Ancient Greece’s Forgotten Mathematician: The Olive Tree

Keywords: radiation-use efficiency, biomass, alternate bearing, floral transition, oil accumulation, phyllotaxy, arbuscular mycorrhizal fungi, evolutionary game theory

Abstract

Living up to 2000 years, the olive tree has long stood as a symbol of resilience and longevity, taking on cultural significance all around the world. This feat requires a toolbox of bioengineering solutions, each with unique foundations in physics, chemistry, and mathematics. At first, it may not seem obvious how Olea europaea uses mathematics, but many of its defining features rely on mathematical principles. The olive tree allocates resources in a manner that helps it thrive. Olive trees follow an alternate bearing pattern and growth distribution that can be modeled mathematically. The oil found in the fruits of the tree is produced linearly, ultimately leading to the storage of energy, regardless of environmental conditions. The leaves of the olive tree show geometric properties that give it many physical advantages (see Extending an Olive Branch: The Olive Tree Interactions with the Physical World and Survival in Extraneous Condition). Furthermore, game theory equations can describe the interactions of the olive tree with fungi, facilitating the creation of mutually beneficial relationships. As unsuspecting mathematicians, olive trees have calculated ways to grow, maintain resources, exploit physical properties, and form relationships, all to thrive in their unique environment.

Introduction

The olive tree has become so synonymous with the Mediterranean climate that the extent of this region is often defined by the tree’s very presence. Yet its iconic status is not merely due to the olives it produces. Rather, it is the result of its remarkable design solutions, rooted in chemistry and physics. Over time, the olive tree has evolved to thrive in coastal areas frequently subjected to various stresses, from harsh droughts to freezing winters. In doing so, it has proved itself a masterful bioengineer in many respects: maximizing dampening in the face of wind, building efficient hydraulic networks, and shaping leaves to avoid overheating while promoting condensation of water droplets. Olea europaea also expertly uses hormones and other cellular regulators to organize its reproduction into alternating years, as well as to protect itself from oxidative, heat, and water stress.

However, a deeper look at these adaptations reveals that mathematics offer a broader and more comprehensive perspective. Mathematics often reside at the core of biological processes, from the diffusion of molecules within tiny cells to the dynamics of entire ecosystems. These mathematical models provide powerful ways to describe and predict how living systems respond to environmental changes–for the olive tree, they underlie many fundamental principles, from resource allocation and leaf geometry to mutualistic interactions.

Modeling Olive Productivity

Olive trees (or any plant for that matter) wield solar energy to power the mechanisms that keep it alive. This includes forming new buds that will turn to fruit, growing longer roots that reach far and wide for water, reinforcing its existing structure to weather the unyielding elements, etc. All these processes are united by a single purpose—to expand and fortify the tree. Under the concept of radiation-use efficiency (RUE), which measures a plant’s ability to convert solar energy into biomass, this growth is directly proportional to the amount of light that is intercepted by the tree’s canopy (Villalobos et al., 2006). Before looking at the overall picture however, let’s start by examining the individual leaf in greater detail, more specifically what happens when sunlight hits a leaf. This interaction is described as the total radiation or total photon flux (PFF) intercepted by each leaf, represented by Equation 1:

where DirRadInter is the direct radiation intercepted by the leaf (a dot product of the leaf normal vector and the light beam vector); TaulDi is the shading coefficient for the direct radiation; DiffRadInter is the diffused radiation intercepted by the leaf; and TaulDiff is the shading coefficient for the diffused radiation (Sghaier et al., 2019).

When looking at the olive tree in its entirety, the total above-ground biomass is a product of photosynthetically active radiation (PAR) and RUE. This biomass is equally divided between vegetative growth (further divided into 30% for leaves and 70% for stems, branches, and trunk) and fruits (which includes oil). By focusing most of the vegetative growth in the supporting organs, the olive tree builds energy reserves that help it withstand less favorable years, while also reducing water loss through its leaves. The distribution of biomass, therefore, not only supports the tree’s resilience but also determines its oil productivity, which can be expressed by the following Equation 2, graphed in Figure 1:

where Y is the oil yield, R_sp the annual incoming PAR, Q_e the fraction of PAR intercepted by the canopy and ε_o, with a value of 0.17 g/MJ, the extrapolated amount of oil produced per unit of intercepted PAR. This simple linear equation allows us to predict the potential yield of an orchard.

