Algorithms and Data Structures (Winter - Spring 2022)¶

Table of Contents

Practice with pandas¶

pandas, the Python Data Analysis Library, is an important resource. pandas will give you many ways to practice your brand of data science, whatever the walk of life.

The pandas.DataFrame type lets you join pandas.Series type columns into a multi-column data table, complete with row and column names of your choice, both re-orderable.

Once you have a DataFrame defined, adding new columns based on the old, getting summary statistics, applying functions, generating visualizations, is all within reach.

In [30]:

import math as clc

In [2]:

clc.sin(clc.radians(90)) # remembering trig

Out[2]:

1.0

In [3]:

# and now for something completely different...
import numpy as np
import pandas as pd

Lets stack up a column of polyhedron names, using a kind of jargon or shorthand.

In [4]:

shapes = np.array(["Tetra", "Cubocta", "Icosa", "Cube", "Octa", 
                    "RT5", "RT5+", "RD", "RT", "PD", "Icosa", "Cubocta", 
                    "SuperRT", "Cube"], 
                    dtype=np.str_) 
shapes

Out[4]:

array(['Tetra', 'Cubocta', 'Icosa', 'Cube', 'Octa', 'RT5', 'RT5+', 'RD',
       'RT', 'PD', 'Icosa', 'Cubocta', 'SuperRT', 'Cube'], dtype='<U7')

So far that's a numpy.ndarray we've created. Now lets bring that into a Series.

In [5]:

shapes_col = pd.Series(shapes, name="Shape")
shapes_col

Out[5]:

0       Tetra
1     Cubocta
2       Icosa
3        Cube
4        Octa
5         RT5
6        RT5+
7          RD
8          RT
9          PD
10      Icosa
11    Cubocta
12    SuperRT
13       Cube
Name: Shape, dtype: object

The vertical Series, a column of some data type (dtype), is the building block of the DataFrame, which sets them side by side in a tabular arrangement.

What are the S and E modules? Lets look at some pictures:

24 S modules, 12 left and 12 right, wedge between Octa 4 and the Icosahedron inscribed inside it.

120 E modules, 120 left and 120 right, comprise the rhombic triacontahedron inside of which, and tangent to its 30 faces, is the unit radius ball.

In [6]:

# geometric constants
phi = (1 + clc.sqrt(5))/2

# volumes of specific tetrahedral wedges
Emod  = clc.sqrt(2)/8 * 1/phi**3
Emod3 = clc.sqrt(2)/8
emod3 = Emod * (phi**-3)
Smod  = (phi**-5) / 2

Sfactor = Smod/Emod

S3 = clc.sqrt(9/8)

# defined to have edges = 2R or 1D
Icosa      = 100 * Emod3 + 20 * Emod
PentDodeca =  84 * Emod3 + 12 * Emod # 348 * Emod + 84 * emod3

# volumes corresponding to our shapes
volumes = np.array([1, 2.5, 2.5 * Sfactor**2, 
                    3, 4, 5, 120 * Emod, 6, 
                    7.5, PentDodeca, Icosa, 20, 
                    20 * S3, 24],  dtype=np.float)

In [7]:

volumes_col = pd.Series(volumes, name="IVM Volume")  # turn np.array into a pd.Series

In [8]:

volumes_col

Out[8]:

0      1.000000
1      2.500000
2      2.917961
3      3.000000
4      4.000000
5      5.000000
6      5.007758
7      6.000000
8      7.500000
9     15.350018
10    18.512296
11    20.000000
12    21.213203
13    24.000000
Name: IVM Volume, dtype: float64

In [9]:

vols_table = pd.DataFrame({"Shape": shapes_col, "IVM Volume":volumes_col})
# vols_table.index = shapes_col  #  the shapes column is the index

In [10]:

vols_table

Out[10]:

	Shape	IVM Volume
0	Tetra	1.000000
1	Cubocta	2.500000
2	Icosa	2.917961
3	Cube	3.000000
4	Octa	4.000000
5	RT5	5.000000
6	RT5+	5.007758
7	RD	6.000000
8	RT	7.500000
9	PD	15.350018
10	Icosa	18.512296
11	Cubocta	20.000000
12	SuperRT	21.213203
13	Cube	24.000000

