## TL;DR
Why does toast tend to land butter-side down? Matthews...
Robert A. J. Matthews is a British physicist, science writer, and p...
Murphy’s Law has its origins in a series of Air Force experiments b...
#### Re-deriving Matthews' “critical‑height” estimate
Matthews m...
If we lived on Mars ( g ≈ 0.38 g ) the critical table height triple...
Would be interesting to replace a toast with a smartphone (similar ...
Einstein is said to have written this line in a letter to Cornelius...
I
Tumbling toast,
Murphy’s
Law
and
I
I
the fundamental constants
Robert
A
J
Matthews
Department of Applied Mathematics and Computer Science, University of Aston, Birmingham B4 7ET
UKt
Received
20
February 1995, in final form
31
March 1995
Abstrmt.
We
investigate the dynamics of
toast
tumbling
from
a
table
to
the floor. Popular opinion
is
that the
6nal
state
is
usually butter-sidedown, andconstitutesprimn
fncie
evidence of Murphy’s Law
(‘If
it
can
go wrong,
it will?.
The
orthodox view,
in
contrast,
is
Lhat
the phenomenon is
essentially random,
with
a
50/50
split of possible outcomes.
We
show
that toast
does
indeed have
an
inherent tendency to
land butter-side down for a wide range
of
conditions
Furthermore, we show that
this
outcome
is ultimately
ascribable to the values of the fundamental constants.
As
such,
this
manifestation
of
Murphy’s Law appears to
be
an
ineluctable feature
of
our
universe.
1.
Introduction
The term Murphy’s Law has its
origins
in dynamical
experiments conducted by the
US
Air
Force in the late
1940s involving an eponymous
USAF
captain
[I].
At
its heart lies the concept that
‘if
something can go
wrong, it will’;
this
has its analogues in many other
cultures
[2],
and is almost certainly of much older
provenance.
The phenomenon of toast falling from
a
table to
land butter-side down
on
the
floor
is popularly held
to
be
empirical proof of the existence of Murphy’s
Law. Furthermore, there is a widespread belief
that it
is
the
result
of
a
genuine physical effect,
often
ascribed to
a
dynamical asymmetry induced by one
side of the toast being buttered.
Quite apart from whether
or
not the basic obser-
vation is true, this explanation cannot be correct.
The mass of butter added to toast (~4g) is small
compared to the
mass
of the typical slice of
toast
(-35g), is spread
thinly,
and passes into the body
of the toast. Its contribution to the total moment
of
inertia of the toast-and
thus
its effect
on
the toast’s
rotational dynamics-is thus negligible.
?Address
for
correspondence:
50
Nomys
Road,
Cumnor,
Oxford,
OX2
9FT
VK;
mail
lM)265.3005@compuxrfe.com
R&umb
Nous
examinons
la
dynamique
du
toast dans sa
chute de la table au plancer. L‘avis populaire tient
ce
que le
toast tombe babituellement c6&
beurri
par
terre
et
que
cela
constitute
le
commencement de preuve de la
loi
de Murphy
(loi de
la
guigne
maxi”).
En
revanche, I’avis orthodoxe
insiste
qui’il s’agit
d‘un
phhom6ne essentiellement
dG
au
hasard, dont
les
rhltats possibles
se
divisent
SOjSO.
Nous
montrons
quele toast a,
en
eret, une
tendance
fondamentale
a
amver cdti beud par terre
dans
des circonstances
diverses
et varik. De plus,
nous
montrons que
ce
r’esultat s’attribue
en
dernike analyse aux
valeun
des constantes
fondamentales.
En
tant
que
tel, cet exemple de la loi de
Murphy semblerait etre
une
caractkistique inkluctable de
notre
univen.
Similarly, the aerodynamic effect of the thin layer
of butter cannot contribute a significant dynamical
asymmetry. It
is
easily shown
that
for air resistance
to
contribute significantly to the dynamics of the
falling toast, the height of fall must be of the order
of
2(pr/pA)d,
where
pr
is the density
of
the toast,
d
is
its thickness and
pA
the density of air. The presence
of butter will contribute only
a
small fraction of
this
total; supposing it to be a generous
25
per cent
and taking the typical values of
pr
N
350
kgm-3,
PA
=
1.3
kgm-’ and
d
-
W2m,
we find that the
toast would have to fall from
a
height over an order
of magnitude higher than the typical table for the
butter
to
have significant aerodynamic effects.
