\}$ and $x=\phi_{1}(x_{0},y_{0})$. Similarly $y$
lies in the interior of another wedge-pod surface and $L$, its normal line at
$y$, must be another element of $\mathcal{L}$. By Lemma 4.2,
$L=L(x^{\prime},y^{\prime})$ for some $y^{\prime}\in
E(\Sigma_{2})\setminus\\{c,d\\}$ and $y=\phi_{2}(x^{\prime},y^{\prime})$. Now,
it is easy to verify that $L=L(x_{0},y_{0})=L(x^{\prime},y^{\prime})$ implies
that $x=x^{\prime}$ and $y=y^{\prime}$. Consequently $y=\phi_{2}(x_{0},y_{0})$
and hence the length of the diameter $xy$ is $2$, as we wished. ∎

######  Theorem 4.5.

The surface $\Im$ is the boundary of a body of constant width $2$.

###### Proof.

Consider the convex hull $\operatorname{conv}(\Im)$ of $\Im$. By Lemma 4.4,
the diameter of $\operatorname{conv}(\Im)$ is $2$. Moreover, by Lemma 4.1,
every point of $\Im$ is the extreme point of a diameter of
$\operatorname{conv}(\Im)$. This implies immediately that $\Im$ is the
boundary of $\operatorname{conv}(\Im)$. Moreover, again by Lemma 4.1, $\Im$ is
smooth with the exception of the $4$ vertices of the tetrahedron $a,b,c,d$ at
which $\Im$ has vertex singularities. Finally, [5, Theorem 3.1.7] implies that
$\operatorname{conv}(\Im)$ is a body of constant width. ∎

##  5\. Parabolic Devices and Robert’s Body of Constant Width

In this section, we will study the case of parabolic pea pod devices; that is,
when the frames of the two pea pod devices satisfy that one of the circles is
a line through $I$ and the confocal quadrics of the pea pod devices are both
parabolas. Furthermore, the spheres of the pea pods rest over the longitudinal
beams.

We begin by giving a couple of definitions. Two sets
$S,T\subset\mathbb{R}^{3}$ are _similar_ if there is a homothecy $h$ and an
isometry $\rho$ such that $\rho h(S)=T$. A tetrahedron with vertices at
$\\{a,b,c,d\\}$ is called _semi-regular_ if the line through the midpoints of
$ab$ and $cd$ is orthogonal to both $ab$ and $cd$.

######  Lemma 5.1.

Given a semi-regular tetrahedron with vertices $\\{a,b,c,d\\}$ and a pair of
confocal quadrics $C_{1},C_{2}$, there is a pair of confocal convex pea pod
devices $\Sigma_{1}$, $\Sigma_{2}$ with longitudinal beams $ab$ and $cd$ and
with convex confocal quadric pea strings similar to $C_{1},C_{2}$.

###### Proof.

Let $C_{1},C_{2}$ be a pair of confocal quadrics. Suppose without loss of
generality that $C_{1}$ lies in the $xz$-plane and $C_{2}$ in the $yz$-plane,
while the foci lie on the $z$-axis. See Figure 7. Suppose
$a^{\prime}b^{\prime}$ is a chord of $C_{1}$; that is,
$a^{\prime},b^{\prime}\in C_{1}$, and $a^{\prime}b^{\prime}$ is contained in
the $xz$-plane and furthermore, assume that $a^{\prime}b^{\prime}$ is
orthogonal to the $z$-axis. Consider the semi-regular tetrahedron
$a^{\prime}b^{\prime}c^{\prime}d^{\prime}$ similar to $abcd$. Then
$c^{\prime}d^{\prime}$ is orthogonal to the $z$-axis and is contained in the
$yz$-plane. If $a^{\prime}b^{\prime}$ is sufficiently small; that is, if
$a^{\prime}b^{\prime}$ is close to the focus of $C_{1}$, then the line through
$c^{\prime}d^{\prime}$ intersects $C_{2}$ in an interval
$c^{\prime\prime}d^{\prime\prime}$ in such a way that
$c^{\prime}d^{\prime}\subset c^{\prime\prime}d^{\prime\prime}$. On the other
hand, if $a^{\prime}b^{\prime}$ is sufficiently close to the focus of $C_{2}$,
then $c^{\prime\prime}d^{\prime\prime}\subset c^{\prime}d^{\prime}$. By
continuity, there is a position of $a^{\prime}b^{\prime}$ for which
$c^{\prime\prime}d^{\prime\prime}=c^{\prime}d^{\prime}$, as we wished. ∎

Since up to similarity there is only one pair of confocal parabolas, we have
the following corollary.

