# Abstract

In this essay, we explore tintinnids’ survival designs within the context of fundamental mathematics principles. Tintinnids use mathematics concepts for everyday functions such as swimming and self-protection. We delve into tintinnids’ various swimming patterns. We take a close look at the helical swimming motion, or more precisely how that motion can be modeled, and what kind of parametrisation can represent it. We also explore their lorica, which characterise tintinnids, and the mathematical concepts that ensure structural stability. Even the world domination tendencies of tintinnids are a result of mathematical laws; more precisely, the laws of statistics. We investigate their spatial distribution, the change in their diversity depending on environmental conditions, and how their diversity and population vary over time. On an individual scale, we investigate their swimming patterns and shell structure.

# Introduction

Tintinnids, which are microscopic unicellular ciliates, are vital organisms in the realm of microzooplankton and the microbial food chain. Their presence all around the world deeply affects the ecosystems of which they are a part. Their success depends on their numerous design solutions that allow them to locate prey, evade and protect themselves from predators, and reproduce.

We will explore the mathematical design solutions that allow tintinnids to thrive in the marine environment. We consider the intricacies of the hexagonal lattice of their lorica which provides them with a strong foundation for their protective home. The material of their lorica also helps them survive in different seasons and influences their population distribution. We investigate the statistics and their strategies for global distribution that allow them to be one of the most abundant organisms in microzooplankton. Finally, we examine the mathematics of their swimming behaviour through an in-depth graphical analysis and an in-depth exploration of their helical swimming path.

These minuscule mathematicians capitalize on a variety of strategies to optimize their survival and allow them to thrive in the aquatic environment.

# Population dynamics

## Tintinnids spatial distribution

Tintinnids are spread all around the world, in coastal waters like arctic waters, as seen in Figure 1. Furthermore, their global biogeography indicates a large heterogeneity in their distribution (Dolan & Pierce, 2012).

**Figure 1: **Distribution of all species of tintinnids in waters (Dolan & Pierce, 2012)

However, most tintinnids species live locally and are not widespread: these species are said to have a patchy distribution (Dolan & Pierce, 2012).

There exists a correlation between the average abundance of tintinnids and their spatial distribution. That correlation consists of a positive relationship, which means that if one of the two parameters increases, the other one will too. This leads to the assumption that more abundant species will be more widespread (Dolan & Pierce, 2012). This assumption has been proven by the very few data available and is illustrated in Figure 2 and Figure 3.

**Figure 2: **Distribution of Tintinnids depending on their average abundance in the Straits of Magellan (Dolan & Pierce, 2012)

**Figure 3: **Distribution of Tintinnids depending on their average abundance in the Southeast Tropical Pacific (Dolan & Pierce, 2012)

However, it is also important to mention that global patterns for spatial distribution and local patterns for spatial distribution will not be the same. Indeed, in a more specific environment, the most widespread species in the world will not necessarily be the most abundant (Dolan & Pierce, 2012). For example, a coastal species will be more abundant in coastal environments than a cosmopolitan species that is present all around the world.

There are multiple habitats in which a tintinnid species or genera can live. The first environment is the cosmopolitan one, where the species live from the Arctic to the Southern Ocean, passing through the Tropic and the nearshore areas (Dolan & Pierce, 2012). The cosmopolitan genera are the most widespread and some are illustrated in figure 4.

**Figure 4:** Cosmopolitan distribution for four genera of Tintinnids (Dolan & Pierce, 2012)

The second tintinnid home is the neritic one, where the species are restricted to nearshore waters. This environment is the most studied since its species are nearer to the shore (Dolan & Pierce, 2012). It is illustrated for a few tintinnids genera in Figure 5.

**Figure 5:** Neritic distribution for four genera of Tintinnids (Dolan & Pierce, 2012)

The third habitat is the warm-temperate one, in which the tintinnids live both in coastal systems and open waters around the world (Dolan & Pierce, 2012). It is illustrated in Figure 6.

**Figure 6:** Warm-temperate distribution of four genera of Tintinnids (Dolan & Pierce, 2012)

The fourth surrounding is the boreal one, in which the ciliates live solely in Arctic and Subarctic waters (Dolan & Pierce, 2012). It is illustrated in Figure 7.

