The longing
Of belonging,
The gain,
Due to pain,
Is the measure
Of the heart sundered,
Is the treasure
Of being in love.
The other day, I came across a tablet which had an unusual shape of a regular hexagon (see image below). I wondered why would a pharma company make a tablet that is hexagonal in shape. The intuitive thing of course was to give the tablet only distinctiveness. But was it the only answer? Perhaps there were more parameters working there. So, I decided to get more gyan on this from my friendly AI.
Me:
Wanted to know how do pharma companies decide on shape, colour, and size of tablets? I understand a tablet can't be more than swallowing power of a normal person. As also, the size would be proportional to the active ingredients inside the tablets. But what about shape and colour? For example, this tablet is in the shape of a regular hexagon. Most tablets are round. Capsules are cylindrical with both ends rounded. Some tablets are flattened cylinders. Any particular theory behind this? Also, why do pharma companies imprint names in the tinniest of fonts? Although, tablets are prescribed to all ages, but the propensity would be for the elderly, would it not? The ones with failing eyesight?
AI:
Good question — there's actually a fair amount of deliberate science (and some non-science, like marketing and patents) behind tablet design.
Shape
- Round/circular is the default because it's the easiest and cheapest to tool and manufacture — round punches wear more evenly, compress more consistently, and are less prone to capping/chipping during high-speed production. That's why most generic tablets are round.
- Oval/capsule-shaped (caplets) are chosen mainly for swallowability. An oval or elongated shape reduces the "throat profile" — the cross-section your throat has to pass — compared to a round tablet of the same volume, so bigger doses (500mg, 1g) are often oval rather than round.
- Unusual shapes (hexagons, diamonds, triangles, your tablet pentagon/hexagon) usually exist for brand differentiation and anti-counterfeiting. A distinctive shape is trademarkable and lets patients/pharmacists visually confirm "yes, this is the real product" at a glance. It also helps patients on multiple medications tell their pills apart by feel/sight, which reduces dosing errors. Sometimes shape is also tied to how the tablet needs to break (e.g., shapes with a scored line for easy halving).
- Capsules are cylindrical with rounded ends purely for swallowing mechanics — the rounded ends reduce friction and the risk of catching in the esophagus, and the gelatin/HPMC shell shape is somewhat fixed by the manufacturing process (dip-molding on pins).
Colour
Colour is almost never functional (no drug requires a particular colour to work). Reasons companies pick specific colours:
1. Brand identity and trademark — colour becomes part of recognizing "my medication," similar to shape.
2. Patient adherence and error reduction — hospitals and patients often rely on colour to distinguish drugs, especially when someone takes many pills. Studies show patients are more likely to stop taking a "generic" if it suddenly changes colour after a formulation switch, because they no longer visually recognize it as the "same" medicine, even though it's identical.
3. Psychological association — there's real research showing colour affects perceived potency and even effect: red/orange tablets are perceived as more stimulating, blue/green as calming or sedative, warm colours as more effective painkillers in placebo studies. Companies do use this in choosing colours for the intended perception (e.g., sedatives are rarely bright red).
4. Masking or matching the active ingredient's natural colour, or hiding slight batch-to-batch colour variation in the raw material.
Size
As you noted — dictated by dose (amount of active + excipients/fillers needed to bind and compress it) and by practical swallowability limits. Companies do try to minimize excipient bulk to keep tablets as small as possible, especially for chronic-use drugs.
The tiny imprint
This one's a genuinely reasonable complaint you're making. A few things going on:
- Regulators (like the US FDA) actually require an imprint code (letters/numbers, sometimes a logo) on most solid oral dosage forms specifically so tablets/capsules can be identified after they're out of the bottle — say, if someone's hospitalized and no one knows what pill it is, or in poisoning/overdose cases. This is a safety requirement, not vanity.
- The imprint is deliberately small because it has to fit on a tablet that's already sized to the minimum swallowable volume — there's very little surface area on something 6-10mm across, and the imprint has to leave room for the score line, logo, and not weaken the tablet's structural integrity (deep engraving can create fracture points).
- You're right that this creates an accessibility gap for elderly patients or anyone with low vision. This is a recognized problem in pharmacy practice — it's part of why pill identification is more reliably done via pharmacist/pill-identifier databases and apps (using shape + colour + imprint together) rather than by patients reading the imprint themselves. The imprint is really designed for someone else — a pharmacist, ER doctor, or poison control — to read, not necessarily the patient day-to-day.
So the short version: shape and colour are largely branding, trademark, and human-factors tools (recognition, adherence, error-avoidance) rather than anything to do with the drug's chemistry; the imprint is a regulatory safety requirement optimized for professional identification rather than patient legibility.
Me:
What is more amenable to ingestion into human systems? Tablets, or liquid potions? All things being equal: for example, degree of difficulty in manufacturing, solubility, etc.
AI:
If you strip away manufacturing cost, shelf-stability, and practicality — and ask purely "which form gets the active drug into the bloodstream more efficiently" — liquids have the edge, for a fairly fundamental pharmacokinetic reason.
Why liquids absorb faster/more completely, in principle
A tablet, once swallowed, has to go through a sequence of steps before the drug can even reach the bloodstream:
1. Disintegration — the tablet breaks into granules
2. Deaggregation — granules break into fine particles
3. Dissolution — those particles dissolve into solution
4. Absorption — only the dissolved drug can cross the gut membrane
A liquid (true solution, not suspension) skips steps 1–3 entirely. The drug is already dissolved, so it's immediately available for absorption the moment it hits the stomach/intestine. This is why dissolution is often the rate-limiting step for tablets — pharmaceutical scientists spend enormous effort optimizing tablet disintegration/dissolution precisely because that's the bottleneck liquids don't have.
This is also why you'll notice:
- Pediatric and geriatric formulations skew liquid — faster onset, easier titration
- Emergency/acute-care drugs (e.g., some cardiac or antiemetic drugs) are often given as liquids, sublingual, or IV rather than solid tablets
- Bioequivalence studies use an oral solution as the theoretical "reference" against which a tablet's relative bioavailability is often benchmarked
Where tablets catch up or win, though
"All things being equal" doesn't fully hold in practice, because:
- Chemical stability: many drugs are far more stable as a dry solid than dissolved in water — hydrolysis, oxidation, and microbial growth are all faster in liquid form. This is why syrups need preservatives and shorter shelf lives, and why some drugs are literally impossible to formulate as stable liquids and only exist as tablets/capsules.
- First-pass metabolism: this is a systemic factor unrelated to formulation for most oral drugs — both tablets and liquids taken orally go through the liver first. (This only differs if you compare oral to sublingual/buccal/IV routes, which bypass first-pass regardless of solid or liquid form.)
- Dosing accuracy: a tablet delivers a fixed, precise dose every time. Liquids depend on the patient measuring correctly (a notoriously error-prone step — think of how inconsistently people use a kitchen spoon vs. a proper oral syringe).
- Gastric irritation and taste: some drugs are unpalatable or damaging to the stomach lining in concentrated liquid form and are deliberately given as enteric-coated or delayed-release tablets to control where and when they dissolve — something a liquid can't easily do. Modified-release tablets can actually outperform liquids for drugs where you want a slow, sustained absorption curve rather than a spike.
- Some drugs are insoluble or unstable in any aqueous vehicle at all, regardless of effort — solubility isn't just an engineering problem, it's dictated by the molecule's chemistry.
Net take
So the honest answer is: liquids win on pure absorption kinetics because they remove the dissolution bottleneck — but "amenable to ingestion" in the everyday clinical sense (accuracy, stability, control of release site/timing, patient convenience) often favors tablets, which is why they dominate the market despite the kinetic disadvantage.
Each season chases out the former,
All seasons lie in wait in the Sky,
For the extant to be chased away:
Spring lies in wait
For Winter to be chased away,
As Summer waits for Spring
To be chased away,
Then Autumn, then Winter,
They are all hiding in the Sky.
One day a season or more will get tired,
Someday a season or more will slow down,
And thus fall far behind,
And, forever lay hidden in the Sky.
Will it be Winter, will it be Spring?
Will it be Autumn, will it be Summer?
Only the Sky can tell:
For It only downs the Veil,
It also holds the Rotating Wheel.
I received this Sunday gyan today:
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The knowledge- Sunday Reading
Studies suggest that walking in a park or near a water body reduces cortisol (stress hormone) by 30%.
Crazy fact: In just 1 hour, more solar energy hits Earth than humanity uses in a whole year. And in <8 hours, enough sunlight hits Earth to power us for a year.
i^i = 0.20787957635….. Imaginary number to the power of imaginary number is a real number be-cause of Euler's formula, which connects trigonometric functions with complex exponential func-tions.
...and more reading
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The third factoid was pretty interesting. I asked help of AI to find how i^i = 0.20787957... I added a bit of my own complexity just for fun. I asked AI thus: What would be i to the power of i? I saw somewhere that it would be 0.207.... This is amazing , is it not? A decimal fraction? What would be tetration of i , say 4 subscript i?
Now, tetration is an maths animal I had just heard about. Tetration of any number would be that number to the power of that number, to the power of that number, to the power of that number...and so on. In the case of 4 subscript i(or, ⁴i), it would be i to the power i, to the power of i, to the power of i, to the power of i, or in other words: i^(i^(i^i)), or,
Yes, i^i is one of the loveliest surprises in complex analysis — a pure imaginary number raised to a pure imaginary power ends up being a real number.
Why i^i is real
Using Euler's formula, i = e^(iπ/2) (writing i in exponential form). So:
i^i =(e^(i*π/2))^i = e^(i*i*π/2) = e^(-π/2)
And e^(−π/2) ≈ 0.2078795...
It's a real number, but it's not a fraction in the sense of a ratio of integers — it's irrational (in fact, transcendental). 0.207... is just its decimal approximation, not a terminating or repeating fraction.
Now the tetration: ⁴i = i^i^i^i
Tetration stacks exponents from the top down: ⁴i = i^(i^(i^i)). Let's build it one level at a time, always using the principal branch:
Level 1: a₁ = i
Level 2: a₂ = i^i = e^(−π/2) ≈ 0.20788 (real, as above)
Level 3: a₃ = i^(a₂). Since a₂ is real, this is:
i^(0.20788) = e^(0.20788 * i*π/2) = cos(0.3266) + i*sin(0.3266) =approx 0.9472 + 0.3208i
Level 4: a₄ = i^(a₃), now with a complex exponent:
i^(a₃) = e^((a₃)*i*π/2))
Working through the complex arithmetic (multiplying a₃ by iπ/2, then exponentiating) gives:
⁴i= approx 0.0502 + 0.602i
Its magnitude is about 0.604, and it sits at an angle of roughly 85° from the positive real axis.
A fun fact: as you keep stacking more i's (⁵i, ⁶i, ...), the tower actually converges to a fixed complex number rather than blowing up or oscillating forever — infinite tetration of i converges to approximately 0.4383 + 0.3606i. That's a much rarer property; most bases don't have convergent infinite towers.
Explaining e^(−π/2): That line comes from calculating i^i.
Here's the reasoning behind it:
Step 1: Write i in exponential form.
Any complex number can be written as e^(iθ), where θ is the angle it makes on the complex plane. The number i sits at 90° (π/2 radians) on the unit circle, so:
i = e^(i*π/2)
Step 2: Raise it to the power i.
i^i = (e^(i*π/2))^i
When you raise a power to a power, exponents multiply:
= e^(i*π/2*i) = e^(-π/2)
since i² = −1. Notice the i's cancel out — that's the key step. What started as an imaginary exponent multiplying another factor of i becomes real, because i times i is just −1.
Step 3: Evaluate the number.
π/2 ≈ 1.5708, so we need e^(−1.5708). Since e ≈ 2.71828, this is 1 divided by e^1.5708, which works out to:
e^(-π/2), approx 0.2078795...
Why this feels surprising: you'd expect an imaginary number raised to an imaginary power to give you something imaginary or at least complex. Instead, the two imaginary-ness factors cancel each other through i² = −1, leaving a plain real decimal — and an irrational one at that, not a fraction, just a number that happens to start 0.207.../2) ≈ 0.2078795...further:
Me:
The fun fact, ' as you keep stacking more i's (⁵i, ⁶i, ...), the tower actually converges to a fixed complex number rather than blowing up or oscillating forever — infinite tetration of i converges to approximately 0.4383 + 0.3606i. That's a much rarer property; most bases don't have convergent infinite towers.' has got me intrigued. Have we found another e-like number as we tend tetration of i towards infinity, albeit in the complex plane?
Claude AI:
Not quite an analogue of e, but let's look at what makes e special versus what makes this fixed point special — the comparison is illuminating.
What actually makes e "e-like"
e is fundamental because it shows up as the unique solution to many independent, unrelated problems: the base where d/dx(eˣ) = eˣ, the limit of (1+1/n)ⁿ, the sum of 1/n!, and so on. It's a universal constant that keeps appearing across completely different branches of math — that ubiquity is what earns it the special status.
What the fixed point 0.4383 + 0.3606i actually is just the solution to one specific equation:
z = i^z
That's it. It's the fixed point of this particular map (raise i to the power of the input, repeat). It's a legitimate and interesting number — but it's not showing up independently all over mathematics the way e does. It's more like asking what does the sequence x, cos(x), cos(cos(x)), ... converge to? (answer: the Dottie number, ≈0.739085, the fixed point of cos(x) = x). That's a genuinely analogous situation — a real fixed point of a specific iterated function, interesting in its own right, but not a universal constant.
Where the real analogy lies
If you want the actual "complex-plane e," you don't need to look for a new number at all — e itself already lives on the complex plane and does the heavy lifting there via Euler's formula, e^(iθ) = cos θ + i sin θ. That's the bridge between e and complex numbers, and it's precisely the tool we used to compute i^i in the first place.
What you have found, though, is something worth appreciating on its own terms: a genuine example of a complex fixed point of an iterated exponential map. Your complex fixed point is the counterpart of that convergence question, just off the real axis. So: a nice number, a legitimate mathematical object, but its significance comes from being a fixed point of this one map — not from popping up everywhere the way e does.
Me:
Can this be geometrically created? Will this point be same point in space wherever we are on earth?
AI Claude:
Good questions — they touch two different ideas, so let's separate them.
Same point wherever we are on Earth?
Yes — but the reason is simpler than it might seem: this point isn't a location in physical space at all. It's a point in the complex plane, which is a purely mathematical construct — a 2D coordinate system where one axis represents real numbers and the other represents multiples of i. It has nothing to do with geography, gravity, or your position on the planet.
So 0.4383 + 0.3606i is the same abstract mathematical object whether you compute it in Bengaluru, Antarctica, or on the Moon — the same way "7" or "π" doesn't change depending on where you are. It's a number, not a place.
Can it be geometrically constructed?
Here there are two very different meanings of construct, and the answer differs for each:
1. Classical straightedge-and-compass construction — almost certainly no.
Classical constructibility (the kind used for bisecting angles or constructing regular polygons) only works for a specific class of algebraic numbers — ones reachable by a finite sequence of square roots. Our fixed point comes from solving z = i^z, a transcendental equation with no closed algebraic form. Numbers like this are expected to be transcendental (not the root of any polynomial with rational coefficients), which puts them firmly outside what compass and straightedge can ever reach — much like you can't classically square a circle because π is transcendental.
2. Iterative geometric construction — yes, and it's genuinely elegant.
You can construct it as a limiting picture through repeated geometric steps:
- Start at any point in the complex plane
- Apply the transformation z → i^z (a rotation by angle proportional to Re(z), combined with a scaling by a factor depending on Im(z))
- Plot the new point
- Repeat
If you connect the successive points, you get a spiral that winds tighter and tighter, converging visually toward 0.4383 + 0.3606i. This is the complex-plane version of a cobweb diagram, the same visualization technique used to show the real Dottie number (fixed point of cos x) spiraling into place. It never exactly arrives in finitely many steps — but it approaches arbitrarily closely, which is a legitimate and beautiful way to construct it geometrically, even though it's not the classical Euclidean sense of the word.
Want me to actually plot that spiral for you? It would visually be a satisfying way to see this whole conversation come together.
Me:
Yes please.
AI Claude:
Here it is:
