Height advantage in hiking

For an outdoorsy, not-so-tall girl, it’s not uncommon to wind up at the back of a pack of significantly taller, male hiking companions. Sweaty and panting, I watch their backpacks recede further away up the trail, and even the sweep guy might abandon his role to bolt around me. In an endurance situation, mental fatigue sends the foggy brain into rhythmic, ineffectual loops. Unable to do mental arithmetic while moving, one can only see that the negative space triangle formed by others’ legs is larger for taller people, and imagine that this reflects some advantage… but how much advantage?

Later, off the trail, pen and paper in hand, one can focus on calculating the magnitude of how much this height advantage adds up to, in terms of explaining how a physically fit person might lag so far behind:

Height of the taller person inches
Walking cadence steps per minute
Stride angle (angle between legs at full extension point) degrees
Height of the shorter person inches
The taller of the two hikers, being inches tall, has an assumed leg length (measured from hip joint pivot point) of inches. Given his stride angle of degrees, he takes steps that are inches long. At his walking cadence of steps per minute, he thus hikes at the rate of miles per hour.

Meanwhile, the shorter person has an assumed leg length of inches. Despite using the same stride angle and walking cadence as her companion (i.e., putting in the same amount of effort), each of her steps is smaller and she therefore covers ground more slowly...merely due to being shorter!

In order to keep up, the shorter person must work harder, by either:
(a) making her small steps more rapidly, at a faster cadence of steps per minute; or
(b) matching her companion's same walking cadence, but making each step longer by widening her stride angle to degrees. (As efficiency-conscious runners well know, increasing step length beyond what is optimal for one's height has a dramatic effect on tiredness.)

Alternatively, if the shorter person exerts only the same effort as her taller companion, she will fall behind miles after one hour of hiking. In such case, the taller person will get to stop and rest minutes every hour, while waiting for the shorter person to catch up.

[Note: We’ve made the simplifying assumption of leg length as a fixed proportion (45%) of overall height – a reasonable constant, given that average ratios of leg length to height, and step length to leg length (a function of stride angle, which correlates positively with speed) enable trackers to infer height from footprints.]

Other factors driving differential physical effort between two companions are undoubtedly afoot during a hike: aerobic fitness, anaerobic endurance, strength-to-weight ratio, movement/form efficiency, backpack contents, stomach contents, sufficiency of recent sleep, injuries, performance of clothing/gear, and who’s chatting more than listening. Still, the point here is that leg length alone has a substantial impact on rate of travel. Regardless of which physical issues contribute to the exertion asymmetry, the optimal solution for both hikers (assuming they value fairness and social interaction) is to “put Herbie in front” — i.e., have the disadvantaged hiker set the pace.

Eli Goldratt’s 1984 classic The Goal vividly illustrates this principle of operational efficiency with….a hiking example! Herbie (the fat kid in the book; in our case, the short hiker) is the bottleneck. When the fast kids hike at their own pace with Herbie in the back of the single-file line of boy scouts, Herbie falls behind. They impatiently wait for him at trail intersections, but only to immediately take off hiking again as soon as he catches up and before he catches his breath. Herbie gets more and more tired, and thus even more physically disadvantaged, since fatigue initiates a negative feedback loop in terms of physical performance. Meanwhile, the fast kids get periodic rest, and so the effort differential increases from both directions. Putting Herbie in the front of the line — combined with distributing his backpack load among the fast kids — ensures that the hikers stay together and evenly spaced, and that the physical effort difference is somewhat lessened. (The effort saved by fast kids hiking slower than their capabilities is less than the effort saved by Herbie avoiding being in chronic, desperate catch-up mode.)

Posted in Interactive calculators, Main, Math is everywhere!, [All posts]

Lighting adequacy calculator

“How many light fixtures will I need for this room, to ensure sufficient lighting?”