######  Corollary 5.2.

Given a semi-regular tetrahedron with vertices $\\{a,b,c,d\\}$, there is only
one pair of convex confocal pea pod devices $\Sigma_{1}$, $\Sigma_{2}$ with
longitudinal beams $ab$ and $cd$ and with confocal parabolic pea strings.

###  5.1. Robert’s Body with Tetrahedral Symmetry

The body of constant width obtained from the regular tetrahedron $abcd$ with
sides of length $2$ by always using pea pod devices with confocal parabolic
pea strings will be called the _Robert’s body_ $\mathcal{R}$ of constant
width. This body was informally described by Patrick Roberts at the web page
[9]. Observe that Robert’s body can not be the extreme body of the Blaschke-
Lebesgue conjecture because Anciaux and Guilfoyle proved in [1] that the body
that minimizes the volume among all 3-dimensional bodies of constant width has
the property that the smooth parts of its boundary are spherical caps or
surfaces of rotation, which is not the case for Robert’s body. See [5, Section
14.2]. The boundary of Robert’s body is smooth with the exception of the $4$
vertices of the tetrahedron, at which the boundary has vertex singularities.

Next we will show that Robert’s body has the symmetry of the tetrahedron
$abcd$. Using the unique pair of confocal pea pod devices $\Sigma_{1}$,
$\Sigma_{2}$ with longitudinal beams $ab$ and $cd$ and with confocal parabolic
pea strings, we obtain the wedge-pod surfaces $(AB)$ and $(CD)$, defined as in
Section 4.

Let $L$ be the axis of the devices; that is, $L$ is the line through the
midpoints of $ab$ and $cd$, and suppose the origin $O$ is the barycenter of
the tetrahedron $abcd$. Let $\rho_{\pi/2}$ and $\rho_{\pi}$ be the rotations
along $L$ by an angle of $\pi/2$ and $\pi$ respectively, and let $\tau_{1}$ be
the reflection through the plane of $\Sigma_{1}$ and $\tau_{2}$ the reflection
through the plane of $\Sigma_{2}$, and let $\alpha$ be the antipodal map.
Since the confocal parabolic pea strings are invariant under $\rho_{\pi}$,
$\alpha\rho_{\pi/2}$, $\tau_{1}$ and $\tau_{2}$, by construction, the same
holds for the wedge-pod surfaces $(AB)\cup(CD)$. Now, using the unique pair of
confocal pea pod devices with longitudinal beams $ac$ and $bd$ with confocal
parabolic pea strings, we obtain the corresponding wedge-pod surfaces $(AC)$
and $(BD)$ and in similar fashion the corresponding wedge-pod surfaces $(AD)$
and $(BC)$. By Corollary 5.2, there is an isometry that sends $(AB)$ to $(AC)$
and $(CD)$ to $(BD)$. All this implies that Robert’s body has tetrahedral
symmetry.

###  5.2. Robert’s Body and the Minkowski Sum of the Meissner Bodies

Although Robert’s body and the Minkowski sum of the two well known Meissner
bodies of constant width (see [5, Section 8]) both have constant width $2$,
both contain the tetrahedron $abcd$, both have tetrahedral symmetry, and the
boundary of both is smooth with the exception of the $4$ vertices of the
tetrahedron where they have vertex singularities, they are not the same body.
We will observe next that they are essentially different, since they differ in
one of their sections, as the following theorem shows.

######  Theorem 5.3.

The Minkowski sum of the two Meissner bodies of constant width $1$ is not the
Robert’s body $\mathcal{R}$.