Fig. 1. The linear relationship between the annual intercepted PAR (Q_e) and the oil yield (Y) in the cultivars Picual and Arbequina. The slope of the linear regression for all data points is ε_o = 0.17 g/MJ. [Adapted from Villalobos et al. (2006)]

Surprisingly, olive trees have a low RUE (0.86 g/MJ on average) compared to many annual crops, such as sycamore (RUE of around 1 g/MJ) and juniper (RUE of around 1.6 g/MJ), meaning that Olea europaea converts sunlight into biomass relatively slowly (Villalobos et al., 2006). While it may seem as though the olive tree is an inefficient plant, it can instead be argued that its lower RUE is a strategy favoring long-term resilience and stability over short-term gains. It sacrifices fast growth, which is often water and nutrient intensive. Furthermore, the low RUE suggests the olive tree employs sun-avoidance strategies, mitigating the potential for photodamage and water loss from the harsh Mediterranean sun.

The Fruits of its Labor: Mathematical Analysis of Olive Fruit

The olive tree is one of the most cultivated tree crops in the world, mostly known for its fruit, which can be processed into delicious oils or eaten (Rosati et al., 2023). The production of olive fruit itself follows many interesting patterns, such as alternate bearing and linear oil accumulation, which can be modeled using simple mathematical equations. These equations provide a straightforward way to analyze the growth of the olive tree and its fruit production. Overall, this allows for a better understanding of the importance of olive fruit in the survival and reproduction of the olive tree.

Fruit Production

Fruit production is essential for the protection and dispersal of seeds for any fruit bearing plant, and the olive tree follows an especially interesting pattern to produce these fruits. Generally, the olive tree undergoes an initial floral transition, which is when the vegetative meristem transitions from forming leaf primordia to forming flower primordia, in turn beginning the flowering process and fruit production (Smoly et al., 2025) (Fig. 2).

Fig. 2. Flowering process in olive trees as time progresses, showing development of inflorescent buds (containing flowering primordia) through to the opening of the flowers (Alagna et al., 2016).

The timing for this switch can depend on many factors, including genotype, environmental conditions, and the plant's internal state. The number of meristems that enter the floral transition essentially determines the amount of fruit that will be produced. However, a higher number of vegetative meristems in the previous year means that there will be less the following year, thus reducing the amount of fruit that is produced. This process of years with more fruit followed by years of reduced fruit loads is called alternate bearing or biennial bearing, and it can be represented mathematically. The number of vegetative meristems (inflorescence) that emerge during the spring depend mainly on two parameters: the number of new meristems between the previous spring and autumn (n), and the percentage of meristems that go through flowering transition (i). The alternate bearing of the olive tree can be quantified using the Alternate Bearing (AB) index which measures the difference in i of the same trees between two consecutive years (Wechsler et al., 2022). The AB index is represented by Equation 3, where i is the percentage of buds undergoing flowering transition.

The AB index quantitatively explains the amount of alternate bearing that occurs year to year: essentially, a higher AB index indicates years of greater difference in the amount of fruit produced (stronger alternate bearing). The alternate bearing tendencies of the olive tree are directly beneficial to its long-term health, mainly due to the responsible allocation of resources, such as energy and nutrients, but it also helps to preserve the quality of fruit so that during bearing years their seed have a better chance of being spread. This is the olive tree’s interpretation of “you can’t pour from an empty cup”: the tree must ensure it is storing resources and filling its own “cup” before reproducing or “filling other cups”. Aside from the alternate bearing of the olive tree, the tree's number of fruits per branch also follows a mathematical distribution (Wechsler et al., 2022). Equation 4 shows the mean number of fruits per branch (x) in relation to the number of new auxiliary buds (n), the average number of fruits on the inflorescences (m), and the average number of final fruits on the branch (f).