Interior designers notoriously ignore this question.   Homeowners and apartment-dwellers don’t know how to project lighting needs.   The result is trial-and-error:  (1) install/plug in the lamps and fixtures you have, (2) wait for nightfall to assess whether the room feels well-lit, (3) order more lights, (4) reassess.   This unscientific approach is inefficient – and can be disastrous in a kitchen renovation, where most light fixtures are hard-wired, and must have been specified upfront in the room design.

I developed this lighting adequacy calculator to eliminate the guesswork!

Some important concepts:

  • Lighting is the single most impactful interior design element in any room.  Change the lighting, change your experience of the space!  Don’t skimp on adequate amounts of beautiful, varied and well-distributed lighting.
  • Wall color has a huge impact on how much light you need.  Dark walls require more light, in order to yield the same effect to your eyes.  (Dark walls with adequate lighting creates a beautifully glamorous ambience.  Consider how many high-end restaurants employ this approach.)
  • Lampshades have a huge impact on how much light you need.  Putting a lampshade over a bulb reduces the amount of light that reaches your eyes.   Theoretically, lampshades are an absurd concept, reducing the efficacy of the light you (and the earth) are paying to produce… in order to produce comfortable and pleasing light.  (Black fabric shades on table lamps and sconces are a hallmark of high-end interior design, adding high drama to a bedroom or living room.)
  • Combine (a) always-on (when the room is occupied) ambient lighting with (b) task lighting.  This gives you lots of optionality to dynamically match lighting with how the room is being used.  (Also, use dimmers liberally and ensure most lights are individually controllable.)
  • Sophisticated and pleasing lighting schemes include light coming from many directions (down from ceiling fixtures, across from floor lamps and sconces, up from table lamps) and of multiple qualities (intense light from bare bulbs, soft light filtered through glass fixtures, mood light escaping from around opaque fabric lampshades).
  • The older you get, the more light you need to feel that a room is well-lit.   Err on the side of more lighting – and especially more task lighting – if you have any trouble reading.  Insufficient lighting makes people feel tired.
  • Lighting requirements depend greatly on what the room is used for. Kitchens require much more light per square foot than do living rooms. Bathrooms need as much light as kitchens, but less of it needs to be task lighting.  Over-lit restaurants with harsh, bare bulb lighting are uncomfortable, unromantic and can easily kill the business.
  • Light produced by a lightbulb is measured in lumens.  Adequate lighting means your lighting scheme is producing adequate lumens (conditioned by wall color and fixture shading). A lightbulb consumes a certain number of watts (electricity) to produce those lumens (visible light output), depending on the type of bulb.  For simplicity, this calculator asks you to input all of your lighting in terms of the nominal watts consumed by the bulb (i.e., the watts listed on the bulb).  The calculator automatically indexes all lighting to “incandescent equivalent watts” using each bulb type’s lumens-per-watt efficiency.

How to use the calculator:  Input the nominal watts for all of your planned or existing light sources.   Compare the resulting “Effective Watts” of your current lighting scheme to the theoretical “Target Effective Watts” figure.  Do you have enough ambient lighting, enough task lighting, and thus enough overall lighting?

Lighting Adequacy Calculator
  Room type:  
  Square feet:  
  Wall color:  
  Nominal watts of light fixture Effective watts Target effective watts

("Effective watts" are incandescent-equivalent watts, adjusted for the shade opacity).

Ambient lighting (always on during occupancy)
  Bare bulb 645
  Light shade 0
  Medium shade 0
  Opaque shade 92
  Low-voltage halogen
  Bare bulb 0
  Light shade 0
  Medium shade 0
  Opaque shade 0
  Bare bulb 63
  Light shade 0
  Medium shade 0
  Opaque shade 0
  Bare bulb 90
  Light shade 0
  Medium shade 0
  Opaque shade 0
  Total ambient light 891 1225
  Ambient light per square foot 2.5 3.5
Task lighting
  Candles (number of wicks) 1  
  Bare bulb 0
  Light shade 0
  Medium shade 0
  Opaque shade 0
  Low-voltage halogen
  Bare bulb 434
  Light shade 0
  Medium shade 0
  Opaque shade 0
  Bare bulb 63
  Light shade 41
  Medium shade 0
  Opaque shade 0
  Bare bulb 0
  Light shade 0
  Medium shade 34
  Opaque shade 0
  Total task light 891 1225
  Task light per square foot 891 1225
Total light 891 1225
Total light per square foot 891 1225
Posted in Design and aesthetics, Interactive calculators, [All posts]

