Identification of Box Scale and Root Placement for Paddy–Wheat Root System Architecture Using the Box Counting Method

Abstract

Root fractal analysis is instrumental in comprehending the intricate structures of plant root systems, offering insights into root morphology, branching patterns, and resource acquisition efficiency. We conducted a field experiment on paddy–wheat root systems under varying nitrogen fertilizer strategies to address the need for quantitative standardization in root fractal analysis. The study evaluated the impact of nitrogen fertilizer heterogeneity on root length and number. We established functional relationships and correlations among root fractal characteristics and root length across different box dimension scales and various root placement angles at 2.5, 5, 10, 20, 40, and 80 box dimension scales. Results indicated that nitrogen fertilizer had a limited impact on paddy–wheat axile roots, with a coefficient of variation below 0.35 among samples. Box dimension scale influenced 3D fractal dimension (FD) and fractal abundance (FA), with strong correlations (>78%) among 3D fractal features and low sample errors (<6%). The linear correlation coefficient exceeded 72% between 3D FA and root length and 50% between FA and FD. Different axile root placement angles significantly impacted planar fractal results, particularly at a 10° angle. This stability was maintained throughout the sampling period, with high correlation coefficients (>0.76 for FA and >0.5 for FD) and low sample errors (<1.5% for FA and <4.5% for FD). In conclusion, for calculating the 3D fractal characteristics of paddy–wheat axile roots during the seedling stage, box dimension scales of 2.5, 5, 10, 20, 40, and 80, as well as 2, 5, 10, 20, 50 and 100 and 3, 6, 12, 24, 48 and 96, were suitable. When computing the planar fractal characteristics of paddy–wheat axile roots during this stage, a 10° placement angle between axile root systems yields lower errors. These findings enhance root quantification methods, standardize root analysis, and promote the comparability of crop root system fractal data across different varieties and regions, thereby advancing root-related research.

1. Introduction

Root systems play a pivotal role in the growth and survival of plants, owing to the diverse and critical functions they perform. These functions encompass nutrient uptake [1,2], mechanical support [3,4], energy storage [5], soil improvement [6], erosion prevention [7], ecological interactions [8,9], and adaptation to various suboptimal growth environments [10,11,12]. Root system architectures (RSAs) pertain to the spatial distribution of root structures within the soil. The quantification of RSAs involves the numerical description and measurement of plant root characteristics, attributes, and parameters, which include root length, root diameter, branching density, root distribution, and other indicators [13,14,15]. Moreover, the quantification of root systems has broad and vital applications in areas such as crop improvement, soil conservation and sustainable utilization, the cultivation of stress-resistant and adaptive crop varieties, and water resource management [16,17,18,19,20,21,22]. Consequently, the quantification of root architectural traits assumes a position of profound significance.

Fractal analysis has become a widely adopted tool for quantifying the structural characteristics of plant roots [23]. This trend began with Tatsumi et al. [24], who first demonstrated the fractal attributes of crop roots. Subsequent studies further elucidated the utility of fractal dimensions in characterizing root complexity and morphology. Eghball et al. [25] highlighted the applicability of fractal dimensions in quantifying root growth complexity, particularly in the context of maize roots under nitrogen stress. Nielsen et al. [26] conducted simulation modeling and fractal analysis to explore the intricacies of legume root systems, investigating the relationship between three-dimensional (3D) fractal properties and their plane projection counterparts. Wang et al. [27] conducted an analysis of rice plant root systems in response to drought stress based on fractal analysis. Dannowski and Block [28] discussed the assessment of below-ground root system structures in diverse plant communities using fractal dimensions and emphasized how structural complexity varies between different community sizes. The application of fractal analysis extends to comparing root morphology in evergreen and deciduous tree species, as demonstrated by Eamus et al. [29]. Additionally, Chen et al. [30] used a root configuration digitizer and fractal theory to examine the effects of different tillage systems on the 3D spatial distribution of root systems. While quantitative analysis of RSAs features based on box dimensions was widely accepted, the box dimension scales throughout the fractal calculation process, which were various in previous studies, have received limited attention in scientific research. This underexplored aspect may have implications for standardized root system quantification.

Root quantitative standardization is a method used for processing and analyzing plant roots to minimize errors that may arise during the acquisition of data related to root characteristic indices. This standardization facilitates the comparison of root data obtained from different experiments or plant specimens. It not only enhances the credibility of data analysis but also ensures the repeatability and comparability of data across diverse conditions. After the importance of standardizing techniques to compare results from different experiments was emphasized by Persson [31], a series of methods based on root image analysis software [32], X-ray CT scanning and 3D root modeling techniques [33], machine learning [34], high-throughput root phenotyping platforms [35], and mathematical modeling [36] were proposed to quantify root characteristics automatically, reducing measurement errors. Furthermore, Bouda et al. [37] contributed an algorithm to reduce quantization errors by translating and rotating on multiple grids and selecting the configuration with the lowest box count. However, previous studies did not establish clear regulations regarding the placement of roots during the calculation of plane fractals, with only a general stipulation that roots should not obstruct each other. This lack of precision in root placement may introduce errors in the data.

