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To preface, let me state that this piece of writing is weird, even by my standards. Please read on at your own peril of losing brain cells.

One can define a âshower thoughtâ as those random thoughts that one gets, seemingly from nowhere, while undertaking a routine task such as showering. These thoughts cover the range from simple to complex, from one of small, personal nonimportance to one of great magnitude that alters the path of history, as when Archimedes came to the realization of the fundamental principle that bears his name. The etymology of the term, perhaps, is lost to time, but it seems very curious, at least to me, just how many of these epiphanies seem to come to us during our time spent spreading shampoo through our hair while showering, or playing with our rubber duckies while bathing, or browsing through random internet articles while commoding. It is to this last activity that I think my greatest number of shower thoughts, though likely on the account that I find commoding to comprise, of these three activities, the longest cumulative duration throughout the course of my day, or indeed, on average over the course of my life to-date, such that naturally the largest number of so-called shower thoughts would arise over the periods when I am sitting hunched over and peering at a small phone screen. Nevertheless, I have come to realize quite some years ago this creative opportunity, and to support and nurture these commode thoughts, I commissioned a 3D-printed low-polygon Thinker statuette, 12 inches in stooped height, to sit atop the tank cover in pensive guidance over my daily or, more accurately, polyquotidian trips to find new bursts of inspiration.

Thank you for your patience; that was a long digression. Let me share with you the latest shower thought I had this afternoon, a version of which probably has crossed your mind at some point or another though perhaps not in the way I will present anon:

Hand sanitizer, napkins, paper towels, and toilet paper have been in the news quite frequently due to the coronavirus pandemic. We have all heard stories of shelves stripped bare of any and all cleaning supplies by a few select rotten apples who have undertaken for themselves the opportunity for arbitrage and price-gouging at the expense of their fellow personsâ health and lives. It is regarding this last item â toilet paper, or TP for short â over which I had the recent polite argument with a friend of opposite gender, in which we each argued in favor of our own respective identified gender as to be the one with lower TP usage. I will not dwell further on recalling our arguments and counterarguments, for it was but a lighthearted conversation with no malice or ill-intent on either side, as we both knew that individual proclivities for cleanliness as well as efficiency of TP use impact greatly the rate of consumption. Maybe I will undertake further study on the analysis of gender-related usage of TP, but researchers at MIT have already conducted a study into this topic,1Â and it is not the point of this document.

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So, to set the scene, I was gazing down past the top of my phone to the TP holder against the wall, within which a half of a roll, with regards to starting diameter, of TP was residing. Let us first define a few terms before continuing further: TP_{sheet} refers to, obviously, the single or double-ply paper pulp for personal cleansing; TP_{card} defines explicitly the cardboard cylinder, *per se*; finally, TP_{roll} is defined as TP_{card} surrounded by TP_{sheet}. Now, I generally like to provide a spare TP_{roll} before the one on the holder runs out, lest arises the awkward situation of hopping out from the bathroom, pants half-down (or up, depending on your level of optimism), in search of a new TProll. I have no strict guidelines, but as a rule of thumb I consider a half of a TP_{roll} remaining to be a good time to jot down a note on my mental checklist for placing a spare alongside the Thinker.

Now, I know that a half of a TP_{roll} with respect to diameter is actually much less than half of the sheets remaining, as there always is an inner empty core taken up by the TPcard, of which the majority of the population finds of little or no use regarding bodily cleaning except in contingencies. Further, even presupposing the limit as the diameter of the TPcardÂ approaches zero, simple mathematics dictates that as the diameter of TProllÂ increases, the diameter of TProllÂ is not directly proportional to the volume of TProll, as the former increases linearly while the latter increases by the square of the radius of TProll. Therefore, let us figure out the answer to these scenarios:

*1) If one-half of a diameter of TProll remaining represents the appropriate time to grab a spare TProll, just how much TPsheet remains compared to the TPsheet of a new TProll?*

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*2) Generalizing Question 1, how does the function of TPsheet remaining and TProll diameter relate in equation and graphical form?*

We need to first determine a suitable candidate of TP manufacturer and model for analysis. This was limited considerably by the available number of manufacturers (n = 1) and models (n = 1) on hand, which turned out to be the intersection of Charmin and double-ply (Figure 1).2Â In terms of specifications, each roll contains 221 double-ply sheets each measuring 115.8 mm x 115.8 mm x 0.25 mm in length, height, and thickness. Therefore, when rolled out completely, the dimensions of the entirety of the TPsheetÂ will be 25.59 m x 115.8 mm x 0.25 mm. Measurement of the outer diameter of TPcardÂ after stripping away TPsheetÂ was tricky in that the thin wall of TPcardÂ makes for a malleable object difficult to accurately measure. Therefore, I simply cut a line perpendicular to the circular openings and rolled the TPcardÂ out, giving me an outer circumference of 133.7 mm which corresponds to an outer diameter of 42.6 mm.

