Infernal Integers

Through me you pass into the city of woe:
Through me you pass into eternal pain:
Through me among the people lost for aye.

Justice the founder of my fabric mov’d:
To rear me was the task of power divine,
Supremest wisdom, and primeval love.

Before me things create were none, save things
Eternal, and eternal I endure.
All hope abandon ye who enter here.

-Dante Alighieri, Inferno

Good evening lycans and gobble-uns. Welcome to Spherical Cow’s continuing Halloween festivities! But don’t allow yourself to get too comfortable, this is merely the assembly point. Account for all your limbs and baggage, and dress light because we are going on an expedition, and tonite I’ll play your Virgil. Ah, now by candlelight, I see the apprehension in your face- the unspoken realization… Yes, heed the poet’s warning well, for our destination is the Valley of Jehoshaphat, Gehinnom, Jehennam, Gehenna, Tartarus, Hades-Hell!

La Porte de l'Enfer, Auguste Rodin 1837

I shall audience no arguments against our descent for the reservation of our reprobation is nontransferable, and where we’re going, people are dying to get in.

“Where is that,” you ask. To which I respond:

“Come now, you must have heard of it. People have been screaming about it. Why, it is all the rage. It’s hot! I tell you- hot! Here, peruse the brochure.” I hand you The Good Book (King James) opened to Revelations 21:8 :

the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.

Yes, our vacation spot – The Second Death. While the climate may not be everyone’s cup of tea, you can’t beat the view. But how bad is heat? Well, our best clue is given by the fact this is a lake of brimstone. Brimstone is an archaic word for sulfur, which became conflated with  the concept of Hell through its odious odor. The historic Valley of Hinnom is the model for Hell in Judaism, Christianity, and Islam. This site essentially served as Jerusalem’s landfill. It had previously been a ritual site for the worshipers of Moloch, an ancient Semitic god, and the ancient Jews felt that its transformation from shrine to junkyard was poetic justice. Like the tire yard of Springfield, constant fires consumed the detritus in the valley. The rot, smoke and heat created an intolerable place – a place of fire and brimstone. Thus Hinnom became legendary. It became Gehinnom then Gehenna, and latter when the Hellenes were converted to Christianity it was forced into Hades, which became Hell.

But how bad is the heat? Well it is a lake of sulfur, and therefore the temperature of Hell must be great enough to melt the element but less than or equal to its boiling point (otherwise it would be a vapor). The melting point of sulfur is 239.38°F, and its boiling point is 832.3°F. From this we can conclude that an average day in Hell has lows of about 240°F and highs around 830°F.  Hellishly hot indeed!

Oh! Leaving so soon? We have only just arrived! Is the heat getting to you? That would explain why you are behaving so rashly. Before you go running back to your Beatrice, you may want to consider this:

Moreover the light of the moon shall be as the light of the sun, and the light of the sun shall be sevenfold, as the light of seven days

-Isaiah 30:26

Isaiah paints a bright picture of Paradise. Perhaps too bright. Did you bring enough sun-block? Let’s check: I propose we keep a tally of Sunshine. He tells us the Moon in Heaven is as bright as the Sun on Earth, so currently the amount of Earth Suns shining on Heaven is one.  Next we learn that Heaven is irradiated each day by sevenfold the light of seven days, that is 7×7=49 Suns. Adding this to the light from the Moon we find that salvation has the equivalent of a blinding 50 suns! To get the requisite 15SPF per sun, you would need to apply 750 layers of Banana Boat.

However, it is not just your sun block budget that would be hurting. A useful thermodynamics law will allow us a clearer view of the sky firmament. The Stefan-Boltzmann law for radiation allows us to compare the absolute temperatures of Heaven(H) and Earth(E) with the amount of radiation being absorbed from the suns.  Without getting too technical, here is the formula simplified. The 50 is representative of the radiation from the 50 suns.


The UN’s Intergovernmental Panel on Climate Change gives the Earth’s temperature as 255K. Setting E=255, we obtain:


Solving for H reveals that the temperature of Heaven is approximately 678K or 760°F! As you can see, sometimes Heaven is hotter than Hell! And even when it isn’t, it’s still hot as hell. Perhaps it would be better to avoid Heaven and Hell altogether and aim strictly for an extended time in Purgatory.

So dear reader, I bid you farewell once again. But the Witching season is not over yet, and the haunting here continues. If you haven’t read part one of my Tricks and Treats, do so now, and be sure to join me again soon as Spherical Cow’s Halloween continues.

You better mind yer parunts, an’ yer teachurs fond an’ dear,
An’ churish them ‘at loves you, an’ dry the orphant’s tear,
An’ he’p the pore an’ needy ones ‘at clusters all about,
Er the Gobble-uns ‘ll git you
Ef you
– James Whitcomb Riley, Little Orphant Annie

This is the second in a series of Halloween posts. If you enjoyed it, you may like the others: Vampire Numbers, Zombi(nacci)The Sum of All Fears


This post was inspired by the classic article, Heaven is Hotter than Hell, published anonymously in Applied Optics (1972). I have made a number of additions and changes to enhance and update the original concept. You can read the original here.


About Webster

Transitionally, I’m a math student and tutor en route to becoming a math professor. Permanently, I’m a mathematics enthusiast. I study mathematics professionally, and as a leisure activity. At the time of writing this, I’m a generalist. I have let to reach the depth of understanding that requires specialization. Though I eagerly await that time, I do enjoy the ‘now’ and find there is bountiful food for thought at any level.
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5 Responses to Infernal Integers

  1. Pingback: Vampire Numbers | Spherical Cow

  2. Annabelle says:

    Hahahaha nice use of Flux = σT^4

  3. Pingback: Zombi(nacci) | Spherical Cow

  4. Pingback: The Sum of All Fears | Spherical Cow

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