**Figure 7: **Boreal distribution of two genera of Tintinnids (Dolan & Pierce, 2012)

The last common environment is the austral one, in which tintinnids are restricted to Antarctic and Subantarctic waters (Dolan & Pierce, 2012). It is illustrated in Figure 8.

**Figure 8:** Austral distribution of two genera of Tintinnids (Dolan & Pierce, 2012)

Tintinnids are certainly not dispersion limited, as observed in how well tintinnid genera are distributed around the world once all their habitats are combined. What helps tintinnids to be this widespread around the world is the capacity to form cysts, which can be transported either actively by other organisms or passively by water or wind currents (Dolan & Pierce, 2012). It is also a possibility that tintinnids get help in their marine world expansion from climate change, ecosystem changes, the construction of canals, and the transport of tintinnids through ballast waters (Dolan & Pierce, 2012). Being distributed all around the world helps tintinnids stay alive by allowing them to have a high diversity that depends on which ecological niche they are located in. Also, the fact that tintinnids live in multiple ecological niches leads to fewer chances of their total extinction, since it is unlikely that all the ecological niches will suffer major changes affecting the tintinnids’ survival at the same time (Dolan & Pierce, 2012).

## Lattitude diversity gradient

Species diversity across the world is influenced by the latitude their diversity is being tested in. This pattern is called the latitudinal diversity gradient. Tintinnids follow a typical latitudinal diversity gradient, as seen in Figure 9 (Dolan & Pierce, 2012).

**Figure 9:** Distribution of global tintinnids according to their latitude (Dolan & Pierce, 2012)

The stereotypical latitudinal gradient seen in Figure 9 (above) shows that tintinnids have a low species richness around the poles. That richness increases toward the equator and reaches a peak at about 15 to 20 degrees from the equator, coming from the north and south poles. There is also a decrease in tintinnid species richness around the equator (Dolan & Pierce, 2012).

This pattern can vary in specific tintinnid genera, that can be more adapted to one environment than to another (Dolan & Pierce, 2012). For example, when comparing the genera Tintinnopsis and Eutintinnus, it is observable that they have different species richness according to the latitude (see Figure 10). This is because Tintinnopsis species are restricted to neritic shallow waters, or more specifically in bays and estuaries from temperate climates, making them more present at a certain latitude, and more diverse at that same latitude. Eutintinnus species are also restricted to neritic waters, but are more specifically found in open waters from temperate to tropical zones, with a predominance for tropical zones. This makes Eutintinnus species more diverse and present at a different latitude than the Tintinnopsis genera (Dolan & Pierce, 2012).

The latitudinal diversity gradient of tintinnids can be generalized because many zooplankton species have a gradient resembling the one of tintinnids. Therefore, tintinnids’ latitudinal diversity gradient can be seen as a model for marine zooplankton (Dolan & Pierce, 2012).

**Figure 10:** Distribution of Tintinnopsis and Eutintinnus genera of tintinnids according to their latitude (Dolan & Pierce, 2012)

## Assemblages of coastal systems

Tintinnid assemblages and patterns are different in every system, which leads to the suggestion that each system has a tintinnid community fingerprint (Dolan & Pierce, 2012). They are described as fingerprints because no two systems will be identical in detail, but they can seem very similar when looking at them from a more general perspective. An example of a common pattern is the one for coastal systems, like the Gulf of Naples and the Hiroshima Bay. In that general pattern, species with hyaline loricae dominate in the summer months, and species with agglutinated loricae dominate in the winter months (see Figure 11). Species with agglutinated loricae dominate during the winter because their lorica offers them an additional layer of protection, compared to the tintinnids with hyaline loricae, which have lighter protection (Dolan & Pierce, 2012; Monti et al., 2019) (see Figure 11). This shift is partly the reason why periods between the summer and winter months are the periods in which species richness is the highest. This is because these months are when both hyaline and agglutinate tintinnid species are present (Dolan & Pierce, 2012).

**Figure 11:** Monthly average abundance in L-1 of agglutinated (black line) and hyaline (dashed line) tintinnids in a water column from July 1998 to July 2016 in the Gulf of Trieste (Monti et al., 2019)

## Long-range evolution of the tintinnid population

During a 2-year study, it was observed that there was more diversity in tintinnid species during the late summer and early fall. The lowest observed number of tintinnid species was during the winter (Verity, 1986). This can be compared to the general pattern for coastal systems since there was more diversity in tintinnids during a transitional period than during the winter season.