Let us assign the variable *n* to be the number of two-ply sheets remaining, the variable *d* to be the diameter of the rapidly shrinking TProll, and the constant *h* to be the height of TProll. Let us assign the constant VsheetÂ to be the volume of a single sheet of two-ply, the constant VcardÂ to be the volume of TPcardÂ with enclosed air, and the variable VrollÂ to be the total volume of the TProll. Given the total volume of TPsheetÂ as 1,248,000 mm3, we know that VsheetÂ is 5,647 mm3, and we previously calculated Vcard to be 165,000 mm3.Â

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We can then relate *d* to *n* by combining the following two equations:

To calculate the diameter of TProll, we can first calculate the volume of TPsheet by multiplying the dimensions provided above to get 740,900 mm^{3}. Using *A = Ïr ^{2}*, we find that the volume of TPcard, including the space filled with air contained within the inner diameter, is 165,000 mm

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With these measurements in place, we can now tackle both questions. Note that we will use actual measurements rather than calculated measurements.

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*1) If one-half of a diameter of TProllÂ remaining represents the appropriate time to grab a spare TProll, just how much TPsheetÂ remains compared to the TPsheetÂ from of a new TProll?*

AÂ TProllÂ begins with 221 sheets of two-ply. If a completeÂ TProllÂ has radius 62.3 mm (diameter 124.6 mm), then one-half of a diameter ofÂ TProllÂ remaining yields a diameter of 62.3 mm. I love simple math! This equates to a volume of one-quarter that of what we started with, or 353,250 mm3. We previously established that the volume ofÂ TPcard, including the air within, to be 165,000 mm3, which leaves only 188,250 mm3Â ofÂ TPsheetÂ remaining: this volume equates to 15.1% that of a completeÂ TPsheet, or **approximately 33 double-ply sheets remaining.**

*2) Generalizing Question 1, how does the function ofÂ **TPsheetÂ **remaining andÂ **TProllÂ **diameter relate in equation and graphical form?*

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This is a fun question, and I will first interpret âTPsheet remainingâ to the number of two-ply sheets remaining, and TProllÂ diameter as the diameter, in mm, of the remaining TProll. Obviously, the former is bound on both ends by the starting number of sheets of 221 and by zero, and the latter is bound on both ends by a starting diameter of 124.6 mm and an ending diameter of 42.6 mm.

Let us assign the variable *n* to be the number of two-ply sheets remaining, the variable *d* to be the diameter of the rapidly shrinking TProll, and the constant *h* to be the height of TProll. Let us assign the constant VsheetÂ to be the volume of a single sheet of two-ply, the constant VcardÂ to be the volume of TPcardÂ with enclosed air, and the variable VrollÂ to be the total volume of the TProll. Given the total volume of TPsheetÂ as 1,248,000 mm3, we know that VsheetÂ is 5,647 mm3, and we previously calculated Vcard to be 165,000 mm3.Â

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We can then relate *d* to *n* by combining the following two equations:

*Vroll = Vcard + nVsheet*

*d = sqrt((4hVroll)/(Ïh))*

Therefore,

*d = sqrt((4h(**Vcard**Â + n**Vsheet**))/(Ïh))*

This simplifies to the following:

*d = sqrt(1814.2 + 62.1n)*

Let us plot this equation in Figure 3 from *d* = 42.6 mm to *d* = 124.6 mm, the full range between used and new TProll. The threshold of half-toilet paper remaining is represented by the red point.

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Limitations of this analysis include poor generalizability due to lack of information regarding the thickness of a TPsheet between various manufacturers, variation among the intrinsic thicknesses of single-ply, double-ply, and triple-ply (generally single-ply TP is comprised of higher pound (thicker paper) than the sheets of double-ply or triple-ply),1Â and variations between dimensions and number of total TP sheets among TP manufacturers and models. Nevertheless, I hope upon reading this document you gain greater consideration of TPsheet remaining in terms of diameter ofÂ TProllÂ so that no further TP mishaps occur.

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No TP was wasted during the course of this analysis.

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**References**

1. MIT. Toilet paper – the interactive user experience of the last 1.5 centuries. http://web.mit.edu/barryk/Public/MIT/2.744/experienceAnalysis/general.html. Published 2006. Accessed July 12, 2020.

2. Amazon.com: Charmin Ultra Soft Bathroom Tissue, 2-Ply, 221 sheets, 30 rolls: Industrial & Scientific. https://www.amazon.com/Charmin-Ultra-Bathroom-Tissue-sheets/dp/B07SX4X27P. Accessed July 12, 2020.

End

Please do not take this post seriously, as it was meant solely for entertainment! If you got a laugh or two out of it, then that is already better than I expect. Now, I could have cut the word count in half without missing anything important, but as I do not plan on publishing it anywhere, the writing is good enough for me – too many shower thoughts and not enough time to write them up! Also, I did not keep track of significant figures in these calculations, but who is counting anyways?

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Last mini-thought – isn’tÂ TPsheetÂ just the most appropriate name? Think about it.

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