photo: Cathode Ray Tube
Sama Al- Shammari
Serendipity has often been a source of ambiguity. Is a lucky accident truly serendipitous? What differentiates serendipity from traditional creativity? The answers to these questions are not clear-cut. It is widely held that serendipity is never exclusively due to chance, but instead also requires intelligent academic preparation and an ability to keenly observe, explore, and ask questions. Traditional creativity typically entails recognising an issue, diverging to brainstorm potential solutions, and then converging on a novel solution to that same initial problem.
In contrast, serendipitous creativity is inherently divergent. Rather than exploring solutions to the problem of interest, it strays toward an entirely different problem or, at times, toward solutions to a problem never considered before. Quéau summarizes these key differences by defining serendipity as “the art of finding what we are not looking for by looking for what we are not finding.”
Interestingly, serendipity itself can be researched. Mathematical notation, in particular, proves useful in distinguishing between serendipitous and non-serendipitous creativity. To this end, Dias de Figueiredo and Campos proposed the four Serendipitous Equations, aimed at capturing pseudoserendipity, serendipity with a metaphor, serendipity without a metaphor, and serendipity with ignorance as a metaphor. In what follows, we explore these four cases, delving into supporting examples and their underlying principles.
Pseudoserendipity
Consider Archimedes’ famous discovery of a method to measure the volume of solid objects. This tale takes us back over 2,000 years, when King Hieron II of Syracuse commissioned a goldsmith to craft a crown from a gold bar. Suspecting that he was deceived, and that the crown contained a mixture of silver and gold, the king tasked Archimedes with determining whether the crown was made of pure gold.
While pondering this problem, Archimedes decided to take a bath. Upon stepping into a tub filled with water, he noticed that water spilled over the edge, and that the amount displaced increased as more of his body submerged. Archimedes realised that a relationship must exist between the volume of an object and the volume of liquid it displaces. This insight enabled him to determine that the king had indeed been cheated and, more importantly, led to the formulation of Archimedes’ Principle, later detailed in his treatise On Floating Bodies.
According to Dias de Figueiredo and Campos’ notation, this unexpected metaphor M (water splashing from the bathtub), combined with Archimedes’ scientific knowledge (KP1), inspired a solution S1 to his original problem P1. This discovery is classified as pseudoserendipitous, rather than serendipitous, because it resulted in a solution to the initial problem, rather than the identification of a new, previously unconsidered one. In set notation, this scenario can be denoted as:
where KM is the knowledge associated with the metaphor, and KN is the new knowledge gained from this metaphor.
Serendipity with a Metaphor
In this case, a new and unexpected metaphor, such as an unforeseen observation, leads the explorer to discover a new problem and, frequently, a solution to that problem. Both are distinct from the problem and solution originally under investigation.
On November 8, 1895, Wilhelm Röntgen, a German physicist, was investigating cathode rays in a darkened room. These rays were typically observed as fluorescence on glass walls or screens inside evacuated glass tubes with electrodes, known as Crookes tubes. In an attempt to better visualise the cathode rays by preventing them from escaping the tube, Röntgen covered the glass walls with thick black cardboard, which was known to be impermeable to cathode rays.
Unexpectedly, Röntgen noticed a faint fluorescent glow on a cardboard screen located several feet away from the apparatus. Knowing that cathode rays could neither penetrate the cardboard nor travel such a distance, Röntgen concluded that he was observing an entirely new and unknown type of radiation, which he famously termed X-rays, with “X” denoting their unknown nature.
Subsequent experiments, including those involving his wife, revealed that X-rays could pass through human tissue but not through bone or metal. To this day, this serendipitous discovery underpins modern medical imaging.
In this scenario, Röntgen was initially exploring cathode rays (P1). Through the unexpected fluorescence (M), he discovered a new phenomenon (P2), leading to a distinct solution (S2, the characterisation of X-ray properties), to a problem he had never set out to solve. This can be represented as:
Serendipity without a Metaphor
At times, discoveries may be serendipitous in nature without the presence of a metaphor or external inspiration. Paul Dirac, widely regarded as a scientific genius, succeeded in unifying Einstein’s theory of relativity with quantum mechanics, two frameworks long thought to be fundamentally incompatible, into what is now known as the Dirac equation.
Through pure mathematical reasoning, Dirac realised that such a unification required an equation linear in both momentum and energy. Achieving this necessitated the use of 4×4 matrices rather than the conventional 2×2 matrices. While mathematically sound, this approach implied something considered absurd at the time: that a single mathematical problem pertaining to a physical real-world application would yield two solutions.
To illustrate, consider the quadratic equation x² = 4, which has two solutions, x = 2 and x = −2. Analogously, Dirac’s equation implied that particles such as electrons could possess both positive and negative energy states, suggesting the existence of antimatter. Many scientists were troubled by this implication, including Dirac himself.
In an independent experiment conduct four years later, Carl Anderson discovered a particle he termed the positron. Unaware of it at the time, this particle, identical in mass and spin but opposite in charge to the electron, corresponded precisely to Dirac’s predicted “anti-electron.”
