Bean There, Done That: Computer-Assisted Design of Bean Sculptures
Dave Pagurek1,∗ and Milica Banic2
1

2

Bean Futurists Society
Institute for Advanced Legume Research,
Bohnestrasse 3000, Munich, Germany(Dated: March 23, 2024)

Chicago’s status as a world-class city is cemented by its iconic bean sculpture. Other cities,
wanting to replicate the success, have muddied the bean waters by introducing their own beanvariations: New York City has a bean sharing similar properties, and Ottawa has a sphere, dubbed
the “Ottawa bean” by locals. Our economic analysis proves their worth, so naturally other cities
will want their own. We present a mathematical model of the space of all bean sculptures, and analgorithm to help cities replace existing landmarks with beans.

FIG. 1: Michaelangelo’s David improved by being turned into a bean sculpture (photo by Wikimedia Commons
Korido, CC BY-SA 4.0)
I.

INTRODUCTION

Public art is important: It can serve as an expression ofculture, heritage, and creativity within a community. It
has the power to stimulate dialogue and provoke thought.
It is a nice thing to go look at with one’s friends§ , or towalk past on one’s way to work. On top of this, successful public art installations can contribute to the economic
vitality of a city by attracting tourists, fostering a senseof place, and enhancing the overall appeal of urban environments. Indeed, successful public art can become an
icon for its host city, putting it on the map in a way thatis not otherwise possible.
Consider as an example “The Bean” (known to nerds
and um ackchuyually types as “Cloud Gate”), a sculpture
by artist Anish Kapoor that was unveiled in Chicago in2006 [1]. Almost 20 years after its construction, the Beanstands as a symbol of Chicago and a success story of∗ davepagurek@gmail.com§ This is true even if one’s friends misrepresent the nature of the

public art to which one is being led.

public art. Having observed the Bean’s impact (perhaps
even feeling threatened by it) New York City, a city withno dearth of art and culture, commissioned another bean
structure from Kapoor, which was completed in 2023.
Naturally one begins to suspect that we are seeing the
beginnings of a revolution within the art world and public
life more broadly.
As the saying goes, twice is a coincidence, but threetimes is a pattern: Recently, a bean-like structure
has been rediscovered by locals in Ottawa, Ontario.
The piece was built in 1966 and originally called “TheSphere” [10], but it has recently enjoyed a surge in
popularity since its rebranding as “The Ottawa Bean”.
Though less curvy than Chicago’s, Ottawa bean is still abean. In fact, it is the simplest bean (that is, the trivial
bean), the result of taking away all possible bends anddimples.
The growing success of this third reflective bean cements the potential of bean structures to revolutionize
public art. It is important to note that the Ottawa Bean2II.

BACKGROUND

A.

FIG. 2: Beans in Chicago, New York City, and Ottawa,
respectively. NYC photo ©Bracht Bug, CC
BY-NC-ND 2.0.

is not by any particularly famous artist† . This point solidifies the intuition that the reflectivity and bean morphology are the key factors in the beans’ appeal, rather
than some arcane feature of Anish Kapoor’s style in particular. This begs the question: How far can we takethis? Can other communities emulate the success of thesethree beans? And how can one create a bean for one’s
own city?
One possibility would be to commission artists, but
communities may be deterred from this option due toartists’ sometimes difficult personalities, and their insistence on being paid for their work. A more pragmatic
option would be to automate the design of the beans;this is a realistic possibility, due to the beans’ simple
forms. Although one could randomize the bean parameters, this kind of approach could be criticized for not
actually being art, due to a lack of inspiration or underlying meaning. We propose an approach in which beans
are designed based on other objects; in particular, here
we base our designs off iconic landmarks, with the intention of eventually replacing them with beans to maximize
the impact of the new beans.
In Section II we present some general background to
motivate the construction of more bean sculptures. In
Section III we outline a mathematical model for beansthat can neatly describe existing beans, but is flexible
enough to accommodate a variety of new beans in a similar style. We accomplish this by formulating beans as asigned distance function for the smooth union of one ormore quadratic Bézier curves. The existing and agreedupon bean sculptures fit nicely into the parameter space(“bean-space”) as single curves, while compound beans
can reproduce structures visually similar to other city
landmarks while still appearing definitively like a bean.
To help along the bean-hopeful, in Section IV we provide
an algorithm to automatically fit existing city structures
into their nearest bean-space equivalent. In Section V weapply the model to some examples, and we conclude in
Section VI.