The non-paradox of choice

Popular wisdom about cause and effect evolves over time from simplistic to sophisticated.  Initially, people assert the existence of black-and-white, monotonic relationships between two variables:  “Money can’t buy you happiness”.   Eventually, an understanding develops that the truth is more nuanced:  “Increasing annual income in the US up to about $75,000 ‘buys’ substantially greater happiness.  Income gains beyond that level promote only marginal increases in happiness.”   A 0/1 conceptualization of the issue is gradually replaced by a dialectical embrace of complexity.  “Either/or” (in the tradition of Aristotle, Kierkegaard, and Kant) evolves into “both/and” (in the tradition of Hegel).

Some causal relationships are clarified quickly, particularly when caveating an over-broad idea is self-serving.  Bacon was revealed as a carcinogen in October 2015.  (In fact, cured and processed meats have long been known to have carcinogenic effects; the popular press picked up the story only when the World Health Organization officially reclassified bacon.)  Within a week, good journalists had elaborated on the story, explaining that the health risks of bacon are a question of dosage – just as with all chemicals to which living organisms are exposed.  Even people who might be expected to have difficulty grasping much about toxicology and pharmacology seem able to get their heads around bacon being trivially-threatening in small portions, relative to the toxic soup in which we all live.  Bacon’s fashionableness is saved by a modest dose of intellectual sophistication.

Other popular sayings are difficult to enrich with truthful complexity because they are so self-gratifying.  “What doesn’t kill you makes you stronger” is a common platitude intended to encourage people who are suffering and rationalize their pain as positive.  People prefer to avoid acknowledging one another’s struggles, and often justify such avoidant behavior by invoking this self-comfortingly dismissive aphorism.  In so doing, they add insult to injury by invalidating real experience and flippantly ignoring the risks and vulnerabilities to further harm which are faced by a downtrodden person.

Adverse events in series lead to exhaustion and a negative feedback loop — just like each knock-out punch makes a boxer weaker and more susceptible to another knock-out punch.  More accurately:  “What doesn’t kill you can make you stronger…up to a point, and if negative experiences are spaced out enough so you have time to recover.”  Research on recovery from tragedy shows that resilience to adversity is a function not of individual willpower, but of social support and family structure; withholding such support via a simplistic folk prophecy makes that prophecy even less likely to be fulfilled.

Irrationalities in decision-making have gotten popular attention since behavioral economics (psychology + economics) went mainstream in the early 2000’s.  Findings of academics like Kahneman, Tversky, and Thaler were popularized by the book “Freakonomics” and its numerous successors.  Last week, I counted four people mention “the paradox of choice” to me, without seeming to reference to the original economic concept, its subsequent elaborations, or recent misgivings.

Academic understanding about consumer choice has followed a familiar evolutionary arc:  original broad premise that more choice is good, disruptive finding that more choice reduces satisfaction and consumption, sophisticated clarifications that more choice is ok when choosers know the domain and options are well organized, further realization that paralyzing anxiety in the face of abundant choice may be an uncommon effect arising only in exceptional situations.  Meanwhile, consumers lag behind, stuck on the seductively simple idea that “lots of choice is bad”.  If there were a monotonically decreasing function linking number of available options with outcome satisfaction, that would provide a handy excuse for lack of self-knowledge, poor decision-making skills, and intentional game-playing.

As applied to the realm of online dating, the “paradox of choice” purports to explain how (the illusion of) immense choice makes people (primarily men) feel unsatisfied with any one individual option.  In actuality, such observed fickleness seems more likely to be the result of not consciously knowing a priori what one wants from an online interaction (due to lack of self-awareness), failure to think through a logical and systematic decision-making approach to filter available options (exacerbated for men by low consequences to them of adverse decision outcomes), and intentional pursuit of ego payoffs as an endgame.