Based on the previous summary, this study posits two hypotheses: (1) The box scale used will impact the calculation results of the 3D fractal dimension of the root system, and (2) the method of placing the root system will affect the calculation results of the planar fractal dimension. This research computes the 3D fractal dimension and planar fractal dimension of the paddy–wheat root system based on the visualization technology of field roots. It thoroughly investigates the impact of box scale and root placement methods on the results. Subsequently, a comprehensive analysis of multiple indicators is conducted to identify the optimal box scale and root placement approach for the precise fractal dimension calculations of the paddy–wheat root system.

2. Materials and Methods

2.1. Description of Experimental Location and Design

The field experiment site is at Babaiqiao (118°59′ E, 31°98′ N) in Nanjing, Jiangsu Province. This region undergoes a subtropical monsoon climate, with an average annual temperature of 15.1 °C and abundant sunlight. Located in the middle and lower reaches of the Yangtze River, the area receives an average annual precipitation of approximately 1000 mm, and the groundwater level is high. This site has a long history of rice–wheat rotation, which has resulted in a relatively compacted plow bottom layer at a soil depth of 10–12 cm due to rotary tillage practices. The soil composition in the 0–10 cm layer includes 3.9% sand, 81.5% silt, and 14.6% clay. The soil’s properties are as follows: organic matter (13.91 g∙kg⁻¹), total nitrogen (1.12 g∙kg⁻¹), total phosphorus (0.66 g∙kg⁻¹), total potassium (12.19 g∙kg⁻¹), available phosphorus (18.18 mg∙kg⁻¹), available potassium (237.25 mg∙kg⁻¹), and pH (5.6).

Wheat seeds (Ningmai13) were sown in a paddy field following the rice harvest, with manual removal of surface straw before sowing. On 6 November 2019, wheat seeds were sown into untilled soil with a plant spacing of 15 mm and row spacing of 200 mm, within small plots measuring 5 × 3 m². This experimental design was replicated three times, with machinery creating drainage ditches between the plots. Additionally, manually opened sowing ditches, measuring 40–50 mm in width and 30–50 mm in depth, were established along the row direction. Fertilizers, including diammonium phosphate at 375 kg∙hm², potassium chloride at 375 kg∙hm², and the urea applications of 70 kg∙hm², 100 kg∙hm², and 130 kg∙hm², representing the three N application rates, i.e., N70, N100, and N130, respectively, were applied along the sowing ditches, maintaining a distance of 50 mm. Abundant rainfall obviated the need for irrigation during the growing seasons. Herbicides and pesticides were applied following standard agricultural practices to mitigate yield loss.

2.2. Visualization of the Paddy–Wheat Root System

The structure of a wheat root system comprises both seminal roots and adventitious roots, which are referred to as axile roots in this manuscript. The geometric interplay between axile roots’ growth direction and length determines their trajectory and depth [38]. Additionally, root growth systems exhibit hierarchy, with axile roots serving as the foundational component of the lower-order root members [39]. This hierarchy establishes the framework for root architecture. The distinct features of axile roots underscore the importance of visualizing and quantifying them as a crucial objective.

Four root zone soils of plants with the average appearance in one plot of each nitrogen application rate were sampled every 14 days from sowing until the 70th day, so twelve samples were collected for visualization of the paddy–wheat root system at each sampling time. A sizable soil core, measuring 180 mm in diameter and 250 mm in height, was extracted using a shovel, ensuring that it encompassed the entire wheat root system and was concentric with the base of the plant stem. The collected soil cores were then transported to the laboratory for digitization [15,35]. The wheat root system data, referred to as wheat axile roots, were obtained through a layered excavation procedure, following the methodology outlined by Chen et al. [40]. An adapted digitizer was employed to capture this data, which was subsequently transferred to Pro-E for 3D visual modeling (Figure 1a).

2.3. Fractal Quantification of the Paddy–Wheat Root System

The 3D fractal dimension (FD) and fractal abundance (FA) of the paddy–wheat root system were calculated, using MATLAB [41], based on the visualization of the paddy–wheat root system and the principles of fractal analysis. The 3D fractal analysis of the paddy–wheat root system was developed based on the box dimension principle. A D0 dimensional sphere was employed to encase the root system with a radius (ε), and the minimum number (N(ε)) required to wrap it as ε → 0 was determined. Subsequently, the fractal dimension was calculated using Equation (1).