† The authors mean no offense to Art Price, progenitor of the Ot-

tawa Bean, but he does not have a Wikipedia page and that’s justa fact.

Why Beans?

We begin with a brief economic argument for prioritizing the construction of bean sculptures. The first
notable bean, the one in Chicago, provides ample reason for other cities to want to follow suit. It is situated in Millennium Park. Prior to construction of thebean, the park had a total of zero annual visitors. After construction began, the city estimated it had 5 million annual visitors. [5] That’s an incalculable increase
from 0. Figure 3 shows how, in literature, references toMillennium Park went through the roof after construction of the bean began. In the first half of 2016, thecity counted 12,859,360 visitors—approximately 26 million visitors annually—making it the “#1 attraction in
the Midwest and among the top 10 most-visited sites in
the U.S.” [5]That’s a lot of visitors, but how does it stack up against
other individual attractions? As a point of comparison,
we look at Ottawa’s Parliament Hill. In a 2007 report, it
had just 3 million annual visitors. [4] A recent restoration
project for Parliament’s Centre Block is budgeted as a
$4.5-5 billion project [8]. Chicago’s bean, meanwhile,costed just $23 million. [2]We crunched the visitor numbers, and came to the following conclusion:
5,000,000>1
3,000,000(1)

We also crunched the numbers for the cost:$23,000,000<1$4,500,000,000(2)

The natural conclusion that Ottawa would come to isthat it would be more financially responsible to replace
Parliament with a bean. As it is only a matter of time
before other cities come to this conclusion, too, we provide an algorithm to replace something like Parliament
with a bean for minimal disruption to the existing space.
B.

Why Replace Landmarks?

We understand that cities may feel incentivized to simply build a new bean in a new location rather than replace
existing landmarks. The main argument against this isan environmental one. A single bean will likely not satiate cities. If cities were to keep expanding every timethey want a new bean, it would contribute to an unprecedented level of urban sprawl that is simply irresponsible
in our present environmental crisis.
The other main reason to replace existing landmarks
instead is because leaving landmarks in place impedes
inevitable progress. We know that everything is chromein the future [9] and replacing landmarks with beans is aclear path to that future.

3

FIG. 4: Examples of different quadratic Bézier segmentswith our bend constraints.

FIG. 3: Google Ngram Viewer data on references toMillennium Park in literature. Interest in MillenniumPark skyrocketed in the leadup to and the introductionof the bean.

III.

BEAN-SPACE MODEL

We use a quadratic Bézier as the base component of abean since it is able to capture both the bent shape of
the Chicago bean, but also in the trivial case where allcontrol points are zero, the spherical Ottawa bean.
A single segment is unable to capture the potential
range of all future beans, so we allow beans to be composed of multiple segments. Simply taking the unionof multiple segments appears rigid and decidedly unbeanlike (see Figure 5, left), so we instead define thesurface in a way that allows us to use a smooth union(see Figure 5, right.) Signed Distance Functions allow
such a smooth union to be defined succinctly, so we chose
this format to represent our bean surfaces. This meansthat we represent the surface of a bean as the isosurface
f (X) = 0, where f describes the signed distance to thesurface. At a high level, f describes the smooth union
between multiple quadratic Bézier segments.

A.

Signed Distance Functions

A signed distance function (SDF) is a function f :Rn 7→ R describing the distance to a surface at a given
point in space. Mapping out the isosurface of f (X) = 0
yields the surface described by the function. This can bedone via the Marching Cubes algorithm to produce a 3D
mesh, or via sphere tracing to produce an image.
SDFs are a flexible surface representation if one wants
to organically join multiple base shapes. While taking
u(d1 , d2 ) = min{d1 , d2 } of two surfaces produces a surface representing the union of the shapes, the smooth
union operation u(d1 , d2 ) = d1 + kg(d2 − d1 )/k) blendssmoothly between its two inputs when they are a distance of k apart, using the kernel g to control the curveof the blending.

FIG. 5: Different values for the smoothness k between
segments: 0, 0.05, and 0.2, respectively.

B.

Formulation

Each quadratic Bézier segment is referenced in f via
a function representing the signed distance to the centerline of the segment, Q(X; C). [6] Here, C ∈ R3×3
describes the three control points to the function. For
brevity, we omit the full definition of Q. We subtract aradius r from B to give the segment thickness.
In practice, we constrain C such that Ci,1 = Ai , Ci,3 =Bi , and Ci,2 = (Ai + Bi )/2 + bn̂, where b ∈ R and n̂ is
˙ 3 − C1 ) = 0. In effect,a normalized vector such that n̂(Cthe middle control point is always halfway between thefirst and last control point, plus an offset normal to theline between them. This ensures a physically plausible
curve with no self-intersections. Examples of segments
fitting these constraints are shown in Figure 4.
We combine each segment using the SDF smooth union
operator U (d1 , d2 ; k), which blends the distance between
two input surface distances when they are a distance kor less away: [7]U (d1 , d2 ; k) = d1 + kg(d2 − d1 )/k(3)

Figure 5 shows the effect of the smoothness parameterk on the final surface.
Using the above, we represent the bean surface f recursively: the final value f (X; n) is defined as the smooth
union of the nth segment with f (X; n − 1), with the basecase being a single segment:(

Q(X; C0 ) − r0 ,i=1
U (B(X; Ci ) − ri , f (X; i − 1), k), i > 1(4)
To summarize, a bean in f has the following parameters:f (X; i) =• n, the number of Bézier segments

4• ri , 0 < i ≤ n, the radius of each segment• Ai , 0 < i ≤ n, the start point for each segment• Bi , 0 < i ≤ n, the end point for each segment• bi , 0 < i ≤ n, the amount of bend in each segment• k, the smoothing between segments
IV.

BEAN OF BEST FIT

value of x that produced the highest-seen value of P (x)can be kept. This is effectively a zero-gradient optimization, since we only evaluate P (x), not dP (x)/dx. Thiscan be useful for mixed-integer optimization problems
where a gradient would not exist. It also has the added
benefit of being simple enough that the authors can implement it themselves, and not pay for an expensive license for optimization software. [3]V.

Given an target image, we want to optimize our beanspace parameters to find a bean that best matches thetarget. The target image T is a 300 × 300-pixel one-bit
image representing a mask of the space we want a bean
to occupy. We define a function P (n, r, A, B, b, k; T ) thatdefines the score of a set of parameters in relation to thetarget image T . Using the bean-space parameters, werender similarly-sized image M of the bean they represent. We then define P as the intersection-over-union
between the pixel grids M and T . Maximizing this number encourages maximum coverage of the target area withminimal overlap with areas we don’t want covered:

RESULTS AND ANALYSISA.

Validation

We found the bean of best fit of known beans in order to validate the bean-fitting algorithm. The expected
result would be a bean matching the shape of the input
bean, as it is already a bean and does not need to be rebeaned. Both should only include one Bèzier segment,
and further, the Ottawa bean should have all its control points equal to 0 to form a sphere. Figure 6 shows
the result of the optimization process when run on the
Chicago and Ottawa beans, where it successfully matches
the shape of the input.

Pi,j Mi,j ∧ Ti,jP (M ; T ) = P

i,j Mi,j ∨ Ti,j(5)

To perform the optimization, we need to use a methodof optimization that does not rely exclusively on gradients, as the parameter n is an integer and therefore
does not have a useful gradient. We pick the MetropolisHastings algorithm to explore the space defined by ourbean parameters as a probabilistic, gradient-free optimization algorithm.
A.