Such estimates lend credibility to the widespread
‘orthodox’ answer
to
the tumbling toast question:
that it is essentially
a
coin-tossing process in which
only the bad outcomes are remembered. Indeed,
there is some experimental evidence
to
support
this.
In
tests conducted for
a
BBC-TV programme
on
Murphy’s
Law
[I], buttered bread was tossed into
the air
300
times in a variety of situations designed
to
reveal the presence of Murphy’s Law.
In
all
tests, the results were statistically indistinguishable
from the
50/50
outcome expected from random coin-
tossing, suggesting that selective memory is the true
explanation of Murphy’s Law.
Tumbling
toast,
Murphy’s
Law
and
the
fundamental
constants
~
173
There are, however, two problems with this. First,
by its very nature Murphy’s Law might contrive to
ruin
any overt attempt to demonstrate its existence
by such probabilistic means.
This
would make experi-
mental verification of its existence very problematic.
A
simple Bayesian probability analysis shows that
there are grave ditficulties with attempts to demon-
strate Murphy’s Law if it is considered to be
a
skewing of an otherwise symmetric probability distri-
bution in the direction of an unfavourable outcome.
Second, and more seriously, Murphy’s Law may be
far more fundamental than a skewing of probability
distributions: it may actually forbid certain favour-
able outcomes from taking place. In the case
of
falling toast,
this
implies that Murphy’s Law might
influence the dynamics of the toast at a fundamental
yet subtle level. If
so,
failure to reveal its presence
by carelessly hurling toast randomly into the
air
would hardly be surprising.
As
we now show, the dynamics of falling toast are
indeed rather subtle, and do depend fairly critically
on initial conditions. Nevertheless,
in
a broad range
of realistic circumstances, the dynamics do lead to a
bias towards
a
butter-side down final state. We pro-
vide both theoretical and experimental evidence for
this conclusion and show that the results have surpris-
ingly deep origins. Specifically, we show that the fall
of toast is a manifestation of fundamental aspects
of
the nature of
ow
universe.
2.
Dynamics
of
falling toast
In what follows we model the tumbling toast problem
as an example of a rigid, rough, homogeneous rectan-
gular lamina,
mass
m,
side
2a,
falling from a rigid
platform set a height
h
above the ground. We con-
sider the dynamics of the toast from an initial state
where its centre of gravity overhangs the table by a
distance
a,,,
as shown in figure
1.
Initially, we ignore
the process by which the toast arrives at this state,
Flgure
1.
The
initial
orientation
of
the
rotating
toast
t’l
i.
,
,
,
,
,
,
,
,
,
,
,
and also assume that it
has
zero horizontal velocity;
the important effect of a non-zero horizontal velocity
is addressed later. Finally we assume a perfectly
inelastic impact with the
floor
with zero rebound.
With these assumptions, the dynamics of the
lamina
are
determined by the forces shown
in
figure
1:
the weight,
mg,
acting vertically downward, the
frictional force,
F,
parallel to the plane of the
lamina and directed against the motion, and the reac-
tion of the table,
R.
The resulting angular velocity
about the point of contact,
w,
then satisfies the
differential equations of motion
m6w
=
R
-
mg
-cos0
m6w2
=
F
-
mg.sin8
m(k2
+
62)G
=
-mg6.
cos0
(1)
(2)
(3)
where
k
is the appropriate radius
of
gyration, such
that
k2
=
d/3
for the rectangular lamina considered
here. Multiplying
(3)
by
2w
and integrating from the
initial conditions
w
=
0
at
0
=
0
leads to:
w2
=(6g/a).[q/(1+3q2)].sin0
(4)
where we have used
6
qa, with
q
(0
<
q
<
1) being
the ‘overhang parameter’. Equation
(4)
is the central
equation of the tumbling toast problem, as it gives
the rate of rotation of the toast once it has detached
from the table from a speciEc state of overhang.
Unless the toast can complete su5cient rotation
on
its descent to the
floor
to bring the buttered
side facing upwards, the toast
will
land buttered-
side down.
Thus
if the toast begins its descent at
an angle
#
to the horizontal, then for it to land
butter-side up again we must have
where
w,
is the freefall
6
rotation rate and
T
the free-
fall time for the height of the table
h,
so
that
w0r
>
(3r/2)
-
4
(5)
T
=
[2(h
-
2a)/~]”~
(6)
The frictional force acting on the lamina
will
prevent
detachment until the lamina
has
rotated through at
least an angle
4,
at which point slipping
occm.