This is an important representation, since it helps translate the number of meristems that undergo the flowering transition state into the actual amount of fruit produced. This provides a better overall picture of how much fruit the olive tree produces from year to year.

Oil Production and Accumulation

Under the constant stress posed by harsh Mediterranean summers, drought conditions, and pollution, olive trees have been able to handle the heat and thrive despite the challenges. To accomplish this impressive feat, the olive tree bioengineered its own solution: energy storage in the form of oils in its fruits. In other words, the very quality that gives olives their distinct taste and makes them famous in kitchens around the work is also essential to their survival (Fig. 3).

Fig. 3. Olive fruit, alongside its infamous oil (Petrocelly, 2024).

Not only is it impressive that energy is chemically stored as oil in the olive fruits, but this storage follows mathematical principles which help to explain the accumulation of oil in relation to fruit maturation. Oil synthesis takes place in the mesocarp (the part that is typically eaten) of the olive, and the accumulation of oil starts after pit hardening is complete, generally in late summer and early autumn. Oil accumulation rate is affected by the ripening duration of the fruit, fruit load, cultivar characteristics, and environmental or agronomical conditions (López-Bernal et al., 2021). Work done by López-Bernal and colleagues (2021) explains the fascinating mathematical approach taken by the olive tree in its oil accumulation. Fruit oil content O_f and the accumulation of oil in the olive fruit is linearly related to the dry weight of the fruit w_f. Fruit dry weight is the weight after all water has been removed; it represents the solid components of the fruit. The linear relationship between O_f and w_f can be shown by Equation 5, where w_f0 is fruit dry weight at onset of oil accumulation and β the amount of oil accumulated per gram of fruit dry weight since the start of oil accumulation:

The linear relation shown by the fruit dry weight and fruit oil content can be visualized as a graph (Fig. 4b).

Fig. 4. Graphical representation of linear relationship between fruit dry weight and fruit oil content. Graph shows range beginning at the end of pit hardening to full fruit maturity, the range where oil accumulation typically occurs (López-Bernal et al., 2021.

Typically, it would be assumed that both w_f0 and β would be subject to environmental and genetic parameters. However, López-Bernal and colleagues (2021) hypothesize that although the w_f0 is cultivar dependent, β is independent of many factors that affect the availability of assimilates for fruit growth, including water status or fruit load. This indicates that the olive fruit, regardless of cultivar and environmental conditions, may accumulate similar amounts of oil per fruit dry weight increase beginning at the start of accumulation. This may provide an essential evolutionary purpose, since olive trees are very often subject to conditions that affect the availability of assimilates, such as drought. Maintaining a linear relationship between fruit dry weight and oil accumulation is thus a design solution: it allows the olive tree to establish an energy and nutrient reserve that remains consistent in the face of a changing environment.

Intrinsic Leaf-and-Shoot Geometry in Olea europaea

While fruit development shows how the olive tree regulates reproduction across time, its structure also reveals an equally ordered spatial logic. In particular, the arrangement of shoots and leaves lends itself to systematic geometric description, with leaf positions forming repeatable patterns along the stem, and leaf shapes following consistent rules.

Phyllotactic Logic in Olive Shoots

Olive shoots are short stem segments that carry a sequence of leaf nodes along an olive branch. On these shoots, leaves attach in characteristic phyllotactic patterns. In an opposite pattern, two leaves share the same node. In an opposite decussate pattern, each successive pair is rotated by about 90° around the stem, so the angles follow about 0°, 90°, 180°, and 270° before repeating with minor variation. Cultivar surveys also report alternate sequences, where single leaves occur one per node along the stem, and whorled sequences, where three or more leaves appear at the same node. These variants are present on olive branches but occur less often than the opposite decussate arrangement in the material examined (Sarwar et al., 2023). Figure 6 shows representative photographs of alternate, opposite, opposite decussate, and whorled insertions, together with a panel that illustrates pinnately reticulate venation for context (Sarwar et al., 2023).

Fig. 6. Examples of leaf insertion types observed on Olea europaea shoots (Sarwar et al., 2023).