Metropolis-Hastings Optimization

Markov Chain Monte Carlo (MCMC) algorithms arealgorithms used to sample probability distributions P (x)that are difficult to sample directly. In our case, beanspace, weighted by similarity to an input image, is sucha difficult distribution.
The Metropolis-Hastings Algorithm is one such
MCMC algorithm. Starting with one sample x (in ourcase, a set of bean parameters), a new candidate sample x′ is selected from an easier-to-sample distribution,
Q(x′ |x), such as by randomly mutating x. A random
number u ∈ [0, 1] is generated: if u < P (x′ )/P (x),the candidate sample is accepted; otherwise, it is rejected. Intuitively, a sample where P (x) is higher thanthe current sample is always accepted, but there is stilla chance of acceptance of lower-valued samples, too, allowing jumps into other areas of the distribution. This
property is useful for exploring highly non-convex spaces.
To use this procedure for optimization, P (x) can betreated as as function to optimize, and a record of the

FIG. 6: Existing beans recreated via our bean-fitting
algorithm: Chicago’s bean (top) and Ottawa’s bean(bottom).
Now that we know our bean-fitting algorithm works onexisting beans, we explore its impact on replacing structures yet to be beaned.
Since the purpose of beaning a city is to place it on
the map, it is natural that cities would want to ensure its
bean is in a significant, easy-to-access location. A logicalplace to start, then, would be to replace a city’s mostnotable landmarks with an equivalent bean. Figure 7
shows some examples of notable Ottawa landmarks: theParliament buildings are maybe the motivation for most5
grade school field trips to Ottawa, but are unlikely to bea stand-out attraction in the eyes of the students visiting.
A bean replacement will increase tourist satisfaction withno downsides.

FIG. 7: Beaned Ottawa landmarks: Maman by Louise
Bourgeois (photo by John Talbot, CC BY 2.0), and theParliament buildings.

B.

Novel Inputs

Figure 8 shows yet more examples of Canadian andworld landmarks replaced by beans. The beans effectively retain the form and character of the originals. The
BN Tower maintains the same erect stature as the CN
Tower (Figure 8, first row.) In Pisa, tourists can still poseand hold up the Beaning Tower the same way they would
have held up the Leaning Tower (Figure 8, second row.)
We believe this tight integration with the environment issure to increase bean acceptance and tourism.

VI.

CONCLUSION

Having established the economic advantage of replacing tired monuments with bean sculptures, we have presented an algorithm for designing bean sculptures on thebasis of these existing landmarks: We establish a modelof bean-space by creating smooth-unioned ensembles ofsimple beans, each parametrized by a quadratic Bézier
curve. The parameters of the hyper-bean are then optimized to achieve the best fit to a given landmark. We
apply our model to some example landmarks and we findgood results, even with a relatively low-dimensional bean
space. These results can be improved upon simply by increasing the number of parameters used in defining thehyper-bean; the cost of computing time for this should
be small compared to the cost of hiring actual people todesign beans.

FIG. 8: Improved world landmarks: the CN Tower(photo by Wikimedia Commons user Wladyslaw, CC
BY-SA 3.0), the Leaning Tower of Pisa (photo by Wikimedia Commons user U3207458, CC BY-SA 4.0), theGreat Pyramid of Giza (photo by Wikimedia Commons user Nina, CC BY 2.5), and the Earth (photo by
NASA/Bill Anders)
We also envision extending this model to create beansfor other purposes, beyond public art installations and/or
upgrades to existing infrastructure and landmarks. For
example, there may be a market for smaller personal

6beans, or even custom beans based on objects with somepersonal significance or sentimental value.
But let not the motivation for the proliferation of beans
be only capitalistic. As a society, we need beans. Their
reflective surface inspires us to take a close look at ourselves. Their smooth contours and cold surfaces aresoothing, a much needed balm in the feverish times ofthis 21st century. In the warped surface of a bean, one
sees one’s surroundings distorted, and one is inspired tosee the world from a new perspective. Our world of rightangles and matte surfaces has gotten us this far, but toprogress further as a species, it is crucial that we embrace a radically different attitude in the decoration of

our public and private spaces. The model we have presented in this work will without a doubt be an important
tool in this effort of beanification.

ACKNOWLEDGEMENTS

The authors thank the R(izzness)atJOPT group chatfor their comments throughout the preparation of thismanuscript. MB thanks Anna Xu and Lyn Vakulenko
for useful discussions. DP thanks the City of Chicago forthe existence of such delights as the Bean and the Rat
Hole.

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