A better – though still accessibly simple – folk truism would be “more choice is good…though at some point, dealing with tons of choice requires a lot of time, knowledge and skill”.   Case in point:

  • For 2 ½ years after a tumultuous divorce, I lived transiently with what fit inside my car.  This included a total of 5 pair of pants (1 nice jeans + 1 ratty old jeans + 1 yoga + 1 hiking + 1 interview pants). That was hard – especially when people noticed me wearing the same one pair of pants to every networking event and professional meeting.
  • When I eventually regained access to my full complement of belongings, I unpacked some 25 more pants into my new apartment.  That was hard, too, since it took time to sort through them each day, and I had the option and burden of selecting pants to precisely match the formality and attitude of any situation.
  • Ultimately, I culled the pants collection down to an optimized personal sweet spot of 16 pairs (1 nice jeans + 2 casual + 1 ratty old jeans + 2 yoga + 1 hiking + 4 winter work + 4 summer work + 1 interview pants).  Choosing among 16 pants yields better outcomes than with just 5, and limiting the choice set to 16 avoids the stress of dealing with 30.
Posted in Decision quality, [All posts]

Fix your internet

Many frustrations with home internet reliability are actually “operator error” — not caused or addressable by the internet service provider.

First, understand how your system works and resolve all issues under your control… before you masochistically call Customer Support!

Click on the image below to open an 8-page PowerPoint with step-by-step instructions to troubleshoot your home Internet experience!
fix your internet

Posted in Business topics, [All posts]

Ventilation adequacy in residential kitchens

Vent hoods are often an afterthought in kitchen renovation projects — the last appliance selected by homeowners.   However, vent hoods are important:

  1. Functionally.  They remove heat, smoke, smells, steam, and fumes from the workspace.
  2. Aesthetically.  The vent hood sits at eye level and is the first thing one may notice when entering the room.

How much ventilation do I need?  Vent hood manufacturers have various formulas for determining what “adequate” ventilation is.  Use the calculator below to bound the answer!

Kitchen ventilation adequacy calculator
Fume/odor generation capacity indicator:
  Gas stove power Btu
  Electric stove width inches
Room dimensions:
  Length feet
  Width feet
  Height feet
  Area 192 square feet
    18 square meters
  Volume 1,728 cubic feet
    49 cubic meters
Throughput measurement equivalency:
  1 cubic foot per minute (cfm) = 1.7 cubic meters per hour (cmh)
Minimum recommended ventilation power
Method CFM CMH   Calculation
A (electric stove size) 360 612   CFM = electric stove width inches * 10
B (gas stove power) 400 680   CFM = gas stove btu / 100
C (room volume) 288 490   CMH = cubic meters * 10
D (room volume - alt method) 432 735   CFM = cubic feet * 15 cycles / 60 minutes
E (room area) 384 563   CFM = square feet * 2 [For 8-ft ceilings, this yields the same result as 15 cycles of room volume per 60 minutes]
F (room volume - for bathroom) 230 392   CFM = cubic feet * 8 cycles / 60 minutes


Installation considerations:

  • Noise
    • Noise comes from both the motor and from air movement
    • More turns in ducting = more noise
    • 6 sones @ 600 cfm is a reasonable noise level
    • For >1000 cfm, you need at external blower.  Blower should be placed beyond >12 ft ducting, for noise abatement.
  • Location
    • 30″ above cooking surface
    • 20-24″ projection from wall
    • Wood-covered hoods must be >36″ above the cooktop, which reduces their effectiveness.  Metal hoods are therefore preferable.
Posted in Design and aesthetics, Interactive calculators, [All posts]

Appointment frequency calculator

Managing weekday appointments at work can be challenging. Most types of service providers don’t offer weekend or evening appointments. And, some work environments and supervisors are unsympathetic toward workers needing to leave work for appointments.

While one person might average 1 appointment per month for basic maintenance/preventative care, someone else could easily have over 5 appointments per month for basic care — just because of different personal circumstances. All personal circumstances that could drive high appointment frequency are legally protected from discrimination. However, managers sometimes don’t connect the dots between personal circumstance, protected class membership, and the need to leave work for appointments.

This problem is more acute for women, for single people who can’t share the burden with a partner, and for those with pets, children, or medical conditions. For example, a worst case scenario would be a single woman with a medical condition who has a pet/child with a medical condition (perhaps necessitating some 9 appointments every month). Furthermore, though employees aren’t required to inform their managers of their personal and medical details (in fact, again, this is protected information), as a practical matter they often must do so, in order to avoid negative perceptions.

Use this calculator to determine your expected average appointment frequency — and to discuss the potential “legitimate” range of weekly work absences (between

Appointment frequency calculator

Note: If you never go to a given type of appointment type "9999".

  Months between appointments
  Primary care
  Eye doctor
  Physical therapist
  Talk therapist
  Specialist 1
  Specialist 2
Personal grooming
  Other 1
  Other 2
  Other 3
Pet care
Child care
  School meeting
  Other 1
  Other 2
  Other 3
Other services
  Car mechanic
  Other 1
  Other 2
  Other 3
  Other 4
  Other 5
Average appointments per week 0.7
Average appointments per month 3.2
Entering 0 indicates you are continuously in this appointment at all times - please enter another value!
Posted in Business topics, Interactive calculators, Social issues, [All posts]

Fabric yardage for custom lampshades

Custom lampshade makers typically require designers to provide substantially more fabric yardage than is actually necessary.  This is result of the slightly involved arithmetic required to calculate how a 3-dimensional shape translates into 2-dimensional surfaces.

Using this calculator to generate a second opinion of (exact theoretical) yardage requirements, you can reduce waste and save money.  (Make sure to increase this theoretical result to a practical yardage amount that contemplates hems, wire-wrapping, selvage, grain direction, pattern placement and bolt width!)

Fabric requirements for conical shades
  Top diameter
  Bottom diameter
  Slant height
  Pattern dimensions:  
a Total radius 18.5
b Radius of inner cutout circle 6.5
c Angle between radial cuts (angle of area discarded) 165
Minimum fabric piece size (x, y)
37     x     21
Posted in Design and aesthetics, Interactive calculators, [All posts]

Ellipse perimeter

Ellipses are far more aesthetically pleasing than “racetrack”-shaped tables. They also have fascinating — and somewhat complicated — geometry.

I created this calculator for two purposes:
(1) Commission elliptical tables from furniture-makers not using CNC machines (who thus need to draw the shape using the string method)
(2) Scale table designs to ensure desired seating capacity (as opposed to guessing based on racetrack tables, as furniture showrooms do)

Perimeter of an elliptical table
Major axis - centerpoint to most distant edge of table (a) inches
Minor axis - centerpoint to closest edge of table (b) inches
  Distance from centerpoint to each of the two foci (f)  a2 - b2  26 inches
  Length of string to draw the ellipse 2a 72 inches
Table perimeter π [3(a+b) - √ (3a+b)(a+3b) ] 193 inches
Number of people at the table inches
Table frontage per person 24 inches
Compare to a racetrack-shaped table (circle of radius b, plus rectangular insert for expansion):
  Circumference of circular table, without insert 2πb 157 inches
  Table frontage per person, without insert 20 inches
  Width of insert (d) inches
  Perimeter of table, with insert 2πb+2d 197 inches
  Table frontage per person, with insert 25 inches

ellipse sketch

Posted in Design and aesthetics, Interactive calculators, Math is everywhere!, [All posts]

Sample size calculator

Appropriate sample size for market research surveys is an endlessly (yet sloppily) debated topic.

Sample size calculator

Click on the above link to open an Excel model that calculates sample sizes and effect sizes, based on desired confidence and power levels. (You will need to enable editing and macros.)

Also, read an in-depth explanation of how to choose the appropriate sample size in this post: www.theideafactoryonline.org/sample-size-theory/

Posted in Business topics, Decision quality, Interactive calculators, [All posts]