Phyllotaxis on an olive shoot can be described with a simple rotation rule on a cylindrical stem. The key number is the divergence angle, the angular turn around the stem from one insertion to the next. In decussate phyllotaxis, the two leaves at the same node are opposite and separated by 180°. From one node to the next the frame rotates by about 90°, which creates four orthostichies, or vertical ranks of leaves aligned above one another. Figure 7 shows this quarter-turn construction in both side view and cross-section, providing a visual key for reading vertical ranks on olive shoots. The same geometric frame also explains other placements recorded in olive when the size of the turn is changed. A half-turn of 180° gives two orthostichies and is called distichy. A one third turn of 120° gives three orthostichies and is a whorl of three. When successive turns are close to 180° a two-ranked appearance reemerges. When the turn is not a simple fraction, the pattern is spiral and vertical ranks are not fixed. In the olive material examined, non-spiral patterns dominate, with decussate phyllotaxis most frequent and distichy or tri-whorls less common on the same branch (Okabe et al., 2019).

Fig. 7. Decussate placement on a cylindrical stem and corresponding cross-sections. This numbering does not indicate age or size, it is only an index that helps track how the decussate pattern repeats around the cylindrical stem. [Adapted from Okabe (2019)].

Olive shoots with opposite decussate phyllotaxis place leaves in fixed vertical ranks, which increases self-overlap compared with spiral arrangements of the same size and spacing. This geometry reduces the fraction of incoming light intercepted per shoot, an effect shown for Mediterranean woody plants and especially relevant under high irradiance (Brites & Valladares, 2005). In olives specifically, leaves further adjust their inclination and azimuth across crown sectors, creating systematic within-crown gradients in leaf angles that redistribute light loads through the day and help avoid excessive exposure while maintaining carbon gain under dry, bright conditions. Taken together, ranked placement and crown-level angle plasticity form a conservative strategy suited to Mediterranean climates, where tempering midday stress and balancing photosynthesis against water risk improves persistence without assuming any particular orchard training or pruning regime (Escribano-Rocafort et al., 2016).

Leaf Geometry and Shape Invariants

Olive leaves are botanically simple: each leaf is a single undivided blade rather than a set of smaller leaflets. In outline, they are lanceolate: much longer than they are wide, with the blade narrowing gradually toward the tip. The margins are entire, so the edges are smooth and lack teeth, and the veins are pinnate reticulate, a pattern with a midrib and side veins that branch and reconnect in a net. Cultivar surveys measure these blades directly and report wide diversity in size within Olea europaea (Sarwar et al., 2023). An Iranian germplasm panel, a collection of diverse olive varieties grown in Iran for breeding and research, recorded leaf length from about 27 to 79 mm and leaf width from about 5 to 23 mm across accessions, which shows substantial variation in elongation across olives (Khadivi et al., 2022). Because the blade is a single smooth outline with a clear midrib, its geometry can be read in intrinsic terms as length L, width W, area A, and perimeter P. To make these measurements concrete on an olive blade, Figure 5 reproduces an annotated example that marks the leaf boundary, maximum length along the midrib, and maximum width perpendicular to it (Blazakis et al., 2017).

Fig. 5. Morphological characteristics of a leaf. a) raw image data. b) Morphological measurements. Red line: leaf boundary; Green line: tip curve; Blue line (segment AB): blade height; Black line (segment CD): blade width; Pink line: petiole (Blazakis et al., 2017).

To quantify how elongated a blade is, the aspect ratio (AR) is used, a scale-free index of form in Equation 6. Specifically,

where L is the maximum blade length measured along the midrib and W is the maximum blade width measured perpendicular to the midrib. A long narrow blade and a short broad blade can have the same area, yet different geometry that affects how the blade is built. A simple way to see this is to model the outline as an ellipse with semi-axes a = L/2 and b = W/2 as in Equation 7, which gives area:

For a fixed blade area, increasing the length-to-width ratio enlarges the semi-major axis a and reduces the semi-minor axis b. The farthest lateral path from midrib to margin is set by b, so a higher ratio shortens the maximum transverse transport distance across the lamina. In the olive tree, this means an elongated leaf concentrates along the midrib and reduces the longest lateral route that water and sugars must traverse between the major vein and the edge. Because hydraulic and phloem resistances increase with path length, shorter lateral routes imply a purely geometric reduction in the longest within-leaf path that conduits must service. A second consequence follows from perimeter-to-area geometry. At equal area, perimeter P increases as the outline becomes more elongated. As a result, a larger fraction of the tissue lies near the margin, where marginal loops of veins close the network. This shifts how the venation network must allocate length between axial and lateral directions (Sack & Scoffoni, 2013).

The Olive Tree and its Fungi: A Mathematical Relationship

The olive tree leaf optimizes its shape for efficient water and nutrient transport—yet exploring the source of these nutrients unearths an unlikely ally: mushrooms! In fact, approximately 80% of land plants are in symbiotic relationships with arbuscular mycorrhizal fungi (AMF) (Steidinger & Bever, 2014; Martin & van der Heijden, 2024). These fungi inhabit plant roots and grow far-reaching hyphae that extract nutrients for the plant. In return, the plant supplies the fungi with organic compounds and vitamins (Martin & van der Heijden, 2024).

Although Olea europaea has perfected many of its own strategies to withstand the drought-ridden and scorching climate of the Mediterranean, it too relies on fungal alliances. In fact, over 80% of the olive tree’s root system is affected by its relationship with AMF (Santilli & Briccoli Bati, 2014). This is not entirely surprising, given all that these fungi do for the tree; their vast networks of hyphae allow extraction of essential nutrients (such as phosphorus) and can even help defend against certain pathogens. They also increase uptake of osmotic compounds, which the olive tree needs to maintain cell turgor in the face of drought (Ouledali et al., 2018) (Fig. 8).

Fig. 8. A comparison of an olive tree with (right) and without (left) mycorrhizal fungi associations, with numerous benefits listed in grey boxes. Note that endomycorrhizal fungi are synonymous with AMF (Land Arch Concepts, 2021).

However, at the center of this 450-million-year-old symbiosis lies an enormous paradox: fungal cheaters. After all, from an evolutionary standpoint, why pay for resources when you can steal them? In fact, many fungal species have evolved to take advantage of plants and their truly mutualistic fungi–weaving their way into their networks as thieves (Kiers & van der Heijden, 2006). So how does the olive tree avoid falling victim to these insidious cheaters? In other words, how does it ensure its fungal symbionts remain worthwhile investments across evolutionary timescales? The answer to this question may exist in an evolutionary game-theory model proposed by Steidinger & Bever (2014) (Fig. 9).

Let us assume that fungal partners can adopt one of two strategies: They can be mutualists (m) if they exchange resources with their host plant, or cheaters (c = 1 - m) if they simply take resources without giving anything in return. On the other hand, host plants can either be discriminators (D) if they defend themselves against cheaters in some way, or givers (G = 1 - D) if they do not. For givers, the payoff is simple: a benefit (B) when associating with a mutualist, or a cost (-K) when associating with a cheater.

Because discriminators invest in cheater defense mechanisms, they incur less of a cost when dealing with them. However, this resistance to cheaters comes with a trade-off: Discriminators also benefit less from mutualists. To describe this trade-off mathematically, let r represent the degree to which the host is discriminatory, and β represent the proportionality between reduction of costs and reduction of benefits for a discriminator. The payoff for discriminators is thus B/(1+r) if they associate with mutualists, or -K/(1+βr) if they associate with cheaters.

From the point of view of mutualistic fungi, they invest resources into their host (-z) to obtain a benefit (b). Cheaters, however, do not make any investments–this allows them to only reap benefits from givers (b), but these benefits may be significantly reduced if they associate with a discriminator. In this case, their payoff is b/(1+αr), where α is the degree to which the cheater is hindered by the discriminator’s defense mechanisms.