The reconstructed RSAs were covered with volume, and the cube was divided into different scales, as shown in Figure 1b,c. The first step was to define the coordinates of grid positions ([x_n−1, y_n−1, z_n−1], [x_n−1, y_n−1, z_n], [x_n−1, y_n, z_n−1], [x_n, y_n−1, z_n−1], [x_n, y_n−1, z_n−1], [x_n, y_n−1, z_n], [x_n, y_n, z_n−1], and [x_n, y_n, z_n]). Then, the grids were counted once when the known points (x₀, y₀, z₀) met the following conditions: x_n−1 ≤ x₀ < x_n, y_n−1 ≤ y₀ < y_n, and z_n−1 ≤ z₀ < z_n. Next, the numbers of grids (N(r)) occupied by the root system at different grid scales (r) were calculated, which all started from the origin of the root system. Finally, log10(K), representing the 3D fractal abundance (FA), and the 3D fractal dimension (FD) were determined using Equation (2).

$$D_0=\underset{\epsilon \to 0}{lim}\frac{\ln N\left(\epsilon \right)}{\ln \left(1∕\epsilon \right)}$$

(1)

$$\mathit{log}10\left(N\left(r\right)\right)=-F\times log10\left(r\right)+log10\left(K\right)$$

(2)

In this study, we conducted a comprehensive investigation into the influence of root placement on the results of planar fractal calculations. We employed square grids with varying side lengths (r = 2.5, 5, 10, 20, 40, and 80 mm) within Pro/E to segment the wheat root system. The number of grids was determined for various root placement methods, and subsequently, the planar FD and planar FA were computed using Equation (2). It is important to note that throughout this study, the root systems were assumed to be rigid bodies without bending. Additionally, four distinct settings were explored for root system placement, specifically focusing on the included angles between the axile roots, which were set at 5°, 10°, 15°, and 20°, as explained in Figure 2.

3. Results

3.1. Axile Root Growth Dynamics in Paddy–Wheat Seedings

Nutrient distribution has inherent heterogeneity within the field soil, which is further exacerbated by uneven nitrogen fertilizer application. It is worth noting that variations in nitrogen supply tend to have a more pronounced effect on the responsiveness of lateral roots (LRs) in comparison to primary roots [42,43,44]. In this study, twelve wheat plants from the same field were collected to form a sample set for each sampling time. Figure 3 shows the dynamics of root length, root number, and coefficient of variation for paddy–wheat RSAs. The fluctuations and increases in the length and number of the paddy–wheat root system were observed with time (Figure 3a,b). Interestingly, the coefficient of variation among root samples for root number was found to be smaller than that for root length, with all values being less than 0.35, as illustrated in Figure 3c,d.

The analysis of root length and root number revealed that the heterogeneity variation between samples at different stages could be acceptable. Furthermore, it suggested that the nitrogen fertilizer application in this experiment had a minimal impact on the axile root system’s root length and number. Consequently, the sample set of twelve wheat roots remained suitable for subsequent analyses.

3.2. The 3D Fractal Dynamics of Paddy–Wheat RSAs at Different Box Dimension Scales during Seedling Stage

A range of box dimension scales were used to assess the fractal characteristics of paddy–wheat RSAs. These scales, including 2.5, 5, 10, 20, 40, and 80; 2, 5, 10, 20, 50, and 100; 3, 6, 12, 24, 48, and 96; 4, 8, and 16; and 10, 20, 40, 80, and 160, were derived from previous studies [23,26,29,30,37] and have been utilized in this manuscript. Figure 4 illustrates the changes in 3D FA, 3D FD, and errors across different box dimension scales over time. The 3D FA values for the paddy–wheat RSAs, like the FD, displayed a dynamic increasing trend, except when ‘r’ was set at 4, 8, and 16. Furthermore, the dynamics of 3D FA and 3D FD for the paddy–wheat RSAs at the box dimension scales of 2.5, 5, 10, 20, 40, and 80; 2, 5, 10, 20, 50, and 100; 3, 6, 12, 24, 48, and 96; and 4, 8, 16 were significantly more pronounced than those observed at the box dimension scales of 10, 20, 40, 80, and 160.

It is noteworthy that error values among samples were significantly higher when the box dimension scales were set at 10, 20, 40, 80, and 160 compared to the other four box dimension scales. These findings emphasize the dynamics of 3D fractal parameters within paddy–wheat RSAs and their relevance to understanding root development and nutrient distribution.

Correlation tests and analyses of differences among FDs and FAs were conducted to examine the impacts of various box dimension scales on the 3D fractal characteristics of paddy–wheat RSAs. The results are presented in

Table 1. Correlation and differential analysis of 3D fractal features for paddy–wheat RSAs at various box dimension scales (1 represents r values of 2.5, 5, 10, 20, 40, and 80; 2 represents r values of 2, 5, 10, 20, 50, and 100; 3 represents r values of 3, 6, 12, 24, 48, and 96; 4 represents r values of 4, 8, and 16; and 5 represents r values of 10, 20, 40, 80, and 160).