In [11]:

vols_table['XYZ Volume'] = vols_table['IVM Volume'] * 1/S3

In [12]:

vols_table

Out[12]:

	Shape	IVM Volume	XYZ Volume
0	Tetra	1.000000	0.942809
1	Cubocta	2.500000	2.357023
2	Icosa	2.917961	2.751080
3	Cube	3.000000	2.828427
4	Octa	4.000000	3.771236
5	RT5	5.000000	4.714045
6	RT5+	5.007758	4.721360
7	RD	6.000000	5.656854
8	RT	7.500000	7.071068
9	PD	15.350018	14.472136
10	Icosa	18.512296	17.453560
11	Cubocta	20.000000	18.856181
12	SuperRT	21.213203	20.000000
13	Cube	24.000000	22.627417

Practice with df.loc[rows, columns].

In [13]:

vols_table.iloc[12]  # entire row

Out[13]:

Shape           SuperRT
IVM Volume    21.213203
XYZ Volume         20.0
Name: 12, dtype: object

In [14]:

vols_table.iloc[0]  # entire row

Out[14]:

Shape            Tetra
IVM Volume         1.0
XYZ Volume    0.942809
Name: 0, dtype: object

df.loc[df['col1'] == value]

In [15]:

vols_table.loc[vols_table['Shape'] == "SuperRT" ] # specific cell

Out[15]:

	Shape	IVM Volume	XYZ Volume
12	SuperRT	21.213203	20.0

Now lets add some constituent modules that may be used to assemble the above shapes.

In [16]:

modules = np.array(["A","B", "T", "E", "S"],
                    dtype=np.str_) 

mods_col = pd.Series(modules, name="Shape")

mod_vols = np.array([1/24, 1/24, 1/24, Emod, (phi**-5) / 2],  dtype=np.float)
mod_vols_col = pd.Series(mod_vols, name="IVM Volume")

mods_table = pd.DataFrame({"Shape":mods_col, "IVM Volume":mod_vols_col})
# mods_table.index = mods_col

In [17]:

mods_table

Out[17]:

	Shape	IVM Volume
0	A	0.041667
1	B	0.041667
2	T	0.041667
3	E	0.041731
4	S	0.045085

In [18]:

mods_table['XYZ Volume'] = mods_table['IVM Volume'] * 1/S3

In [19]:

mods_table

Out[19]:

	Shape	IVM Volume	XYZ Volume
0	A	0.041667	0.039284
1	B	0.041667	0.039284
2	T	0.041667	0.039284
3	E	0.041731	0.039345
4	S	0.045085	0.042507

And now it's time to assemble the full table.

In [20]:

pd.concat([mods_table, vols_table])

Out[20]:

	Shape	IVM Volume	XYZ Volume
0	A	0.041667	0.039284
1	B	0.041667	0.039284
2	T	0.041667	0.039284
3	E	0.041731	0.039345
4	S	0.045085	0.042507
0	Tetra	1.000000	0.942809
1	Cubocta	2.500000	2.357023
2	Icosa	2.917961	2.751080
3	Cube	3.000000	2.828427
4	Octa	4.000000	3.771236
5	RT5	5.000000	4.714045
6	RT5+	5.007758	4.721360
7	RD	6.000000	5.656854
8	RT	7.500000	7.071068
9	PD	15.350018	14.472136
10	Icosa	18.512296	17.453560
11	Cubocta	20.000000	18.856181
12	SuperRT	21.213203	20.000000
13	Cube	24.000000	22.627417

In [21]:

CH = pd.concat([mods_table, vols_table])

In [22]:

CH = CH.reset_index(drop=True)

In [23]:

CH.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 19 entries, 0 to 18
Data columns (total 3 columns):
 #   Column      Non-Null Count  Dtype  
---  ------      --------------  -----  
 0   Shape       19 non-null     object 
 1   IVM Volume  19 non-null     float64
 2   XYZ Volume  19 non-null     float64
dtypes: float64(2), object(1)
memory usage: 584.0+ bytes