Fig. 9. A summary of the evolutionary game theory model for fungi-host mutualism—in this case, the host is the olive tree. A) A payoff matrix, which displays payoffs for every possible combination of fungal strategy (mutualist or cheater) and host strategy (discriminator or giver). Note that matrix entries are written as x; y (where x represents fungal payoff and y represents host payoff). Fitness (W) for each strategy is calculated using a weighted sum of the strategy’s two possible payoffs. B) Fitness benefit for the host (W) of associating with a mutualist or cheater fungi, as a function of the host’s degree of discrimination (r). C) Fitness benefit for the fungi (W) of adopting a mutualist or cheater strategy when inhabiting a plant with degree of discrimination r. In both B) and C), r is the critical r-value above which mutualist fungi have a competitive advantage over cheater fungi [Adapted from Steidinger & Bever (2014)].

This model sheds light onto the delicate balance the olive tree must reach to reap the benefits of associating with fungi without being overwhelmed by cheaters. r is defined in Equation 8—it represents the critical value above which the tree discourages cheaters enough that mutualists gain the competitive advantage. It notably depends on α, which is the effectiveness of the tree’s cheater defense mechanisms:

A crucial evolutionary implication thus emerges from Equation 8: in order to rely so heavily on fungi, the olive tree must defend against cheaters, but only to a degree that still allows it to receive enough nutrients from mutualistic fungi.

Although research into the specific mechanisms that the olive tree employs to strike this balance is limited, there exist many possible explanations. For example, olive tree roots may grow directionally toward areas of higher nutrient concentrations. When the tree associates with fungi, areas of higher nutrient concentrations correspond to clusters of mutualistic fungal hyphae. Because root growth tends to be inhibited by these hyphae, when roots try to grow through them, it is possible that the carbon that was allocated for root growth instead gets transferred to the fungi–setting up a reward system that favors only the mutualistic fungi that extract nutrients for the tree (Douglas, 2008).

Moreover, due to the relatively low number of fungal species Olea europaea has been found to associate with, the olive tree may have a high degree of discrimination (r) relative to other species (Calvente et al. 2004; Santilli & Briccoli Bati, 2014). This means that the olive tree is more conservative in its fungal partner choice–it doesn’t mind losing out on some benefits, so long as it shuts down cheaters. This aligns with the olive tree’s strategy of occasionally sacrificing efficiency in favor of safety (see Olive Physics), or sacrificing reproduction in favor of maintaining resources (see Olive Chemistry), which pays off in the unforgiving environment it has learned to thrive in.

Conclusion

The olive tree transforms sunlight into matter—it optimizes biomass accumulation, which helps it weather the harsh Mediterranean climate. Olive fruit development follows simple rules that stabilize reproduction across variable years. Oil content increases linearly with fruit dry mass once accumulation begins, which makes energy storage consistent across cultivars and water status. Alternate bearing spreads reproductive costs by adjusting how many buds enter floral transition from year to year. At the organ scale, olive leaves and shoots express an equally clear geometric logic. Elongated blades reduce lateral transport distances at fixed area, and opposite-decussate phyllotaxis keeps leaves in fixed vertical ranks that share light within a shoot and temper midday exposure without external tuning. The same allocation logic extends belowground, where roots favor mycorrhizal partners that reciprocate and reduce support for those that do not. Taken together, the equations that underlie Olea europaea’s energy conversion, resource allocation, leaf and shoot geometry, and mutualistic interactions reveal the strategy that has allowed this tree to thrive in such a risky environment: a conservative use of structure and resources that seeks to maximize efficiency, but never at the cost of safety.

This expert mathematician of the natural world can inspire many design ideas. From fruit behavior, a storage system can treat capacity as a linear function of structural mass once accumulation begins and can schedule production with an alternate-bearing style cycle that alternates output and recovery to protect long-term capacity. From leaf and shoot geometry, a light-harvesting design can use elongated platforms when transverse transport is costly and can arrange small canopies or device tiers with a quarter-turn rotation to promote light sharing and reduce midday peaks without fine control. From root-fungus interactions, a hydraulic system that uses adapted roots as pipes can use the fungal inhibition of root growth to circumvent problematic areas—guiding the biomimetic pipes along some pre-determined path.