This
minimum value of
r$
follows from the
usual
condition
F
=
pR,
where
p
is the coefficient
of
static
friction between the lamina and the table edge.
From (l),
(2)
and
(4)
we find
4>
arctanIp/(l
+9v2)1
(7)
To
calculate the free-falling angular rotation rate
w,,
we must deal with the post-slipping regime. At the
instant
of
slipping, the centre of rotation of the
lamina
is
a distance
aq
from the centre of gravity,
and the rotational rate is given by
(4).
A
point on
the
shorter,
non-overhanging section of
lamina
at a
distance a(q+
E),
0
<
E
<<
1
from the
CG
will thus
have a rotationally-induced horizontal component
of velocity aew.
sin
#
away
from
the table. Slipping
will
bring
this
point
vertically
over
the
table edge,
so
174
R
A
J
Matthews
that contact between table and toast is broken, the
latter then tumbling about
its
CG at a rotational
rate
U,
essentially unchanged from the original
value. Although irrezularitv
in
the surface of the
This
was found to be
For
bread
[qo],,hs
-
0.02
For toast:
IdAk
-0.015
(12)
,
toast can prevent Gediate post-slip detachment,
confirm that the value
of
w,
can
be
taken
as that induced by the initial overhang torque
of
mgoq,,.
Thus the free-falling toast rotates at a rate
Both bread and toast are thus relatively unstable
to
tumbling from overhanging positions. Crucially,
neither can sustain overhangs anywhere near as
large as the critical value given in (10).
This
implies
that laminae with either composition do not have
sufficient angular rotation
to
land butter-side up
d
=
(6g/Q)'[%/(1+3d)]~n$
where the value of the critical overhang parameter
q,,
and slip angle
$
at which detachment takes place may
be determined experimentally. To place a lower limit
on the overhang needed to avoid
a
butter-side down
final
state, we insert
(8)
in
(5),
set
$
=
a/2
and solve
the resulting quadratic equation for
qo:
where
a
=
7?/12(R
-
2)
and R
=
h/Q
For conventional tables and slices of toasts, we have
h
-
75cm,
ZQ
-
lOcm leading to
R
-
15,
a
-
0.06
and thus a lower limit on the critical overhang
parameter of
qo
>
0.06 (10)
if the toast is to complete sufficient rotation to avoid a
butter-side down
final
state.
3.
Experimental results and implications
An experimental determination of
qo
holds the key to
establishing whether or not the fall of toast constitutes
a manifestation of Murphy's Law. Tests were carried
out using a lamina derived from a standard white loaf
(supplied by Michael Cain
&
Co.,
Oxford Road,
Cumnor, Oxford). The lamina was cut into a rectan-
gle
of
lOcm
x
7.3cm
x
1.5cm
(so
that
2~
=
IOcm),
and placed
on
a
rigid flat and level platform
of
kitchen Contiboard, used to model the surface of a
clean, uncovered table.
Measurements of the value of the coefficient of
static friction
p
between the lamina and the platfom
were made by measuring the angle of the platform at
which sliding just began; the tangent of
this
angle is
then equal to
p.
Test were carriedout
on
both bread
and toast, leading to
For bread:
[pjOb,
-
0.29
For
toast:
[plobs
N
0.25
(11)
Measurements of the value of the critical overhang
parameter
qo
were then made by placing the lamina
over the edge of the Contiboard and determining
the least amount of overhang of the
2a
=
lOcm
edge at which detachment and free-fall took place.
following free-fall from a table-top. In other words,
the material properties of slices of toast and bread
and their size relative
to
the height of the typical
table are such that, in the absence of any rebound
phenomena, they lead to a distinct bias towards a
butter-side down landing. But before this can be
taken as confirmation of popular belief, however,
some practical issues must be addressed.
4.
The effects
of
non-zero horizontal velocity
So
far, we have ignored the means by which the toast
comes to be in the overhang condition shown in figure
1. This
is
clearly of practical importance, however, as
the toast will typically leave the table as the result
of
sliding
off
a tilted plate, or being struck by a hand
or arm. The consequent horizontal velocity may
dominate the dynamics
if
the gravitational torque
has insacient time to induce signiticant rotation.
In
this case, the toast will behave like a simple
projectile
off
the edge of the table, keeping its
butter-side up throughout the flight.
This
raises the
possibility that, while dynamically valid, the butter-
side down phenomenon may only
be
witnessed for
an
infeasibly small range
of
horizontal velocities. To
investigate
this
range, we fmt note that the time for
an
initially horizontal lamina of overhang parameter
q
to acquire inclination
$
follows from
(8):
r($)
=
[Q(l
+3d)/'%%l1''~($)
(13)
where
=
2$'/2
for small
$
(14)
If the lamina has a horizontal velocity
VH
as
it
goes
over the edge
of
the table, the time during which it
is susceptible to torque-induced rotation is
-Q/VH.
During
this time its average overhang parameter
qo
will
he of the order
0.5,
and it will acquire a down-
ward
tilt
through the torque of order
$.
If
this
angle is small, the dynamics of the lamina can be
considered those of a projectile. By
(13)
and the
smaU
angle approximation in (14), this implies that
the effects of torque-induced rotation, and thus
tumbling motion,
will
be negligible for horizontal
Tumbling
toast,
Murphy’s
Law
and
the
fundamental constants
~
175
velocities above about
v,
-
(3gu/7$)”2
-
1.6ms-’ (with
$
-
5’)
(15)
At
speeds considerably below this value (below, say,
VH/S
-
3SOmms-’)
the torque-induced rotation
should still dominate the dynamics of the falling
toast, and the butter-side down phenomenon should
still be observed.
This
conclusion is supported by
observation. Furthermore, the relatively higb value
of
VH
ensures that the butter-side down phenomenon
will
be observed for
a
wide range of realistic launch
scenarios, such as
a
swipe of the hand
or
sliding
off
an inclined plate (which, by (11),
will
have to be
tilted downward
by
at least
-
arctan(0.25)
-
14”).
It therefore appears that the popular view that toast
falling
off
a table has an inherent tendency to land
butter-side down is based in dynamical fact.
As
we
now show, however,
this
basic result has surprisingly
deep roots.
5.
Tumbling toast and the fundamental
interactions
We have seen that the outcome
of
the fall
of
toast
from a table
is
dictated by two parameters: the
surface properties of the toast, which determine
qo,
and the relative dimensions of the toast and table,
which determine
R.
The latter is, of course, ulti-
mately dictated by the size of
humans.
Using an
anthropic argument, Press
[3],
has
revealed an
intriguing connection between the typical height of
humans and the fundamental constants of nature. It
centres
on
the fact that bipedal organisms like
humans are intrinsically less stable than quadrupeds
(e.g. giraffes), and are more at risk of death by
toppling. This leads to
a
height limitation
on
humans set by the requirement that the kinetic
energy injected into the head by a fall will be insuff-
cient to canse major structural failure and death.
This height limitation
on
humans in turn implies
a
limit
on
the height
of
tables. We now deduce this
limit
using
an anthropic argument similar to that
of
Press.
We
begin
by considering
a
humanoid organism to
be a cylindrical mass of polymeric material of height
LH
whose critical component is
a
spherical
mass
Mc
(the head) positioned at the top of the body. Then,
by Press’s criterion, the maximum size of such an
object is such that
f
.(Mcv&/~)
<NEB
(16)
where
urd
-
is the fall velocity, f(-O.l)
is
the fraction
of
kinetic energy that goes into breaking
N
polymeric bonds of binding energy
EB.
and the
fracture is assumed to take place across
a
polymer
plane
n(-
100) atoms thick,
so
that
N
N
n(Mc/fnp)2/3
(17)
LH
-
(n/f
)(Mc/~P)~’~ .EdMcg
EB
-
qa2m,c2
(19)
where
mp
the
mass
of the proton.
Thus
the height of
the humanoid will be of the order
(18)
A
simple Bohr-atom model shows that
where
01
is the electronic fine structure constant,
me
is
the mass of the electron,
c
the speed of light, and q for
polymeric materials is
-3
x
The acceleration
due to gravity,
g,
for
a
planet can also be estimated
from 6rst principles, using an argument based
on
balancing internal gravitational
forces
with
those
due to electrostatic and electron degeneracy effects
[4].
This leads to
g
-
(4.G/3~~)(a/a~)”~m~/aga
(20)
where p(-6) is the radius of the polymeric atoms
in units of the Bohr radius
ao.
and
aG
is the
gravitational fine structure constant
Gmg/Ac.
We
also have
Mc
-
4rRzp0/3
(21)
where
Rc
is the radius of the critical component
(-LH/20)
and
po
is the atomic mass density
p0
-
A~,/(PU,)’
(22)
where
A(-
100) is the atomic mass of the polymeric
material. Substituting these relations into our
original criterion for
LH
gives, alter some reduction,
(23)
L~
<
K
.
(a/aC)’/J.
a.
where
K
(3nq/f)‘/’p2A-’’‘
-
50
Inserting the various values, we find that
this
6rst-
principles argument leads
to
a
maximum safe height
for human of around
3
metres. Although the estimate
of
LH
is
pretty rough and ready, its weak dependency
on
the uncertainties in the various factors in
(23)
makes it fairly robust. The resulting limit has a
number
of
interesting features. The estimate of its
value agrees well with the observation that a fall
onto the
skull
from
a
height of
3
m is very likely to
lead to death; interestingly, even the tallest-ever
human,
Robert
Wadlow (1918-1940), was-at
2.72m-within this bound. The limit
on
height
is
also universal, in that it applies to all organism
with human-like articulation
on
any planet. Most
importantly, however, it puts an upper lit
on
the
height of
a
table used by such organisms: around
LH/2,
or 1.5m.
This
is
about twice the height of
tables used by humans, but still
only
half that
needed to avoid a butter-side down ha1 state
for
176
RAJ
Matthews
tumbling toast: rearranging
(9)
we fmd
(24)
and inserting the observed value
1)
N
0.015
given
in (12) leads to
R
-
60
and
h
-
3
metres. The limit
(23)
thus implies that
all
human-like organisms
are doomed to experience tumbling toast landing
butter-side down.
6.
Conclusions
Our
principal conclusion
is
a surprising one, given
the apparently quotidian nature of the original
phenomenon:
all
human-like organisms are destined
to experience the ‘tumbling toast’ manifestation
of Murphy’s Law
because
of the values
of
the
fundamental constants in our universe.
As
such, we
have probably confirmed the suspicions of many
regarding the innate cussedness
of
the universe. We
therefore feel we must conclude this investigation
on
a more optimistic note. What can human-like-and
thus presumably intelligent-organisms do to avoid
toast landing butter-side down?
Building tables
of
the -3m height demanded by
(24)
is
clearly impracticable. Reducing the
size
of
toast
is
dynamically equivalent, but the required
reduction
in
size
(down
to squares
-2.5cm
across)
is
also
nnsatisfactoq.
The best approach is somewhat counter-intuitive:
toast
seen
heading
off
the table should be given a
smart swipe forward with the hand. Similarly, a
plate
OK
which toast is sliding should
be
moved
swiftly downwards
and
backwards, disconnecting
the toast from the plate. Both actions have the effect
of minimising the amount of time the toast is exposed
to the gravitationally-induced toque, either by giving
the toast a large (relative) horizontal velocity or by
sudden disconnection
of
the point of contact.
In
both cases, the toast
will
descend to the floor keeping
the butter side uppermost.
We end by noting that, according to Einstein, God
is subtle, but He
is
not malicious. That may be
so,
but
His
inhence
on
falling toast clearly leaves much
to
be
desired.
Acknowledgements
It is a pleasure to
thank
Professor Ian Fells and Robin
Bootle for providing background
on
Murphy’s
Law.
References
[l]
Bootle
Rand
FeUs
I
1991
QED:
Murphy>
Luw
121
Bootle
R
1995
personal
wmmuniation
[3]
Press
W H
1980
Am.
J.
Phys.
48 597-8
141
Davies
P
C
W
1982
The
Accidental
Universe
@ondon,
BBC)
(Cambridge:
Cambridge
University
Press)
44-9
Einstein is said to have written this line in a letter to Cornelius Lanczos (a Hungarian physicist and mathematician), around 1942. However, exact attributions vary, and it also appears in recollections of conversations (e.g., by Abraham Pais, in Subtle is the Lord...)
Einstein is believed to have said this while discussing the mysteries of nature and the universe, especially in relation to quantum mechanics. He was expressing the idea that the universe, while deeply complex and perhaps not easily understood ("subtle"), is not inherently deceptive or malevolent. It reflects his belief in an underlying order and rationality in nature.
If we lived on Mars ( g ≈ 0.38 g ) the critical table height triples; Martian colonists with Earth‑size kitchen furniture would less often experience butter‑side‑down toast!