Fractal Traits		Fractal Abundance					Fractal Dimension
Sampling Time	Scale	1	2	3	4	5	1	2	3	4	5
14	1	1					1
	2	0.991	1				0.969	1
	3	0.983	0.996	1			0.959	0.996	1
	4	0.951	0.972	0.967	1		0.652	0.726	0.711	1
	5	0.945 *	0.95 *	0.955 *	0.897 *	1	0.884 **	0.871 *	0.882 *	0.446 **	1
28	1	1					1
	2	0.991	1				0.906	1
	3	0.989	0.991	1			0.934	0.944	1
	4	0.957	0.95	0.96	1		0.789	0.689	0.738	1
	5	0.955	0.968 *	0.948 *	0.859 *	1	0.735 **	0.859 **	0.785 **	0.311 **	1
42	1	1					1
	2	0.954	1				0.811 *	1
	3	0.921	0.975	1			0.778	0.989	1
	4	0.705	0.823	0.825	1		0.127 *	0.443	0.453	1
	5	0.9 **	0.888 **	0.862 **	0.516 **	1	0.758 **	0.817 **	0.799 **	−0.038 **	1
56	1	1					1
	2	0.983	1				0.795	1
	3	0.977	0.993	1			0.792	0.979	1
	4	0.95	0.943	0.942	1		0.375 *	0.11	0.162	1
	5	0.927 *	0.955	0.961 *	0.847	1	0.742 **	0.921 **	0.948 **	0.018 **	1
70	1	1					1
	2	0.986	1				0.812	1
	3	0.984	0.993	1			0.816	0.987	1
	4	0.897	0.931	0.935	1		0.072 **	0.366 *	0.407 **	1
	5	0.905	0.858	0.846	0.737	1	0.785 **	0.638	0.588 **	−0.059	1

** indicates highly significant differences (p < 0.01), * indicates significant differences (p < 0.05).

.

Throughout the sampling period, we observed consistently high correlations in FD among different box dimension scales, except for a minor exception at 42 days after sowing. At 14 days after sowing, significant differences in FA were detected between box dimension scales of 10, 20, 40, 80, 160 and others (p < 0.05). Similar differences were noted 28 days after sowing, but they were significant except for box dimension scales of 2.5, 5, 10, 20, 40, and 80 (p < 0.05). Subsequently, differences between box dimension scales of 10, 20, 40, 80, 160 and others became more pronounced (p < 0.01). However, at 70 days after sowing, no significant differences were observed in FA. Conversely, when examining FD, strong correlations (0.778–0.996) were observed throughout the sampling period among three box dimension scales—2.5, 5, 10, 20, 40, and 80; 2, 5, 10, 20, 50, and 100; and 3, 6, 12, 24, 48, and 96—except during the 42 day period, where significant differences were evident between box dimension scales of 2.5, 5, 10, 20, 40, and 80 and 2, 5, 10, 20, 50, and 100 (p < 0.05). No differences were detected for the three box dimension scales in other sampling periods.

Across the sampling period, relatively weak correlations (0.018–0.446) existed between FDs for box dimension scales of 4, 8, and 16 and 10, 20, 40, 80, and 160, with significant differences noted, except at 70 days after sowing (p < 0.01). Strong correlations were observed between FDs for box dimension scales of 10, 20, 40, 80, and 160 and box dimension scales of 2.5, 5, 10, 20, 40, and 80; 2, 5, 10, 20, 50, and 100; and 3, 6, 12, 24, 48, and 96, with highly significant differences observed at 28, 42, and 56 days after sowing (p < 0.01).

To confirm the suitability of box dimension scales, the relationships between root length, 3D FD, and 3D FA at various growth stages were examined (

Table 2. Linear regression of root length and 3D fractal features for paddy–wheat RSAs at various box dimension scales (where L represents root length, X₁ represents 3D FD, and X₂ represents 3D FA).

Scale	Sampling Time	Regression Function	R2	Regression Function	R2	Regression Function	R2
2.5~80	14	L = 654.35X2 − 1371.61	0.85	X2 = 1.37X1 + 1.42	0.71	L = 703.64X1 − 234.37	0.37
	28	L = 1010.35X2 − 2355.94	0.91	X2 = 1.74X1 + 1.08	0.5	L = 1462.56X1 − 949.68	0.32
	42	L = 1420.79X2 − 3592.24	0.72	X2 = 1.11X1 + 1.87	0.58	L = 1018.59X1 − 286.46	0.17
	56	L = 1986.39X2 − 5032.77	0.83	X2 = 1.95X1 + 0.91	0.63	L = 2986.83X1 − 2472.94	0.34
	70	L = 1868.64X2 − 4940.65	0.82	X2 = 2.07X1 + 0.8	0.64	L = 3128X1 − 2567.83	0.34
2~100	14	L = 696.74X2 − 1463.96	0.88	X2 = 1.33X1 + 1.48	0.65	L = 735.2X1 − 229.48	0.36
	28	L = 1001.8X2 − 2313.35	0.93	X2 = 2.18X1 + 0.62	0.68	L = 2043.63X1 − 1538.68	0.55
	42	L = 1458.58X2 − 3664.9	0.85	X2 = 1.4X1 + 1.57	0.76	L = 1824.11X1 − 1142.56	0.52
	56	L = 1813.11X2 − 4728.63	0.85	X2 = 2.09X1 + 0.79	0.83	L = 3383.06X1 − 2850.99	0.56
	70	L = 1825.1X2 − 4753.67	0.88	X2 = 2.01X1 + 0.85	0.53	L = 3839.45X1 − 3312.05	0.61
3~96	14	L = 616.87X2 − 1240.37	0.84	X2 = 1.36X1 + 1.45	0.71	L = 667.57X1 − 166.41	0.38
	28	L = 941.88X2 − 2133.62	0.93	X2 = 2.01X1 + 0.8	0.66	L = 1735.37X1 − 1210.92	0.51
	42	L = 1239.29X2 − 3013.08	0.76	X2 = 1.37X1 + 1.59	0.84	L = 1432.76X1 − 742.68	0.45
	56	L = 1663.29X2 − 4301.39	0.87	X2 = 1.98X1 + 0.89	0.85	L = 2952.93X1 − 2423.72	0.6
	70	L = 1685.73X2 − 4363.49	0.86	X2 = 2.02X1 + 0.87	0.76	L = 3065.62X1 − 2498.86	0.53
4~16	14	L = 894.93X2 − 2060.9	0.83	X2 = 0.9X1 + 1.94	0.21	L = 207.82X1 + 308.22	0.01
	28	L = 1005.26X2 − 2334.72	0.82	X2 = 1.03X1 + 1.85	0.32	L = 454.25X1 + 145.99	0.05
	42	L = 1692.95X2 − 4408.71	0.76	X2 = 0.85X1 + 2.18	0.51	L = 746.94X1 + 58.32	0.1
	56	L = 2493.05X2 − 6870.5	0.9	X2 = 1.59X1 + 1.37	0.23	L = 3174.81X1 − 2572.91	0.13
	70	L = 2072.75X2 − 5519.67	0.86	X2 = 1.11X1 + 1.97	0.3	L = 1509.84X1 − 541.19	0.1
10~160	14	L = 366.28X2 − 457.79	0.75	X2 = 1.61X1 + 1.21	0.9	L = 518.31X1 + 51.27	0.53
	28	L = 651.81X2 − 1167.24	0.86	X2 = 2.23X1 + 0.7	0.89	L = 1383.77X1 − 650.53	0.7
	42	L = 587.12X2 − 868.92	0.48	X2 = 1.62X1 + 1.34	0.91	L = 821.18X1 + 51.71	0.32
	56	L = 1000.53X2 − 2037.08	0.7	X2 = 1.93X1 + 1.03	0.93	L = 1964.48X1 − 759.36	0.51
	70	L = 749X2 − 1220.5	0.47	X2 = 1.82X1 + 1.16	0.93	L = 1055.23X1 − 12.82	0.26

). The analysis showed a weak correlation between root length and 3D FD. However, a strong linear correlation was observed between root length and 3D FA for most box dimension scales, except when ‘r’ values were set at 10, 20, 40, 80, and 160. Similarly, a significant linear association was found between 3D FA and 3D FD, except for box dimension scales with ‘r’ values of 4, 8, and 16. The analysis above indicated that for the fractal quantification of the axile root system of paddy–wheat, the appropriate box dimension scales are 2.5, 5, 10, 20, 40, and 80; 2, 5, 10, 20, 50, and 100; and 3, 6, 12, 24, 48, and 96.

3.3. The Fractal Dynamics of Paddy–Wheat RSAs at Different Placement Angles during the Seedling Stage

The impact of root placement angles (5°, 10°, 15°, and 20°) on planar FD and FA in paddy–wheat RSAs was investigated during the seedling stage. Using fixed box dimension scales (2.5, 5, 10, 20, 40, and 80), the roots were assumed to be rigid and non-bending, growing in ideal environmental conditions. Results (Figure 5) showed that FA increased over time for various angles, with the lowest values being at 5°. The planar FD significantly increased over time at 5° and 10° angles but fluctuated at 15° and 20°. Notably, errors were highest at a 5° angle compared to others when evaluating dynamic errors between root system samples at different angles.

These findings enhance our understanding of how root placement angles influence planar fractal characteristics in paddy–wheat RSAs during seedling growth, with implications for more efficient agricultural practices.

A thorough analysis of planar FA and FD values at different sampling points to identify optimal placement angles was performed (

Table 3. Correlation and significance analysis of planar fractal features at different placement angles and sampling times.

		Planar Fractal Abundance (Planar FA)				Planar Fractal Dimension (Planar FD)
		5°	10°	15°	20°	5°	10°	15°	20°
14	5°	1				1
	10°	0.983	1			0.954	1
	15°	0.969 **	0.98	1		0.816	0.818	1
	20°	0.967 **	0.986	0.991	1	0.861	0.881	0.909 *	1
28	5°	1				1
	10°	0.976	1			0.89	1
	15°	0.981	0.981	1		0.78	0.852	1
	20°	0.964 *	0.956	0.984	1	0.478	0.412 *	0.457	1
42	5°	1				1
	10°	0.764	1			0.645	1
	15°	0.917 **	0.827 *	1		0.508 *	0.365	1
	20°	0.922 **	0.803	0.907	1	0.447 **	0.144 *	−0.045	1
56	5°	1				1
	10°	0.971	1			0.773	1
	15°	0.984	0.98	1		0.689 *	0.703 *	1
	20°	0.95	0.967	0.969	1	0.154 *	0.444 *	0.513	1
70	5°	1				1
	10°	0.959	1			0.497	1
	15°	0.979	0.937	1		0.132 **	0.253 **	1
	20°	0.929	0.891	0.913	1	−0.123 **	−0.008 **	0.011	1

** indicates highly significant differences (p < 0.01), * indicates significant differences (p < 0.05).

). Our findings consistently showed a high correlation (>0.764) among FA values across various placement angles throughout the study. Notably, at 14 days after sowing, highly significant differences (p < 0.01) were found between FA values at the 5° placement angle and those at 15° and 20° angles. A significant difference (p < 0.05) was also noted at 28 days after sowing between the 5° and 20° angles. At 42 days after sowing, the FA values at the 5° angle displayed highly significant differences (p < 0.01) compared to both the 15° and 20° angles, with a significant difference (p < 0.05) between the 10° and 20° angles.

For FD values, a notably high correlation (>0.816) was observed among different placement angles at 14 days after sowing, with significant differences (p < 0.05) between the 15° and 20° angles. At 28 days after sowing, strong correlations (0.78–0.89) existed among FD values for the 5°, 10°, and 15° angles, with no significant differences. However, the 20° angle displayed a lower correlation (0.412–0.478) with FD values at these angles and a significant difference (p < 0.05) compared to the 10° angle. At 42 days after sowing, the 20° angle exhibited a lower correlation with FD values at other angles, with no significant differences compared to the 15° angle. Lastly, at both 56 and 70 days after sowing, significant differences were observed in FD values between the 15° and 20° angles and those at the 5° and 10° angles.

These results provide valuable insights into the relationship between placement angles and planar fractal characteristics in paddy–wheat RSAs during the seedling stage, offering guidance for optimizing agricultural practices.

The correlations between root length and its planar fractal features were established at various growth stages. A linear relationship was observed between root length and planar FA. Additionally, the correlations between 3D FA and planar FA were examined, as well as 3D FD and planar FD, subjecting these relationships to rigorous significance analyses. The results highlighted a high linear correlation between 3D FA and planar FA, except at the 5° placement angle. Conversely, the correlation between 3D FD and planar FD was generally modest, with a significant disparity noted from 28 days after sowing to 56 days after sowing. Revisiting the definitions of FA, FD, root length, and their interrelationships based on prior research, we reaffirmed that a 10° placement angle is well suited for calculating the planar FD and FA of axile root systems of paddy–wheat.

4. Discussion

Root length represents the cumulative extension of a plant’s roots in the soil, reflecting its soil exploration range and nutrient and water absorption capacity [45]. Root number refers to the number of roots within a plant’s root system and is a critical parameter in plant ecology, growth, and adaptability [46]. These parameters are crucial for evaluating soil resource utilization efficiency across various plant varieties or populations, and the relevant topics encompass environmental responses, soil stability, and ecosystem sustainability [47,48,49]. Additionally, nitrogen fertilizer is pivotal in enhancing crop yield and quality by influencing root growth and nutrient absorption. It is commonly observed that lateral roots are more sensitive to variations in nitrogen supply than primary roots [42], and primary root growth in Arabidopsis is usually reported to be relatively insensitive to [43] or even stimulated by [44] the normal range of nitrate concentrations. The results of this study suggest that the nitrogen application rate set in the field experiment may not have a significant impact on the length and number of the axial root system of paddy–wheat (Figure 3a,b).

Nitrogen fertilizer was applied primarily to the surfaces of no-till field plots after sowing. However, this method may not ensure the full impact of the applied nitrogen fertilizer on paddy–wheat root systems. Moreover, inherent soil fertility combined with the application of nitrogen fertilizer may maintain wheat root systems within a normal range of nitrogen concentrations during their growth. Therefore, it is reasonable to quantitatively assess the distribution characteristics of paddy–wheat axile roots by selecting root samples from different nitrogen fertilizer treatments simultaneously, treating them as a single sample set. Furthermore, differential analyses of various samples revealed minimal differences (Figure 3c,d), further confirming the validity of using 12 root samples from different nitrogen fertilizer treatments at each sampling time as a sample set.

Root fractal analysis describes root system branching and spatial patterns [50,51,52]. A higher value of fractal analysis indicates greater complexity linked to enhanced nutrient absorption and soil resource utilization. It provides valuable insights into root system complexity and nutrient efficiency. Further, conducting a 3D fractal analysis of root systems enables a comprehensive understanding of the spatial structure and exploratory capabilities of roots [53]. In this study, the impact of varying box dimension scales on dynamic 3D fractal characteristics in paddy–wheat RSAs was quantified based on 3D fractal analysis and prior research [25,26,27,28,29,30], which was important but ignored before. Results revealed increasing 3D FD and FA in root systems over time. Changes in box dimension scales significantly affected the values of 3D root fractal characteristics in paddy–wheat RSAs (Figure 4). Also, the relationship models among FA, FD, and root length at various box dimension scales were established (Table 2). Wang et al. discovered a correlation between FA values and root length traits in well-watered conditions for rice [27]. Walk et al. dynamically modeled roots of various architectures and found an association between FA and root length [54]. Chen et al. investigated the relationship between root fractal characteristics and root length features under various cultivation methods. Their study revealed a statistically significant linear correlation between FA and root length [30]. Given the strong correlation between FA and root length, which was supported by previous research findings indicating a significant correlation between root length and FA, the box dimension ranges 1, 2, and 3 were suitable for quantifying 3D root fractal characteristics in paddy–wheat RSAs.

Root systems’ subterranean growth has made investigating their spatial distribution challenging. Most studies focus on clean, scanned root data without specific placement requirements. This research highlighted how root placement methods affect planar root system fractal calculations (Figure 5 and Table 3). Placement can alter root density, including length, quantity, and branching patterns. This study quantitatively assessed correlations between planar root system fractal attributes and root length and explored the relationship between planar and 3D fractals (

Table 4. Linear regression of root length and planar fractal features at various placement angles (where L represents root length, X₁ represents planar FD, and X₂ represents planar FA).

Angle	Sampling Time	Regression Function	R²	Regression Function	R²	Regression Function	R²
5°	14	L = 709.84X2 − 1413.09	0.93	X2 = 1.54X1 + 0.99	0.84	L = 1028.21X1 − 636.47	0.69
	28	L = 839X2 − 1752.35	0.94	X2 = 1.73X1 + 0.82	0.68	L = 1361.57X1 − 996.75	0.56
	42	L = 1369.97X2 − 3297.28	0.95	X2 = 0.95X1 + 1.85	0.34	L = 1046.26X1 − 445.77	0.21
	56	L = 1408.22X2 − 3387.74	0.96	X2 = 1.78X1 + 0.78	0.73	L = 2452.41X1 − 2219.03	0.67
	70	L = 1783.74X2 − 4574.13	0.95	X2 = 2.04X1 + 0.45	0.72	L = 3284.58X1 − 3303.27	0.56
10°	14	L = 690.6X2 − 1434.65	0.93	X2 = 1.61X1 + 0.97	0.83	L = 994.65X1 − 628.59	0.62
	28	L = 929.09X2 − 2101.8	0.91	X2 = 1.8X1 + 0.79	0.51	L = 1383.04X1 − 1018.05	0.32
	42	L = 1322.32X2 − 3218.15	0.62	X2 = 0.91X1 + 1.97	0.35	L = 63.5X1 + 805.8	0
	56	L = 1780.47X2 − 4651.27	0.92	X2 = 1.49X1 + 1.26	0.38	L = 1894.91X1 − 1448.97	0.18
	70	L = 2135.3X2 − 5835.29	0.87	X2 = 0.96X1 + 1.99	0.14	L = 304.23X1 + 739.23	0
15°	14	L = 774.13X2 − 1719.64	0.87	X2 = 1.47X1 + 1.16	0.53	L = 746.85X1 − 355.55	0.2
	28	L = 1044.5X2 − 2471.23	0.97	X2 = 2.3X1 + 0.28	0.34	L = 2287.51X1 − 2051.09	0.3
	42	L = 1641.4X2 − 4302.38	0.89	X2 = 1.02X1 + 1.91	0.32	L = 923.35X1 − 247.88	0.09
	56	L = 1967.36X2 − 5300.07	0.94	X2 = 1.21X1 + 1.7	0.18	L = 1445.05X1 − 802.23	0.06
	70	L = 2556.62X2 − 7180.9	0.93	X2 = −0.06X1 + 3.33	0	L = 1992.31X1 − 3600.25	0.07
20°	14	L = 784.01X2 − 1744.37	0.9	X2 = 1.88X1 + 0.76	0.65	L = 1148.11X1 − 776.74	0.35
	28	L = 1178.29X2 − 2877.63	0.95	X2 = 0.82X1 + 2.04	0.04	L = 707.76X1 − 182.18	0.02
	42	L = 1581.86X2 − 4940.05	0.91	X2 = 0.55X1 + 2.48	0.06	L = 153.55X1 + 698.69	0
	56	L = 2255.52X2 − 6224.38	0.92	X2 = 0.05X1 + 3.13	0	L = −1146.58X1 + 2733.69	0.05
	70	L = 2666.38X2 − 7525.43	0.9	X2 = −0.2X1 + 3.49	0.01	L = −2110.5X1 + 3684.59	0.16

