scholarly journals Catalan solids derived from three-dimensional-root systems and quaternions

2010 ◽  
Vol 51 (4) ◽  
pp. 043501 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Ramazan Koç
Keyword(s):  
Three Dimensional ◽  
Root Systems

scholarly journals Platonic solids generate their four-dimensional analogues

2013 ◽  
Vol 69 (6) ◽  
pp. 592-602 ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Coxeter Groups ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Root Systems ◽  
Point Of View ◽  
Platonic Solids ◽  
Four Dimensions ◽  
Potential Applications ◽  
Regular Convex ◽  
Quasi Crystals

This paper shows how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone.Viathe Cartan–Dieudonné theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, theF4root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_{1}\oplus I_{2}(n) which induces I_{2}(n)\oplus I_{2}(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

scholarly journals Effect of parameter choice in root water uptake models – the arrangement of root hydraulic properties within the root architecture affects dynamics and efficiency of root water uptake

2014 ◽  
Vol 18 (10) ◽  
pp. 4189-4206 ◽  
Author(s):  
M. Bechmann ◽  
C. Schneider ◽  
A. Carminati ◽  
D. Vetterlein ◽  
S. Attinger ◽  
...  
Keyword(s):  
Root System ◽  
Water Uptake ◽  
Root Length ◽  
Hydraulic Properties ◽  
Three Dimensional ◽  
Root Systems ◽  
Root Water Uptake ◽  
Hydraulic Lift ◽  
Hydraulic Redistribution ◽  
Young Root

Abstract. Detailed three-dimensional models of root water uptake have become increasingly popular for investigating the process of root water uptake. However, they suffer from a lack of information on important parameters, particularly on the spatial distribution of root axial and radial conductivities, which vary greatly along a root system. In this paper we explore how the arrangement of those root hydraulic properties and branching within the root system affects modelled uptake dynamics, xylem water potential and the efficiency of root water uptake. We first apply a simple model to illustrate the mechanisms at the scale of single roots. By using two efficiency indices based on (i) the collar xylem potential ("effort") and (ii) the integral amount of unstressed root water uptake ("water yield"), we show that an optimal root length emerges, depending on the ratio between roots axial and radial conductivity. Young roots with high capacity for radial uptake are only efficient when they are short. Branching, in combination with mature transport roots, enables soil exploration and substantially increases active young root length at low collar potentials. Second, we investigate how this shapes uptake dynamics at the plant scale using a comprehensive three-dimensional root water uptake model. Plant-scale dynamics, such as the average uptake depth of entire root systems, were only minimally influenced by the hydraulic parameterization. However, other factors such as hydraulic redistribution, collar potential, internal redistribution patterns and instantaneous uptake depth depended strongly on the arrangement on the arrangement of root hydraulic properties. Root systems were most efficient when assembled of different root types, allowing for separation of root function in uptake (numerous short apical young roots) and transport (longer mature roots). Modelling results became similar when this heterogeneity was accounted for to some degree (i.e. if the root systems contained between 40 and 80% of young uptake roots). The average collar potential was cut to half and unstressed transpiration increased by up to 25% in composed root systems, compared to homogenous ones. Also, the least efficient root system (homogenous young root system) was characterized by excessive bleeding (hydraulic lift), which seemed to be an artifact of the parameterization. We conclude that heterogeneity of root hydraulic properties is a critical component for efficient root systems that needs to be accounted for in complex three-dimensional root water uptake models.

scholarly journals Simulating rhizodeposition patterns around growing and exuding root systems

2021 ◽  
Author(s):  
Magdalena Landl ◽  
Adrian Haupenthal ◽  
Daniel Leitner ◽  
Eva Kroener ◽  
Doris Vetterlein ◽  
...  
Keyword(s):  
Root Architecture ◽  
Three Dimensional ◽  
Temporal Distribution ◽  
Root Systems ◽  
Distribution Patterns ◽  
Three Dimensions ◽  
Root Branching ◽  
Architecture Model ◽  
Spatio Temporal ◽  
Model Approach

1AbstractIn this study, we developed a novel model approach to compute the spatio-temporal distribution patterns of rhizodeposits around growing root systems in three dimensions. This model approach allows us for the first time to study the evolution of rhizodeposition patterns around complex three-dimensional root systems. Root systems were generated using the root architecture model CPlantBox. The concentration of rhizodeposits at a given location in the soil domain was computed analytically. To simulate the spread of rhizodeposits in the soil, we considered rhizodeposit release from the roots, rhizodeposit diffusion into the soil, rhizodeposit sorption to soil particles, and rhizodeposit degradation by microorganisms. To demonstrate the capabilities of our new model approach, we performed simulations for the two example rhizodeposits mucilage and cit-rate and the example root system Vicia faba. The rhizodeposition model was parameterized using values from the literature. Our simulations showed that the rhizosphere soil volume with rhizodeposit concentrations above a defined threshold value (i.e., the rhizodeposit hotspot volume), exhibited a maximum at intermediate root growth rates. Root branching allowed the rhizospheres of individual roots to overlap, resulting in a greater volume of rhizodeposit hotspots. This was particularly important in the case of citrate, where overlap of rhizodeposition zones accounted for more than half of the total rhizodeposit hotspot volumes. Coupling a root architecture model with a rhizodeposition model allowed us to get a better understanding of the influence of root architecture as well as rhizodeposit properties on the evolution of the spatio-temporal distribution patterns of rhizodeposits around growing root systems.

scholarly journals Controls on creek margin stability by the root systems of saltmarsh vegetation, Beaulieu Estuary, Southern England

2019 ◽  
Vol 2 (1) ◽  
pp. 21-38 ◽  
Author(s):  
Yining Chen ◽  
Charlotte Thompson ◽  
Michael Collins
Keyword(s):  
High Resistance ◽  
Three Dimensional ◽  
Ct Scanning ◽  
Root Systems ◽  
Distribution Patterns ◽  
Root Diameter ◽  
Diameter Distribution ◽  
Southern England ◽  
Quantitative Measurements ◽  
Atriplex Portulacoides

The retreat of cliffs (lateral expansion) within tidal creeks results in a net loss of saltmarshes, but this retreat process can be retarded by root systems. To understand the interaction between root presence and bank sediment, quantitative measurements of two saltmarsh species root systems (Atriplex portulacoides and Juncus maritima) were carried out in a saltmarsh in Southern England, and their relationships with bank stability were examined. Computed Tomography (CT) Scanning techniques were used to investigate three-dimensional root architecture. The data obtained (e.g., root volume, diameter, and distribution patterns of roots) were examined alongside more traditional root density measurements. The volumetric percentage, ratio between horizontal (lateral) and vertical roots (H/V ratio), and root diameter distribution are discussed in relation to their influence on bank sediment erosion threshold and shear strength. The results suggest that Atriplex portulacoides is more effective than Juncus maritimus in stabilising banks. This is because root systems that provide a high resistance to flow-induced erosion are better than those that provide a high resistance to gravity-induced erosion in stabilising cliff banks. This conclusion is relevant to future saltmarsh protection and re-establishment.

scholarly journals Assessment of Common Scab-Inducing Pathogen Effects on Potato Underground Organs Via Computed Tomography Scanning

2008 ◽  
Vol 98 (10) ◽  
pp. 1118-1125 ◽  
Author(s):  
L. Han ◽  
P. Dutilleul ◽  
S. O. Prasher ◽  
C. Beaulieu ◽  
D. L. Smith
Keyword(s):  
Computed Tomography ◽  
Common Scab ◽  
Surrounding Medium ◽  
Three Dimensional ◽  
Structural Complexity ◽  
Ct Scanning ◽  
Root Systems ◽  
Bacterial Disease ◽  
Ct Scanner ◽  
Computed Tomography Scanning

Common scab caused by Streptomyces scabies is a major bacterial disease of potato (Solanum tuberosum). Its best known symptom is superficial lesions on the surface of progeny potato tubers, observed at harvesting. In this study, effects of S. scabies on space occupancy by underground organs and on structural complexity of root systems are investigated during growth via computed tomography (CT) scanning. Two groups of potato plants were grown in a greenhouse in middle-sized plastic pots. Using a high-resolution X-ray CT scanner formerly used for medical applications, their underground organs and surrounding medium (sieved and autoclaved homogeneous sand) were submitted to CT scanning 4, 6, and 8 weeks after planting. For one group, sand was inoculated with the common scab-inducing pathogen (S. scabies EF-35) at potting. Space occupancy by underground organs was estimated via curve fitting applied to histograms of CT scan data, while three-dimensional skeletal images were used for fractal analysis. Root systems of diseased plants were found to be less complex than those of healthy plants 4 weeks after planting, and the relative growth rates derived from space occupancy measures were of different sign between the two groups from week 4 to week 8.

Computed tomography scanning for three-dimensional imaging and complexity analysis of developing root systems

2005 ◽  
Vol 83 (11) ◽  
pp. 1434-1442 ◽  
Author(s):  
Melinda Lontoc-Roy ◽  
Pierre Dutilleul ◽  
Shiv O. Prasher ◽  
Liwen Han ◽  
Donald L. Smith
Keyword(s):  
Computed Tomography ◽  
Fractal Dimension ◽  
Ct Scan ◽  
Root System ◽  
Complexity Analysis ◽  
Three Dimensional ◽  
Root Systems ◽  
Soil Conditions ◽  
Advanced Analysis ◽  
Scan Data

To improve our understanding of the role of root systems in soil-based resource acquisition by plants and eventually model it completely, root system complexity must be quantified, in addition to other morphometric traits. In this note, we introduce a new approach in which computed tomography (CT) scan data are collected on crop root systems in three-dimensional (3-D) space nondestructively and noninvasively, thus allowing for repeated measurements and a relevant complexity analysis of root systems. The experimental crop is maize ( Zea mays L.). Four potted seedlings were CT scanned under wet soil conditions on the day of emergence, and each of the two following days. Specifically, a high-resolution X-ray CT scanner formerly used for medical purposes produced 3 × 500 CT images of 0.1 mm thick cross-sections for each seedling. The fractal dimension of each root system on each day was estimated on a skeletonized 3-D image reconstructed from CT scan data. We found that the mean fractal dimension value was not significantly greater than 1 on day 1 (1.015 ± 0.015), contrary to days 2 and 3 (1.037 ± 0.015, 1.065 ± 0.016). Our results, including original 3-D images, provide support for a novel type of root system studies based on the collection and advanced analysis of CT scan data.

scholarly journals Three-dimensional in vivo analysis of water uptake and translocation in maize roots by fast neutron tomography

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Christian Tötzke ◽  
Nikolay Kardjilov ◽  
André Hilger ◽  
Nicole Rudolph-Mohr ◽  
Ingo Manke ◽  
...  
Keyword(s):  
Fast Neutron ◽  
Water Uptake ◽  
Three Dimensional ◽  
Root Systems ◽  
Neutron Tomography ◽  
Deuterated Water ◽  
Time Resolved ◽  
In Vivo Analysis ◽  
Uptake And Transport

AbstractRoot water uptake is an essential process for terrestrial plants that strongly affects the spatiotemporal distribution of water in vegetated soil. Fast neutron tomography is a recently established non-invasive imaging technique capable to capture the 3D architecture of root systems in situ and even allows for tracking of three-dimensional water flow in soil and roots. We present an in vivo analysis of local water uptake and transport by roots of soil-grown maize plants—for the first time measured in a three-dimensional time-resolved manner. Using deuterated water as tracer in infiltration experiments, we visualized soil imbibition, local root uptake, and tracked the transport of deuterated water throughout the fibrous root system for a day and night situation. This revealed significant differences in water transport between different root types. The primary root was the preferred water transport path in the 13-days-old plants while seminal roots of comparable size and length contributed little to plant water supply. The results underline the unique potential of fast neutron tomography to provide time-resolved 3D in vivo information on the water uptake and transport dynamics of plant root systems, thus contributing to a better understanding of the complex interactions of plant, soil and water.

scholarly journals Parameterizing complex root water uptake models – the arrangement of root hydraulic properties within the root architecture affects dynamics and efficiency of root water uptake

2014 ◽  
Vol 11 (1) ◽  
pp. 757-805 ◽  
Author(s):  
M. Bechmann ◽  
C. Schneider ◽  
A. Carminati ◽  
D. Vetterlein ◽  
S. Attinger ◽  
...  
Keyword(s):  
Root System ◽  
Water Uptake ◽  
Plant Resistance ◽  
Hydraulic Properties ◽  
Three Dimensional ◽  
Root Systems ◽  
Root Water Uptake ◽  
Hydraulic Lift ◽  
Excessive Bleeding ◽  
Single Root

Abstract. Detailed three-dimensional models of root water uptake have become increasingly popular for investigating the process of root water uptake. However they suffer from a lack of information in important parameters, especially distribution of root hydraulic properties. In this paper we explore the role that arrangement of root hydraulic properties and root system topology play for modelled uptake dynamics. We apply microscopic models of single root structures to investigate the mechanisms shaping uptake dynamics and demonstrate the effects in a complex three dimensional root water uptake model. We introduce two efficiency indices, for (a) overall plant resistance and (b) water stress and show that an appropriate arrangement of root hydraulic properties can increase modelled efficiency of root water uptake in single roots, branched roots and entire root systems. The average uptake depth of the complete root system was not influenced by parameterization. However, other factors such as evolution of collar potential, which is related to the plant resistance, root bleeding and redistribution patterns were strongly affected by the parameterization. Root systems are more efficient when they are assembled of different root types, allowing for separation of root function in uptake (short young) roots and transport (longer mature) roots. Results become similar, as soon as this composition is accounted for to some degree (between 40 and 80% of young uptake roots). Overall resistance to root water uptake was decreased up to 40% and total transpiration was increased up to 25% in these composed root systems, compared to homogenous root systems. Also, one parameterization (homogenous young root system) was characterized by excessive bleeding (hydraulic lift), which was accompanied by lowest efficiency. We conclude that heterogeneity of root hydraulic properties is a critical component of complex three dimensional uptake models. Efficiency measures together with information on critical xylem potentials may be useful in parameterizing root property distribution.

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