In [24]:

# df['new'] = pd.Series(dtype='int')
CH['Comments'] = pd.Series(dtype='str_')

In [25]:

CH

Out[25]:

	Shape	IVM Volume	XYZ Volume	Comments
0	A	0.041667	0.039284	NaN
1	B	0.041667	0.039284	NaN
2	T	0.041667	0.039284	NaN
3	E	0.041731	0.039345	NaN
4	S	0.045085	0.042507	NaN
5	Tetra	1.000000	0.942809	NaN
6	Cubocta	2.500000	2.357023	NaN
7	Icosa	2.917961	2.751080	NaN
8	Cube	3.000000	2.828427	NaN
9	Octa	4.000000	3.771236	NaN
10	RT5	5.000000	4.714045	NaN
11	RT5+	5.007758	4.721360	NaN
12	RD	6.000000	5.656854	NaN
13	RT	7.500000	7.071068	NaN
14	PD	15.350018	14.472136	NaN
15	Icosa	18.512296	17.453560	NaN
16	Cubocta	20.000000	18.856181	NaN
17	SuperRT	21.213203	20.000000	NaN
18	Cube	24.000000	22.627417	NaN

In [28]:

CH.iloc[0, -1] = '24 make a Tetra'
CH.iloc[1, -1] = 'AAB = BAA = Mite'
CH.iloc[2, -1] = '1/120 RT5'
CH.iloc[3, -1] = '1/120 RT5+'
CH.iloc[4, -1] = '(φ**-5) / 2'
CH.iloc[5, -1] = "edges D, from 4 IVM balls"
CH.iloc[6, -1] = 'some faces flush with Octa 4'
CH.iloc[7, -1] = 'some faces flush with Octa 4'
CH.iloc[8, -1] = 'Duo-Tet, face diagonals = D'
CH.iloc[9, -1] = 'Dual of Cube, edges D'
CH.iloc[10, -1] = '120 T mods'
CH.iloc[11, -1] = '120 E mods'
CH.iloc[12, -1] = 'Rhombic Dodeca, long diagonals = D'
CH.iloc[13, -1] = 'some vertexes shared with RD'
CH.iloc[14, -1] = 'Pentagonal Dodeca, dual of Icosa'
CH.iloc[15, -1] = 'edges D'
CH.iloc[16, -1] = 'edges = D, 1F, 12-balls around nuclear ball'
CH.iloc[17, -1] = 'icosa of edges D + dual PD'
CH.iloc[18, -1] = 'face diagonals = 2D, 2F'

In [29]:

CH

Out[29]:

	Shape	IVM Volume	XYZ Volume	Comments
0	A	0.041667	0.039284	24 make a Tetra
1	B	0.041667	0.039284	AAB = BAA = Mite
2	T	0.041667	0.039284	1/120 RT5
3	E	0.041731	0.039345	1/120 RT5+
4	S	0.045085	0.042507	(φ**-5) / 2
5	Tetra	1.000000	0.942809	edges D, from 4 IVM balls
6	Cubocta	2.500000	2.357023	some faces flush with Octa 4
7	Icosa	2.917961	2.751080	some faces flush with Octa 4
8	Cube	3.000000	2.828427	Duo-Tet, face diagonals = D
9	Octa	4.000000	3.771236	Dual of Cube, edges D
10	RT5	5.000000	4.714045	120 T mods
11	RT5+	5.007758	4.721360	120 E mods
12	RD	6.000000	5.656854	Rhombic Dodeca, long diagonals = D
13	RT	7.500000	7.071068	some vertexes shared with RD
14	PD	15.350018	14.472136	Pentagonal Dodeca, dual of Icosa
15	Icosa	18.512296	17.453560	edges D
16	Cubocta	20.000000	18.856181	edges = D, 1F, 12-balls around nuclear ball
17	SuperRT	21.213203	20.000000	icosa of edges D + dual PD
18	Cube	24.000000	22.627417	face diagonals = 2D, 2F

For further reading: