Fractal dimension analysis with popcorn grains and popped popcorn grains

In this manuscript, a very simple experiment was mounted to explore the fractal dimension of a popcorn grain before (popcorn grains) and after (popcorn itself) the popping process.

Keywords
fractal dimension; popcorn

Authorship

Eduardo do Carmo ^** Correspondence email address: eduardo.carmo@unila.edu.br

Universidade Federal da Integração Latino-Americana, Instituto de Ciências da Vida e da Natureza, Foz do Iguacu, PR, Brasil.Universidade Federal da Integração Latino-AmericanaBrasilFoz do Iguacu, PR, BrasilUniversidade Federal da Integração Latino-Americana, Instituto de Ciências da Vida e da Natureza, Foz do Iguacu, PR, Brasil.

https://orcid.org/0000-0001-8796-1856

Marcelo Goncalves Hönnicke

https://orcid.org/0000-0003-3276-4323

* Correspondence email address: eduardo.carmo@unila.edu.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (3)
Tables (1)
Formulas (1)

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Figure 1
Representation of the procedure employed to determine the initial (before popping) and final (just after popping) physical constants (volume and mass) of the different sets of popcorn grains (Zea Mays L.) with different grain amounts in order to determine the fractal dimension. (a) Initial volume determination of the 5 different sets, used in the experiment, with the popcorn grain amounts ranging from 50 (first set) to 800 (last set). (b) Mass determination procedure (by using a digital scale). As an example, the 4th set with 400 popcorn grains is shown. (c) Popping procedure: (i) Popcorn grains embedded in soybean oil inside a pan previous to the popping process and; (ii) Popped popcorn grains (popcorn itself) inside the pan just after being popped (in specifics, the 2nd set with 100 popcorn grains). (d) Final volume and mass determination of the popcorn (in specifics, part of the 5th set with 800 popcorn grains).

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Figure 2
Procedure employed to find the characteristic lengths. The volumes measured, for each different set for the popcorn grains (a) and the popped popcorn grains (popcorn itself) (b), were matched with cubes with the same volumes (mass-volume equivalent cubes): V₁−V₅, for the popcorn grains and

V_{1}^{'} - V_{5}^{'}

, for the popcorns. Each different cube, corresponding to each different measured volume, has a different characteristic length (cube edges: a₁−a₅ for the popcorn grains and;

a_{1}^{'} - a_{5}^{'}

for the popcorns). For the popcorn volumes (b), since we missed pictures, the measured volumes

V_{2}^{'} - V_{4}^{'}

on the graduated cylinders are not shown. The volume

V_{5}^{'}

correspond to 3.77 times the volume shown in the picture. All the schematic cubes are shown in relative scale, i.e., for example, a₅∼2.5a₁,

a_{5}^{'} \sim 3.16 a_{1}^{'}

and, consequently,

a_{5}^{'} \sim 7.07 a_{1}

Thumbnail

Figure 3
Log-log plot of the different masses (m₁−m₅ and

m_{1}^{'} - m_{5}^{'}

, for the popcorn grains and popcorns, respectively, in grams) versus the different mass-volume equivalent cube edges (a₁−a₅ and

a_{1}^{'} - a_{5}^{'}

, for the popcorn grains and popcorns, respectively, in cm) obtained from each different experiment set (five in total, with popcorn grain amounts varying from 50 to 800 – Table 1). (a) Before the popping process (popcorn grains). (b) After the popping process (popped popcorn grains, i.e., popcorn itself). Filled and opened circles are the experimental data, the continuous lines (in red) are the fitted data from which the fractal dimension (D) is extracted. Error bars are also shown. Where they do not appear they are smaller or, at the same size, of the circle representing the experimental data.

Thumbnail

Table 1
Masses and volumes of the five different experiment sets (with different grains amount) measured to determine the fractal dimension (D) of the popcorn grains (before the popping process) and of the popped popcorn grains (after the popping process – popcorn itself). Cube edges for the mass-volume equivalent cubes are also shown. The errors in the masses and in the volumes are given by the measuring instruments (last digit of the digital scale and half of the smallest division of the graduated cylinders, respectively). The errors in the cube edges are given by the propagation of uncertainties of the measured volumes.

Thumbnail

(1)

Figure 1 Representation of the procedure employed to determine the initial (before popping) and final (just after popping) physical constants (volume and mass) of the different sets of popcorn grains (Zea Mays L.) with different grain amounts in order to determine the fractal dimension. (a) Initial volume determination of the 5 different sets, used in the experiment, with the popcorn grain amounts ranging from 50 (first set) to 800 (last set). (b) Mass determination procedure (by using a digital scale). As an example, the 4th set with 400 popcorn grains is shown. (c) Popping procedure: (i) Popcorn grains embedded in soybean oil inside a pan previous to the popping process and; (ii) Popped popcorn grains (popcorn itself) inside the pan just after being popped (in specifics, the 2nd set with 100 popcorn grains). (d) Final volume and mass determination of the popcorn (in specifics, part of the 5th set with 800 popcorn grains).

Figure 2 Procedure employed to find the characteristic lengths. The volumes measured, for each different set for the popcorn grains (a) and the popped popcorn grains (popcorn itself) (b), were matched with cubes with the same volumes (mass-volume equivalent cubes): V₁−V₅, for the popcorn grains and $V_{1}^{'} - V_{5}^{'}$ , for the popcorns. Each different cube, corresponding to each different measured volume, has a different characteristic length (cube edges: a₁−a₅ for the popcorn grains and; $a_{1}^{'} - a_{5}^{'}$ for the popcorns). For the popcorn volumes (b), since we missed pictures, the measured volumes $V_{2}^{'} - V_{4}^{'}$ on the graduated cylinders are not shown. The volume $V_{5}^{'}$ correspond to 3.77 times the volume shown in the picture. All the schematic cubes are shown in relative scale, i.e., for example, a₅∼2.5a₁, $a_{5}^{'} \sim 3.16 a_{1}^{'}$ and, consequently, $a_{5}^{'} \sim 7.07 a_{1}$ .

Figure 3 Log-log plot of the different masses (m₁−m₅ and $m_{1}^{'} - m_{5}^{'}$ , for the popcorn grains and popcorns, respectively, in grams) versus the different mass-volume equivalent cube edges (a₁−a₅ and $a_{1}^{'} - a_{5}^{'}$ , for the popcorn grains and popcorns, respectively, in cm) obtained from each different experiment set (five in total, with popcorn grain amounts varying from 50 to 800 – Table 1). (a) Before the popping process (popcorn grains). (b) After the popping process (popped popcorn grains, i.e., popcorn itself). Filled and opened circles are the experimental data, the continuous lines (in red) are the fitted data from which the fractal dimension (D) is extracted. Error bars are also shown. Where they do not appear they are smaller or, at the same size, of the circle representing the experimental data.

Table 1 Masses and volumes of the five different experiment sets (with different grains amount) measured to determine the fractal dimension (D) of the popcorn grains (before the popping process) and of the popped popcorn grains (after the popping process – popcorn itself). Cube edges for the mass-volume equivalent cubes are also shown. The errors in the masses and in the volumes are given by the measuring instruments (last digit of the digital scale and half of the smallest division of the graduated cylinders, respectively). The errors in the cube edges are given by the propagation of uncertainties of the measured volumes.

Popcorn grains amount	Total masses (m₁−m₅) before popping (g)	Total masses ( $m_{1}^{'} - m_{5}^{'}$ ) after popping (g)	Total volumes (V₁−V₅) before popping (cm³)	Cube edges (a₁−a₅) before popping (cm)	Total volumes ( $V_{1}^{'} - V_{5}^{'}$ ) after popping (cm³)	Cube edges ( $a_{1}^{'} - a_{5}^{'}$ ) after popping (cm)
50	9.2 ± 0.1	10.5 ± 0.1	11 ± 2	2.2 ± 0.1	120 ± 10	4.93 ± 0.14
100	18.2 ± 0.1	22.1 ± 0.1	22 ± 2	2.8 ± 0.1	320 ± 20	6.84 ± 0.14
200	36.7 ± 0.1	44.0 ± 0.1	42 ± 2	3.5 ± 0.1	780 ± 20	9.21 ± 0.08
400	71.5 ± 0.1	85.7 ± 0.1	88 ± 2	4.4 ± 0.0	2320 ± 60	13.24 ± 0.08
800	139.6 ± 0.1	168.4 ± 0.1	160 ± 2	5.43 ± 0.02	3770 ± 60	15.56 ± 0.11

(1)

a \propto m^{\frac{1}{D}}

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[1] * Correspondence email address: eduardo.carmo@unila.edu.br