Robert A. J. Matthews is a British physicist, science writer, and professor known for blending rigorous research with playful curiosity. He gained international attention and the 1996 Ig Nobel Prize in Physics for showing that toast really does tend to land butter-side down, thanks to the laws of physics and not just bad luck.
A fellow of several scientific societies, Matthews has written widely for publications like New Scientist, The Times, and BBC Focus, and currently serves as a visiting professor at Aston University. His work often explores probability, statistics, and the surprising patterns hidden in everyday life.
Would be interesting to replace a toast with a smartphone (similar aspect ratio, much higher density and friction) and analyze the results.
Murphy’s Law has its origins in a series of Air Force experiments back in the late 1940s, when engineers were trying to understand how much force the human body could handle during rapid deceleration - essentially, what happens to pilots in a crash.
Captain Edward Murphy, an engineer working on the project, got frustrated after discovering that a technician had wired some sensors incorrectly during a test. He supposedly muttered something along the lines of, "If there’s a way to mess it up, he’ll find it" The team started repeating the line as a kind of inside joke, and before long it morphed into the now-famous saying: "Anything that can go wrong, will go wrong"
## TL;DR
Why does toast tend to land butter-side down? Matthews shows it’s not just bad luck or selective memory, it’s physics.
Toast typically begins to rotate as it tips off a table, but the height of most tables (~75 cm) is just enough for it to rotate about half a turn, landing butter-side down. This isn’t due to butter's weight or aerodynamics, which are negligible. Rather, it's about the torque and time during the fall.
In this paper, Matthews builds a detailed dynamical model of the toast’s tumble and shows experimentally that under realistic conditions (with little horizontal velocity), there's a built-in bias toward a butter-side down landing.
Surprisingly, this bias is linked to fundamental constants: table height is constrained by human height, which in turn is limited by biomechanical stability and molecular bond strength (à la Press, 1980). Given the values of fundamental constants, the result is universal - all intelligent, human-like beings are doomed to drop toast butter-side down.
To avoid this fate? A little push adds horizontal velocity, reducing rotational torque, a counterintuitive fix but backed by physics.
#### Re-deriving Matthews' “critical‑height” estimate
Matthews mentions that air resistance becomes dynamically relevant only if the toast falls a height
$$
h_{\text{crit}} \sim 2\left(\frac{\rho_r}{\rho_A}\right)d,
$$
where $\rho_r$ is the density of toast, $\rho_A$ is the density of air, and $d$ is the slice thickness. The derivation proceeds as follows.
#### Model the forces
| Symbol | Meaning |
|--------|---------|
| $m = \rho_r A d$ | mass of a rectangular slice (planform area $A$) |
| $W = mg$ | weight of the slice |
| $F_D = \tfrac{1}{2} C_D \rho_A A v^2$ | quadratic drag on a broad face (thin plate) |
Drag becomes important once it rivals the weight:
$$
F_D \gtrsim W
\quad \Longrightarrow \quad
\tfrac{1}{2} C_D \rho_A A v_c^2 \approx \rho_r A d\,g.
$$
Solving for the *critical velocity*:
$$
v_c^2 \approx \frac{2\rho_r d\,g}{C_D \rho_A}. \tag{1}
$$
Ignoring drag up to height $h$, pure free fall gives:
$$
v^2 = 2 g h.
$$
Setting $v = v_c$ from (1) yields:
$$
2 g h_{\text{crit}} = \frac{2\rho_r d\,g}{C_D \rho_A}
\quad \Longrightarrow \quad
h_{\text{crit}} = \frac{\rho_r d}{C_D \rho_A}. \tag{2}
$$
Wind-tunnel data for thin plates give $C_D \approx 0.5$. Substituting into (2) doubles the right-hand side:
$$
h_{\text{crit}} \approx 2\left(\frac{\rho_r}{\rho_A}\right)d,
$$
the expression quoted by Matthews.
For typical kitchen values:
$$
\rho_r = 350\;\text{kg\,m}^{-3}, \quad
\rho_A = 1.3\;\text{kg\,m}^{-3}, \quad
d = 0.015\;\text{m},
$$
one finds:
$$
h_{\text{crit}} \approx 2 \times \frac{350}{1.3} \times 0.015 \approx 8.1\;\text{m}.
$$
A standard kitchen table ($\approx 0.75$ m) is an order of magnitude shorter, confirming that aerodynamic forces can be ignored for everyday toast drops.