The more abundant species of tintinnids between 26 species and 2 years of research between January 1981 and January 1983 were either summer-dominant species or species that lived throughout each season, but whose abundance peaks occurred in the summer months (Verity, 1986).

There is also a study that kept track of tintinnid abundance for a much longer time period, from July 1998 to July 2016 (Monti et al., 2019). Figure 12 is the illustration of their findings.

**Figure 12:** Long-term fluctuations of total titinnid abundance in L-1 in a water column from July 1998 to July 2016 in the Gulf of Trieste (Monti et al., 2019).

In this study, it was found that tintinnid abundance was at its highest from September to October and it was at its lowest from January to April (see Figure 13) (Monti et al., 2019). This is yet another confirmation of the general pattern for coastal systems.

**Figure 13: **Monthly average abundance of tintinnids in L-1 in a water column from July 1998 to July 2016 in the Gulf of Trieste (Monti et al., 2019)

In the above figure (figure 13), the white rectangles represent the dispersion of the data (25% to 75%). The black lines in the middle of those rectangles represent the median. The black bars represent the non-outlier range (Monti et al., 2019).

# Mathematics of Tintinnid Swimming

## Graphical Analysis of Swimming Behaviors

The investigation of tintinnid swimming behavior is a relatively recent addition to our knowledge about tintinnids. Prior to the exponential increase in the use of technology that came with the 21st century, we were essentially unable to study the patterns of how these microscopic creatures move. Now, however, video observation and computational mathematical models have allowed us to push the bounds of the study of tintinnids.

In one quantitative analysis of tintinnid swimming patterns, a study from the University of Lille Nord de France focused on the swimming behavior of the ciliate Strobilidium sp. This was done using cinematographic techniques with a density of 20 ciliates/ml under four experimental food conditions, varying from 121 to 15125 cells/ml of the dinoflagellate Gymnodinium sp. The study recorded and analyzed 100 trajectories per experiment, classifying the swimming patterns of Strobilidium into three categories: “helix,” “non-helix,” and “break.” “Helix” refers to a helical swimming pattern, “non-helix” refers to a straighter swimming pattern, and “break” refers to a rest period taken by the tintinnid between bouts of movement These states were identified through automated recognition of helices based on values of trajectory angles. The symbolic analysis of these states allowed for the discrimination between different food concentration experiments, providing a more complete characterization of swimming behavior (Vandromme et al. 2010).

**Figure 14: **The graphical representation of swimming trajectories under different food concentrations. A – Low food, B – Medium food, C – High food, D – Extremely high food. These trajectories are presented in an x-y coordinate system measured in millimeters (Vandromme et al. 2010)

The study found that helical swimming patterns initially increased with food concentration, then decreased, accompanied by an increase in the number of breaks. However, non-helical motions were more prevalent at the highest food concentrations due to the tintinnid’s preference for a mixed diet. The overabundance of prey around a tintinnid will actually cause them to attempt to escape to find more varied nutrients in order to ensure their survival (Yang et al. 2019).

**Figure 15:** Scale dependence of NGDR (net to gross displacement ratio) and swimming velocity. The NGDR and velocity curves were normalized to the velocity at the smallest scale, and the results were presented as log-log plots to highlight power-law trends (Vandromme et al. 2010)

This graphical representation allows for an understanding of how swimming velocity and NGDR change across different scales, providing insight into the swimming dynamics of Strobilidium under varying food conditions. NGDR is essentially the arc length of the swimming trajectories of tintinnids, indicating a faster swimming speed. In B we see how the differences in the changes of velocity of the tintinnid tie Figure 14 and Figure 15 together; the decrease of velocity, essentially the willingness of the tintinnid to stay in the same place, is much more pronounced in experiment 2 and 3 than in 1 and 4. This supports the knowledge that tintinnids have a higher swimming velocity when there is a shortage of food as well as when there is too much food nearby, and supports the fact that tintinnids will slow their swimming speed when prey is at sweet-spot concentrations.

This study also employed a sophisticated approach combining deterministic and stochastic elements. This methodology is exemplified through two key figures in the study.

**Figure 16:** The Effects of numerical smoothing on tintinnid movement tracking. A – Tintinnid swimming trajectory before smoothing, B – Tintinnid swimming trajectory after smoothing. Positions x and y are in mm (Vandromme et al. 2010)

Figure 16 illustrates the impact of smoothing on a trajectory. Before smoothing, the trajectory appears jagged and less discernible. This stemmed from the fact that the devices that they were using to record the movements of the tintinnid had a pixel size that was larger than the tintinnid itself. This led to a jerky motion in the recorded trajectories with a “staircase”-like appearance and many angles of 90°. To address this, they smoothed the trajectories using a running average over a window of 7/30 seconds. This involved taking an average over 7 consecutive values, which includes 3 steps before and 3 steps after the current position (Vandromme et al. 2010). This specific value was chosen as it was the minimum necessary to effectively smooth the small-scale discretized angles. After smoothing, the helical pattern becomes clearly visible, facilitating the automatic recognition of helices by separate software. This figure is pivotal in demonstrating how raw data is processed to highlight the helical swimming patterns of the tintinnids. The transformation from a rough trajectory to a smoother, more defined helical pattern is crucial for the mathematical model to accurately represent the ciliate’s movement. Figure 16 is a perfect example, therefore, of the math and data processing required to identify key swimming patterns in ways that we as engineers can understand.

**Figure 17:** Log-log plots of probability densities for various tintinnid swimming states. Break: tintinnids not moving, Helices: tintinnids moving in a helical swimming pattern, Non-helices: tintinnids not moving in a helical swimming pattern (Vandromme et al. 2010)

Figure 17 presents a comprehensive plot of residence times of probability densities for all states in each experiment. The experiment was divided into sub-figures representing different states (break, helix, non-helix) observed across all exp.1-exp.4. This figure juxtaposes the experimental data (exp.pdf) with the exponential (Markovian) model. The exponential model is fitting for this biological observation as the nature of tintinnids is to change their swimming behavior in order to avoid predators and catch prey. To create this exponential model for tintinnids, the expected residence time in any swimming state was used, and the exponential decay was assumed to be perfect. Even though we know that tintinnid behavior will not always follow the pattern of perfect decay, the representation of the prediction created by the exponential model correlates extremely well with the observed data from the tintinnids (Vandromme et al. 2010). With the knowledge of this ratified correlation, we can use this mathematical model to predict tintinnid behavior in the future without wasting time collecting experimental data. This detailed representation is also essential in understanding the probabilistic nature of the tintinnids’ swimming states under different conditions. The figure not only showcases the residence times across different states but also provides insight into how these times differ with regard to repetition in the constantly changing marine ecosystem. This comparative analysis helps in validating the mathematical model and understanding the underlying stochastic nature of the tintinnid movement. Figure 17, therefore, offers a comprehensive analysis of the stochastic nature of these patterns and the study’s innovative approach to modeling tintinnid movement.

The exploration of tintinnid swimming behavior, leveraging modern technology and advanced mathematical models, marks a significant advancement in marine biology. The integration of cinematographic recording, data smoothing, and statistical analysis exemplifies the power of interdisciplinary approaches in unraveling the mysteries of microscopic marine life. This research not only enhances our understanding of tintinnid behavior but also sets a precedent for future studies in marine microecology, showcasing the synergy between technology, mathematics, and biology.

## Helical Swimming Motion

Helices can be seen all around us in nature. DNA is a double helix, proteins self-assemble into alpha helices, plants grow stems in helices, and tintinnids have a helical swimming pattern (Figure 18). Helices are so prevalent in nature because they are a fantastic way to condense molecules or other materials into a tight space. However, this may not be the case for why tintinnids exhibit helical swimming. Many microscopic aquatic unicellular organisms swim in a helical path. This is because it is most hydrodynamical favourable due to the low Reynolds number environment (Febvre-Chevalier & Febvre, 1994). Swimming straight through this almost “sticky” water would cause too much drag on the cells. Instead, by swimming in a helical motion they minimize drag and optimize energy usage.

**Figure 18:** An ensemble of various helices found in nature. The top left corner depicts the double helix of DNA (2007). The bottom left corner has a goat whose horns are helices (Siren, 2014). To the right, there is a picture of a plant stem growing in a helical pattern (2006). Finally, the far right is an image of the tintinnid’s helical swimming motion (Montagnes, 2012)

A helix is a type of three-dimensional curve that is infinitely differentiable and turns around a stable axis. Helices share key characteristics as illustrated in Figure 19. Specifically, a helix is defined by the radius, pitch, curvature, and torsion. While there are different shapes and types of helices, they all share these characteristics. The radius is the distance from the curve to the axis. The pitch of the helix is the height of one turn. The curvature of the helix describes how quickly it is bending in the plane. Imagine the curvature as how “tight” the circle or the curve is. Finally, the torsion describes how the curve is bending out of the plane, which is how much it stretches up or down. Torsion is the difference between a closed loop (torsion=0) or a practically straight line (torsion=1).

**Figure 19:** Illustration of a circular helix with various parameters labelled including radius (in red), Pitch (in yellow), the axis of rotation (in dark green), curvature (in light green), the torsion (in light blue) (Created by Bailey, 2023)

There are many different types of helices. A double helix is made of two helices with the same axis that differ only by a translation along the axis. This is what we see in DNA. A circular helix is one with a constant radius, constant curvature, and constant torsion. For a conic helix, sometimes also known as a conic spiral, the radius either increases or decreases as a function of time. Most often, we will use a circular helix to model the pattern of tintinnid’s swimming behaviour, as seen in Figure 19 above. As tintinnids are living organisms, they will very rarely (if ever) perfectly model the mathematic definitions of helices. It is possible that at any given point in time, depending on the circumstances, tintinnids could exhibit any one of the above helix types. For example, two tintinnids swimming together lengthwise have been observed to create a similar pattern to a double helix (Dolan, 2012).

Helices are chiral meaning that their mirror image is non-superimposable. Chirality is a phenomenon that appears in many situations. It can be evidenced in the chemistry of compounds, the wings of some birds, the right and left hand of humans, and many more. Helices chirality means that they can be right- or left-handed. One can determine the handedness by looking along the helix’s axis. If the spiral moves clockwise, away from the observer, then it is a right-handed spiral. On the other hand, if the spiral moves counterclockwise, towards the observer, then it is a left-handed helix. This chirality cannot be changed and is not determined by the observer. Rather, this is a property of the object and no matter which way the observer looks at a right-handed helix, it will not appear as a left-handed one and vice versa. Helix rotation direct (handedness) in tintinnid swimming behaviour can vary not only between individual species but between individual tintinnids. That being said, a single tintinnid will almost never change the handedness of its helical path (Montagnes, 2012).

** Figure 20:** Illustration of left- and right-handed helices (Rajarshi, 2016)

To describe helices mathematically, we can observe their parametrisation. The following parametrisations describe circular helices with a constant radius. It is important to note that circular/general helices will very rarely be found in nature. That being said, they are useful for modeling the general behaviour and characteristics of helices.

A right-handed helix with a constant radius can be described mathematically by the following parametrisation in Cartesian coordinates (Equation 1). A graphical interpretation of this parametrisation can be seen in Figure 21.

**Equation 1: **𝑥(𝑡)=cos(𝑡), 𝑦(𝑡)=sin(𝑡), 𝑧(𝑡)=𝑡

The same helix can be parametrised in cylindrical coordinates as follows (Equation 2):

**Equation 2: **𝑟(𝑡)=1, 𝜃(𝑡)=𝑡, ℎ(𝑡)=𝑡

A more general description of a circular helix with constant radius a and pitch of 2π*b* follows in Equation 3.

**Equation 3: **𝑥(𝑡)=𝑎cos(𝑡), 𝑦(𝑡)=𝑎sin(𝑡), 𝑧(𝑡)=𝑏𝑡

All right-handed helices will have the same form as above and will only deviate in terms of rotations, translations, and scaling. Left-handed helices will have a similar parametrisation to one of the parameters being negative.

** Figure 21:** Graphical illustration of a helix parametrized by a cosine for its x-component and sine for its y-component (Benutzer, 2007)

As mentioned, for simplicity, circular helices can be used to model helices in nature. However, this is not a perfect model. In tintinnids, the radius, curvature, pitch, and torsion will not be kept constant. In fact, these parameters vary strategically in order to make the most energy-efficient helix (Barros & Ferrández, 2009). In 1756, Pierre Louis Morea de Maupertuis put forth the idea of “least action” that describes when something in nature changes, the amount of “action” needed for that change is as small as possible (Barros & Ferrández, 2009). For helical structures in nature to work well, they should be the best path for a certain kind of energy action. When a helix is given more energy, the parameters such as radius or pitch, etc. will change to accommodate this new energy, making it the best trajectory. Furthermore, these changes will be as small as possible such as to not use more energy. Simply put, in general, when things change in nature, they follow the easiest and most efficient paths. This concept extends to all helices in nature including tintinnid swimming motions.

The phenomenon of least action explains the change in tintinnids’ swimming motions when they detect chemical signals. For example, when they detect preferrable prey, they decrease the energy of their helix, decreasing swimming speed, radius, pitch, and torsion and increasing curvature, making the helix resemble a closed loop. Conversely, when they detect a toxic chemical signal, the energy of their helix increases to escape as quickly as possible. They increase speed, radius, pitch, and torsion and decrease curvature as if they were almost swimming in a straight line.

Tintinnid’s helical swimming pattern is a significant design solution as it allows them to use their energy efficiently and be adaptable to their environment. Their helical swimming utilizes the efficiency inherent in helices. The dynamic adjustments in parameters like speed, radius, pitch, torsion, and curvature allow tintinnids to optimize their energy usage based on environmental cues, such as the presence of prey or the detection of toxic substances. This adaptability allows tintinnids to navigate their environment efficiently and is vital for their survival. Tintinnid’s helical swimming behaviors exemplify the ways in which nature employs mathematical principles for optimal biological design.

**Hexagonal lattice**** in the Lorica**

If we examine the surface of the Lorica (the tintinnid shell) using transmission electron microscopy, a hexagonal lattice becomes clear in Figure 22. Transmission electron microscopy uses a high-energy electron beam to visualize the structure of a sample (in this case, the lorica) at an exceedingly small scale. When the beam interacts with the lorica, the incident electrons will either scatter or hit a fluorescent screen at the bottom of the microscope, and an image of the structure can be revealed (Nanoscience Instruments, n.d.). Not only are the molecules which make up the lorica organized in a crystal lattice but the molecules themselves are also made up of hexanes (Agatha & Simon, 2012). The question therefore arises: why hexagons?

**Figure 22: **Transmission electron micrograph of lorica surface, showing a hexagonal crystal lattice (Agatha & Simon, 2012)

A lattice needs to be composed of shapes that can fit together without any gaps, in other words, ones that can be tessellated. There are only 3 shapes that can be tessellated in two dimensions: squares, triangles, and hexagons, so these will be the only shapes we compare in discussing the ideal shape for a lattice. There are two factors to consider when asking why the hexagon “outcompetes” the square and the triangle; optimality and stability (Puu, 2005). Optimality refers to a shape’s ability to fit a maximum area into a minimal perimeter. The shape that is maximally optimized is the circle, however, it cannot be tessellated, so the hexagon is the “next best” when compared to the square and the triangle. Stability refers to a shape’s ability to remain stable and unaffected in the face of random changes.

Hexagons do not only appear in the form of hexanes or crystal lattices, though. One way to quantitatively examine a hexagonal structure is on a much larger scale; through spatial economics. Hexagonal shapes appear in market areas, and German economist August Lösch theorized that the emergence of hexagonal shapes in market areas was due to their optimality. Optimality, in this case, is thought of more concretely in terms of transportation cost. Though it may seem like transportation cost is an irrelevant measure, the cost of transportation would be minimal when the shape is optimized, so the cost of transportation gives us a way to quantitatively analyze and compare these shapes (Puu, 2005). It was found that when considering transportation costs, a hexagonal market shape only had a 1.4% advantage (in terms of savings) over the square shape, and the square had a 7.8% advantage over the triangle. These advantages are quite small, which supports the conclusion that the emergence of hexagons in nature is likely not due to their optimality but rather to their stability.

The logic supporting the idea that stability is the main factor depends on the concept of transversality. Transversality is a mathematical concept that characterizes intersections of manifolds (more specifically, whether these intersections are transverse). Manifolds are a generalization of the idea of a curved surface, but since this case only involved 2D shapes, (squares, triangles, hexagons), we can just think of manifolds as any ℝ^{n} space (for instance a line or a plane). If an intersection is transverse, the condition in equation 4 must be met (Puu, 2005).

dimensions of manifold 1+ dimension of manifold 2 = dimension of their intersection + dimension of surrounding space

**Equation 4**: the condition that must be met in order for manifolds 1 and 2 to intersect transversally

For example, two planes, p1 and p2 (p1 having dimension 2 + p2 having dimension 2 in 3D space (a subspace of dimension 3) intersecting at a line (dimension 1) can be represented as 2+2=3+1, which is true, therefore this intersection is transverse (this case will hereafter be referenced as case 1).

If another plane is added to the same vector space of dimension 3 already occupied by p1 and p2, calling this plane p3 (another subspace of dimension 2) it will intersect with the line formed by the intersection of p1 and p2 (a line / subspace of dimension 1 as previously noted). Since we are operating within the same 3-dimensional vector space, we have 2+1=3+x, that is (dimension of p3) + (dimension of the intersection of p1 and p2) = (dimension of vector space) + x, where x is the dimension of the intersection of all three planes p1, p2, p3. If we assume x=0, meaning the intersection has dimension 0, the equation still holds (calling this case 2). This will be the case wherein p3 intersects with the already-formed intersection of p1 and p2 at a point.

However, if we try to add a fourth plane, p4, with the previous intersection of dimension zero we have 2+0=3+y, that is (dimension of p4) + (dimension of the existing intersection of p1, p2, and p3) = (dimension of vector space) + y, where y is the dimension of the intersection of all 4 planes, p1, p2, p3, and p4. And in order for this equation to be true, the intersection of all four planes (represented with the letter y in the above equation) would have to be negative, and this is not possible. This case will hereafter be referred to as case 3.

If we now think of the ways in which the different tessellations meet at a point as shown in Figure 23, we can examine whether or not they meet transversally. In fact, a hexagonal tessellation is the only one that intersects transversally since 3 hexagons intersect transversally at a point (like in case 2) whereas 4 squares intersect non-transversally at a point (like in case 3) and 6 equilateral triangles would also intersect non-transversally, following the same logic from case 3.

**Figure 23:** From left to right, diagram of a hexagonal tessellation, square tessellation, and triangle tessellation

But why does it matter whether or not the intersections are transverse or not? Transversality assures that a tessellation can undergo random changes or misspecifications and still be structurally sound. What it essentially proves is that the manifolds are not parallel or tangent to each other at the intersection. This is important because if manifolds are parallel or tangent at the point where they meet, any small disturbance can cause them to merge or separate causing the whole lattice to be deformed. Hexagons are the only shape that intersect transversally, this therefore suggests that the reason for which hexagons are so prevalent in nature is because of their stability.

# Conclusion

Tintinnids’ survival designs integrate mathematical principles into their everyday functions and their overall survival. From the patterns of their helical swimming motion to the structural soundness of their lorica, tintinnids showcase the application of fundamental mathematics in their biological design. The examination of their spatial distribution and population dynamics further emphasizes the role of mathematics, specifically statistics, in understanding their global presence and abundance.

Tintinnids face challenges with varying environmental conditions across habitats, yet tintinnids are present across the world, in multiple different ecosystems. Their ability to form cysts for widespread dispersal contributes to tintinnids’ abundance, ensuring the continuation of the species. Furthermore, distinct species are found in various parts of the world, each species having adapted to preferred conditions based on their environment such as an ideal temperature. Their capacity to adjust to specific environments, utilize different distribution strategies and demonstrate seasonal abundance peaks allows tintinnids to thrive in a wide range of ecological conditions.

The mathematical analysis of tintinnid swimming behavior, facilitated by modern technology and advanced models, enhances our understanding of how they change their swimming patterns based on the environment around them. The exploration of helical swimming motion further underscores the adaptability of tintinnids to their environment. The mathematical parametrization of helices, though not a perfect model, provides a framework to understand the efficiency and adaptability of tintinnid swimming patterns. The graphical analysis illustrates how tintinnids’ swimming changes based on the concentration of prey around them. Furthermore, the concept of least action describes why tintinnids dynamically adjust their swimming parameters: to create an efficient helix and optimize their energy usage for survival.

Tintinnids rely on their lorica as a primary defense mechanism, serving as a durable shield against external threats and predators. The lorica’s strength can be attributed to its hexagonal lattice structure and the mathematical principle of transversality. This geometric arrangement enhances the overall resilience and durability of the lorica, showcasing the mathematical principles present in their survival strategies.

Tintinnids exemplify the intersection between biology and mathematics, highlighting how they capitalize on mathematical principles for survival. The subtle yet extensive influence of mathematical patterns is present in so many aspects of the tiny tintinnid, underscoring the depth to which these organisms have incorporated mathematical efficiency into their survival strategies.

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