Dirac’s abstract, metaphor-free and mathematical reasoning had thus serendipitously revealed both a new problem and its solution, entities whose existence had not yet been empirically observed to exist, nor were even believed to exist by scientists. This case can be denoted as:
Serendipity with Ignorance as a Metaphor
Ignorance and incorrect knowledge can, paradoxically, act as powerful catalysts for impactful discoveries. A classic example is Christopher Columbus’s “discovery” of the Americas. Guided by his navigational expertise (KP1) and his goal of sailing westward to reach Asia (P1), Columbus relied on false assumptions, including an underestimation of the Earth’s circumference and incorrectly assuming Asia to be larger than it actually is.
As a result, his voyage led him not to Asia but to the Americas. This serendipitous encounter constituted a new solution (S2) that simultaneously gave rise to a new problem (P2): how to reach and navigate these newly discovered lands. Importantly, neither the problem nor its solution could have been formulated prior to the discovery itself, as Columbus was entirely unaware of the Americas’ existence. Dias de Figueiredo and Campos express this scenario as:
References
About Eureka! (n.d.). Bellarmine University. Retrieved January 08, 2026, from https://www.bellarmine.edu/learningcommunity/eureka/about/
Anguera De Sojo, Á., Ares, J., Martínez, M. A., Pazos, J., Rodríguez, S., & Zato, J. G. (2014). Serendipity and the discovery of DNA. Foundations of Science, 19(4), 387–401. https://doi.org/10.1007/s10699-014-9348-0
Figueiredo, A., & Campos, J. (2001). The serendipity equations. Social Science Research Network. https://www.semanticscholar.org/paper/The-Serendipity-Equations-Figueiredo-Campos/0f1740ecdd7da608c45aad572a1b6421decbba65
Kragh, H. (2016). Anti-worlds. https://pressbooks.pub/simplydirac/chapter/anti-worlds/
Nondestructive evaluation physics: X-ray. (n.d.). Retrieved January 10, 2026, from https://www.nde-ed.org/Physics/X-Ray/discoveryxrays.xhtml
Wilhelm Conrad Röntgen and the discovery of x-rays | Spectroscopy Online. (2026, January 11). https://www.spectroscopyonline.com/view/wilhelm-conrad-r-ntgen-and-discovery-x-rays-0
أنماط الاكتشاف غير المقصود
تتناول هذه المقالة مفهومَ السِّرنديبيَّة بوصفه نمطاً خاصَّاً من الإبداع، لا يقوم على المصادفة الخالصة بل يتطلَّب استعداداً معرفيَّاً وقدرةً على الملاحظة الدَّقيقة وطرح الأسئلة، وتُميِّز بين الإبداع التَّقليديِّ الذي ينطلق من مشكلة محدَّدة ويبحث عن حلٍّ لها والإبداع السِّرنديبيِّ الذي ينحرف عن المسار المتوقَّع ليقود إلى مشكلة جديدة أو حلٍّ لم يكن في الحسبان.
وتعرض المقالة إطاراً تحليليَّاً اقترحه (دياس دي فيغيريدو) و(كامبوس) يقوم على معادلات السِّرنديبيَّة الأربع لتمييز أربعة أنماطٍ رئيسة:
السِّرنديبيَّة الزَّائفة أو الوهميَّة يمثِّلها اكتشاف أرخميدس لمبدأ الإزاحة، إذ قادته ملاحظةٌ غير متوقَّعة إلى حلِّ المشكلة الأصليَّة لا إلى مشكلة جديدة.
السِّرنديبيَّة مع استعارة معرفيَّة كما في اكتشاف رونتغن الأشعَّةَ السِّينيَّة، إذ أدَّت ملاحظةُ وميضٍ غير متوقَّع إلى الكشف عن ظاهرةٍ جديدة وعن مشكلةٍ وحلٍّ لم يكونا ضمن هدفه الأوَّل.
السِّرنديبيَّة دون استعارة معرفيَّة تتجلَّى في عمل بول ديراك الرِّياضيِّ الخالص إذ أفضى التَّفكير التَّجريديُّ إلى التَّنبُّؤ بالمادَّة المضادَّة قبل اكتشافها تجريبيَّاً، ليكشف مشكلةً وحلَّاً جديدين دون محفِّزٍ خارجيٍّ.
السِّرنديبيَّة مع الجهل بوصفه استعارةً يمثِّلها وصول كريستوفر كولومبوس إلى الأميركيتين نتيجة افتراضات خاطئة، وقد كان الجهل محرِّكاً لاكتشاف غير مقصود أوجد مشكلةً وحلَّاً جديدين معاً.
يخلص النِّقاش إلى أنَّ السِّرنديبيَّة ليست حظَّاً عابراً بل عمليَّةً معرفيَّةً معقَّدة تتقاطع فيها الصُّدفة مع الجاهزيَّة العلميَّة والقدرة على إعادة تأطير الملاحظات غير المتوقَّعة.
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