###### Proof.

Let $M_{1}$ and $M_{2}$ be the two Meissner bodies of constant width $1$
constructed over the regular tetrahedron
$\\{\frac{a}{2},\frac{b}{2},\frac{c}{2},\frac{d}{2}\\}$, and let
$M=M_{1}+M_{2}$. Let $H$ be the plane that contains the side $ab$ and passes
through the midpoint of the side $cd$. Note that in both cases, reflection
about $H$ leaves $M$ invariant. This implies that $H\cap M$ has constant width
and hence that $H\cap M=(H\cap M_{1})+(H\cap M_{2})$. On the other hand, note
that $(H\cap M_{1})$ is a Reuleaux triangle whereas $(H\cap M_{2})$ is the
figure of constant width depicted in Figure 11 for some parameters
$r_{1},r_{2}>0$ and $r_{1}+r_{2}=1$. Let us call this figure the
$r_{1}$-figure of constant width $1$. The Reuleaux triangle is thus the
$1$-figure of constant width $1$. On the other hand, the section
$H\cap\mathcal{R}$ shown in the second picture of Figure 11 is also an
$r$-figure for the parameter $r>0$, where $r$ is the radius of the principal
circle of the parabolic pea pod device. It is clear now that the Minkowski sum
of the Reuleaux triangle with the $r_{2}$-figure, $r_{2}>0$, is never an
$r$-figure of constant width $1$. ∎

$a$$b$$r_{1}$$r_{2}$$a$$b$

Figure 11. The $r_{2}$-figure and a section of Robert’s body

###  5.3. Classic Meissner Bodies as Peabodies

Take $C_{e}$ and $C_{h}$, a confocal ellipse and hyperbola respectively, and
let the focus of the ellipse come closer and closer to the center. At the
limit situation, $C_{e}$ becomes a circle and $C_{h}$ a line orthogonal to the
circle through its center. This pair of curves may be considered as confocal
quadrics in our construction.

Let $abcd$ be the regular tetrahedron of side $2$. Using the unique pair of
confocal pea pod devices $\Sigma_{1}$, $\Sigma_{2}$ with longitudinal beams
$ab$ and $cd$ and with confocal pea strings, the circle and the orthogonal
line, we obtain the wedge-pod surfaces $(AB)$ and $(CD)$ as in Section 3.5. We
can see that the two circles of $\Sigma_{1}$ coincide and the center is the
midpoint of $cd$. So, in this case, the collection of disks of the pea pod of
$\Sigma_{1}$ consists of points, and hence the corresponding wedge-pod surface
$(AB)$ is, in this degenerate case, the arc of a circle with its center at the
midpoint of $cd$ from $a$ to $b$. On the other hand, the hyperbolic pea pod
device $\Sigma_{2}$ is such that its quadric pea string is the line through
$cd$ and therefore all the disks of the pea pod of $\Sigma_{2}$ have their
center at $cd$. This implies that the wedge-pod surface $(CD)$ is a surface of
revolution along $cd$. As can be seen, this is precisely the surgery procedure
given in the construction of the Meissner bodies. See [5, Section 8]. Finally,
observe that there is a continuous deformation along the collection of
peabodies of constant width from the most symmetric such body, the Robert’s
body, to the Meissner bodies. This is because, by Lemma 5.1, we can use any
pair of confocal quadrics to construct the pod surfaces $(AB)$ and $(CD)$.

##  6\. General Meissner Peabody Polyhedra

We will finish this paper by extending this construction to a more general
one. In [7], Montejano and Roldan used metric embeddings of self-dual graphs
to construct bodies of constant width, called Meissner polyhedra. Let $G$ be a
metric embedding of a self-dual graph and let $ab$ and $cd$ be a pair of dual
edges of $G$. It is easy to see that $\\{a,b\\}$ and $\\{c,d\\}$ are the
vertices of a semi-regular tetrahedron. Using Corollary 5.2, we can see that
there is only one pair of convex confocal pea pod devices $\Sigma_{1}$,
$\Sigma_{2}$ with longitudinal beams $ab$ and $cd$ and with confocal parabolic
pea strings. This allows us to obtain the wedge-pod surfaces $(AB)$ and $(CD)$
and perform surgery along two dual singularity edges of the ball polyhedra
$B(G)=\bigcup_{v\in V(G)}B(v)$. By performing this procedure for any pair of
dual edges of $G$, we can assemble, exactly as in Section 4, a peabody of
constant width $\mathcal{R}_{G}$ by replacing a small neighborhood of all
edges of $G$ with sections of an envelope of spheres as before. Note that by
the discussion in this last two sections, every symmetry of $G$ is also a
symmetry of $\mathcal{R}_{G}$.

### Acknowledgments

Luis Montejano and Deborah Oliveros acknowledge support from CONACyT under
project CONACyT 282280 and from PAPIIT-UNAM under project IG100721.

## References

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