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Closing in on a square wave with the Fourier series under #Python
#!/usr/bin/env python3
import numpy as n,matplotlib.pyplot as p
x=n.linspace(0,2*n.pi,1000)
def s(x,t):
k=n.arange(1,2*t,2)
return 4/n.pi*(n.sin(k[:,None]*x)/k[:,None]).sum(0)
N=12
colors=p.cm.rainbow(n.linspace(0,1,N))
for i,t in enumerate(range(1,N+1)):
p.plot(x,s(x,t),color=colors[i],label=f'{2*t-1} harmonics')
p.title('Fourier Series Approximation of a Square Wave')
p.grid()
p.legend()
p.show() -
A "Marsrise” but it’s taken from Earth with an 8” f/8 Newtonian. #astronomy
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9E: And it came to pass that the trial concluded with judgment wholly against Uriel, to the surprise of none save perhaps Sar Lailah herself. Uriel was led through a veritable house of torments unto the lowest chambers beneath the palace of Rimmon, and there was he cast into a wet cell, wherein he shivered because of the cold. And Lailah was permitted to pay one final visit unto him within the dungeons beneath the Temple of Azul before the sentence should be carried out. And during the first moments they embraced one another in silence, though iron bars stood between them. Then Lailah looked round about carefully, assuring herself that their brief right of private counsel yet remained inviolate. And Lailah said unto him, “Guilty! And sentenced unto death. I can scarcely believe it, Uriel. Tell me then, what is thy plan?” But Uriel answered, “Plan? There is no plan. Look about thee, Lailah! Behold this security.”
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9D: “But in truth, a rock and a tree are likewise beyond good and evil, and so also are all those who are wholly suborned unto the laws of cause and chance. For rocks, and trees, and those who are not permitted or else refuse to make choice, are amoral. Not immoral, mark thou well, but amoral. For unto such beings ethics hath no application.” And Malphas said unto him, “Then dost thou deny that Emperor Rimmon possesseth authority to make decrees concerning morality?” But Uriel answered, “It is not merely that he ought not decree morality, but rather that he in truth cannot do so. For such a thing is a profound confusion of kinds, even as though one should speak of a triangle having four sides.” And the prince shrugged, and stood down from further questioning, being content to leave all these matters hanging unresolved before the tribunal. Then Sar Lailah pondered things a short span, and declared unto Rimmon, “The defense resteth.”
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9C: “Such piety is as counterfeit as the love found in many, yea, perhaps even in most marriages among the people of Adan, even as I have already declared. How much more precious is the love shared between two souls wholly free either to remain together or to depart, yet who nevertheless choose to abide with one another! And if this seemeth perilous unto thee, know this: to live dangerously is the very thing which lendeth excitement and meaning unto our lives. For only through conflict do we burn with the desire to create a new destiny.” And after these things, Lailah turned and signaled unto Malphas that he might begin his own examination of the defendant. Then Malphas said unto Sar Uriel, “A new destiny? Doth not such a teaching stand contrary to all that we have received concerning the inexorable workings of fate?” And Uriel answered him, saying, “I tell you with full knowledge that Azul once declared unto Ayat that to take fate into one’s own hands is to rise beyond good and evil.
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9B: For unto him the chief matter was not the outcome itself, seeing it had surely been determined beforehand, but rather that the people of the city who attended within the court should behold with their own eyes the injustice of the proceeding. And Lailah said, “Ser Uriel, notwithstanding all the evidence which hath been brought forth concerning the charge of impiety toward Azul, thou still maintainest thine innocence. How can this thing be?” And Uriel answered her, saying, “For most wives in Kemen, the decision to remain within marriage is no great exercise of will, seeing that the only alternative before them is starvation. And in like manner, it cannot be a meaningful choice to worship Azul within Adan, where the alternative is a tribunal and execution, even as this proceeding now unfoldeth before us.” And Lailah said unto him, “Then dost thou contend that any piety compelled by force of law is invalid?” And Uriel answered, “Precisely so.
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9A: And Lailah contended that this new written law had been established solely for the purpose of opposing the teachings of Uriel. And Lailah said before the tribunal, “Never in history hath a man been condemned for deeds made unlawful only after the deed was done. And even if such a thing should now be permitted in this instance, there yet remaineth the matter of jurisdiction. “For Uriel was apprehended just beyond the walls of Shalem, in lands ruled absolutely by my father, Kar Melchizedek. And he hath declared that the scepter is given without repentance, save only in the event of war, of which there hath been none. Therefore the very legality of this tribunal standeth in question.” But the objections of Lailah were permitted only for a short space. For Rimmon wearied quickly of each argument, and bade her pass unto another matter, and then unto another still. And the injustice of the proceedings began greatly to weigh upon Lailah; nevertheless, Uriel accepted all these things with calmness.
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99: “Why then dost thou not summon another halo and depart from this place?” But Uriel answered her, saying, “This matter is a game, and every game possesseth rules. The working of the bubble requireth cooperation between male and female among the Elohim. And there are certain things which Azul and I perform together in Kemen in a spirit of unity. But concerning this tribunal, we are most certainly not united. Nevertheless, be not afraid, Lailah. I am confident that thou shalt acquit thyself well.” Now Lailah had never been instructed in the art of advocacy in such matters; nevertheless, she had been a faithful disciple of Uriel. And the vigor of her defense would have been sufficient to secure his release in nearly any other court, whether upon the Earth or elsewhere in Kemen. And at the first, Lailah directed her defense toward the fact that the Code of Azul had come into existence only after Uriel began to teach in Shalem, and after his doctrines had spread abroad beyond the borders of that city.
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98: Indeed, even before his union with Ayat, Uriel alone would have been sufficient to contend with the prince in matters of intellect. But Malphas perceived not that he was being thoroughly overcome by the rhetoric of Uriel. Nevertheless, Uriel suspected within himself that none of these things would matter in the end. For surely the judgment of the trial had been determined beforehand. And it came to pass after a time that Sar Malphas altogether exhausted his arguments. And he declared unto his father that he had no further questions to put before Uriel. Then Emperor Rimmon commanded that there should be a brief pause before the defense might begin to present its arguments. And during this interval Lailah came unto Uriel, that they might speak together privately for a moment. And Lailah said unto him, “I believe I can contend with this Malphas well enough; nevertheless, why must I do so at all? For at times thou hast summoned a halo and sent me forth across Kemen in the twinkling of an eye.”
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97: And Uriel said unto him, “In like manner as men and wen were separated from the lower beasts, that they might enjoy a nearer relationship unto the Elohim, even so hath Ayat set apart her disciples from among wen, that they might dwell together in unity and nourish one another.” Then Malphas said, “If we pursue thy reasoning, we arrive at a thing too shameful even to be named, a thing accounted utterly unnatural by all our ancient traditions.” But Uriel answered him, saying, “It is natural for the body to possess an odor. Verily, our frequent bathing is itself an unnatural thing. Yet no man counseleth we should return unto the condition of beasts in matters of personal cleanliness. And I speak of these things because this is the manner wherein the Fallen Angels regard that which men call unnatural.” And Malphas continued on in like manner, yet he contended against one who was both eloh and man, even an advanced composite being. And Ser Uriel possessed an answer suitable unto every challenge set before him.
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