and

Table 5. Linear regression of 3D fractal features and planar fractal features of different placement angles (where X₁ represents planar FD, X₂ represents 3D FD, Y₁ represents planar FA, and Y₂ represents 3D FA).

Angle	Sampling Time	Regression Function	R²	Regression Function
5°	14	Y2 = 0.93Y1 + 0.35	0.81 **	X2 = 0.68X1 + 0.32	0.4
	28	Y2 = 0.78Y1 + 0.74	0.92	X2 = 0.42X1 + 0.58	0.37 **
	42	Y2 = 0.73Y1 + 0.91	0.76 **	X2 = 0.68X1 + 0.28	0.53 **
	56	Y2 = 0.66Y1 + 1.13	0.91	X2 = 0.44X1 + 0.58	0.57 **
	70	Y2 = 0.77Y1 + 0.78	0.77	X2 = 0.52X1 + 0.48	0.41 **
10°	14	Y2 = 0.94Y1 + 0.24	0.86	X2 = 0.72X1 + 0.24	0.43 *
	28	Y2 = 0.86Y1 + 0.42	0.88	X2 = 0.34X1 + 0.67	0.13 **
	42	Y2 = 0.88Y1 + 0.43	0.77	X2 = 0.74X1 + 0.24	0.48 **
	56	Y2 = 0.82Y1 + 0.58	0.85	X2 = 0.58X1 + 0.41	0.45 **
	70	Y2 = 0.92Y1 + 0.26	0.69	X2 = 0.32X1 + 0.77	0.08 **
15°	14	Y2 = 1.03Y1 − 0.09	0.77	X2 = 0.53X1 + 0.46	0.13 **
	28	Y2 = 0.97Y1 + 0.09	0.93	X2 = 0.62X1 + 0.35	0.15 **
	42	Y2 = 0.87Y1 + 0.39	0.71	X2 = 0.65X1 + 0.35	0.26 **
	56	Y2 = 0.92Y1 + 0.23	0.89	X2 = 0.66X1 + 0.34	0.35 **
	70	Y2 = 1.14Y1 − 0.46	0.79	X2 = 0.01X1 + 1.17	0 *
20°	14	Y2 = 1.08Y1 − 0.22	0.86	X2 = 1.03X1 − 0.08	0.37
	28	Y2 = 1.11Y1 − 0.36	0.96	X2 = 1.06X1 − 0.13	0.32 **
	42	Y2 = 1.04Y1 − 0.12	0.8	X2 = 0.41X1 + 0.66	0.05 **
	56	Y2 = 0.98Y1 + 0.05	0.75	X2 = -0.04X1 + 1.2	0 **
	70	Y2 = 1.21Y1 − 0.67	0.79	X2 = 0.08X1 + 1.09	0.01

** indicates highly significant differences (p < 0.01), * indicates significant differences (p < 0.05).

). As found by the previous research, root system FA was significantly positively correlated with root system length, and there was a significant correlation between planar fractal characteristics and 3D fractal characteristics [25,29]. The study concludes that, for calculating planar fractal dimensions of axile root systems in paddy–wheat RSAs during 14 to 70 days, a 10° angle between roots is the most suitable placement method. This finding significantly contributes to standardizing root quantification methods, recognizing the profound impact of root placement on planar fractal calculations, and ensuring the sustainable development of agriculture.

5. Conclusions

Our research indicates that applying nitrogen fertilizer had minimal impact on the paddy–wheat axile roots in the field, with low variation among samples. The choice of box dimension scale significantly influenced 3D fractal features, showing strong correlations and minimal errors. Notably, there was a robust linear correlation between 3D FA and root length and a moderate correlation between FA and FD. The positioning angle of axile roots had a substantial impact on planar fractal results, particularly at a 10° angle, which consistently yielded high correlations and low errors throughout the sampling period. In conclusion, using specific box dimension scales and a 10° positioning angle for axile root systems enhances the precision of root quantification methods, standardizes root analysis procedures, and facilitates the comparability of crop root system fractal data across different varieties and regions, contributing to advancements in sustainable agriculture and root analysis research.