Hidden Detail in Crotch Solves 500-Year-Old Leonardo Da Vinci Mystery
To draw the Vitruvian Man in 1490 – an illustration of the 'ideal' human body – the Renaissance man may have relied on a mathematical ratio not formally established until the 19th century.
It's one of the most iconic images of all time, and yet for more than 500 years, no one could figure out why da Vinci chose such specific proportions for the arms and legs.
A London dentist thinks he's solved the mystery at last.
Related:
Rory Mac Sweeney has found a crucial hidden detail, tucked in the Vitruvian Man's crotch: an equilateral triangle that he thinks may explain "one of the most analyzed yet cryptic works in art history."
The Vitruvian Man is partly inspired by the writings of Roman architect Vitruvius, who argued that the perfect human body should fit inside a circle and square.
Da Vinci's drawing uses a square to precisely contain a 'cruciform pose', with arms outstretched and legs in. The circle, meanwhile, encompasses a posture where the arms are raised and the legs are spread.
A popular explanation is that da Vinci chose the Vitruvian Man's proportions based on the Golden Ratio Theory, but the measurements don't quite match up.
According to Mac Sweeney, "the solution to this geometric mystery has been hiding in plain sight".
"If you open your legs… and raise your hands enough that your extended fingers touch the line of the top of your head… the space between the legs will be an equilateral triangle," da Vinci wrote in his notes for the Vitruvian Man.
When Mac Sweeney did the math on this triangle, he found that the spread of the man's feet and the height of his navel created a ratio of around 1.64 to 1.65.
That's very close to the tetrahedral ratio of 1.633 – a uniquely balanced geometric form, officially established in 1917.
The ratio is used to establish the optimum way to pack spheres. If four spheres are connected as closely as possible into a pyramid shape, for instance, then the height to base ratio from their centers will be 1.633.
Perhaps Mac Sweeney recognized the significance of that number because of a similar triangular principle used in dentistry.
Imagined on the human jaw, Bonwill's triangle dictates the optimal positioning for jaw function, used since 1864. Its ratio is also 1.633.
Mac Sweeney doesn't think that's a coincidence.
Similar to minerals, crystals, and other biological packing systems found in nature, Mac Sweeney thinks the human jaw naturally organizes around tetrahedral geometries, which maximize mechanical efficiency.
If the tetrahedral ratio is repeated around our bodies, Mac Sweeney thinks that is because "human anatomy has evolved according to geometric principles that govern optimal spatial organization throughout the universe."
If Mac Sweeney is right, Da Vinci may have stumbled across a universal principle while drawing the Vitruvian Man.
"The same geometric relationships that appear in optimal crystal structures, biological architectures, and Fuller's coordinate systems seem to be encoded in human proportions," writes Mac Sweeney, "suggesting that Leonardo intuited fundamental truths about the mathematical nature of reality itself."
Whether other scientists agree with Mac Sweeney remains to be seen, but the fact that da Vinci mentioned the equilateral triangle in his notes suggests that what lies between the Vitruvian man's legs is important.
The study was published in the Journal of Mathematics and the Arts.
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Jul 25 2025 This Week in Cardiology
Please note that the text below is not a full transcript and has not been copyedited. For more insight and commentary on these stories, subscribe to the This Week in Cardiology podcast , download the Medscape app or subscribe on Apple Podcasts, Spotify, or your preferred podcast provider. This podcast is intended for healthcare professionals only. In This Week's Podcast For the week ending July 25, 2025, John Mandrola, MD, comments on the following topics: The group at the University of Leeds in the UK have published an interesting paper on endurance athletes, cardiac fibrosis, and ventricular arrhythmia. They cheekily named it Ventoux to correspond to a recent Tour de France stage finishing on Mount Ventoux. I've climbed Ventoux. It took me 1 hour 42 minutes. Tadej and Jonas did it this week in 54 minutes—a record by a minute. Imagine liking, no, loving, the notion of climbing a 10% gradient for well more than hour and you will understand the mindset of older endurance athletes. Thinking this is fun is a sort of disease. People who do this day in and day out for decades have an affliction. I know because I do. We are like rodents who, given a wheel in our cage, will run on it, for no purpose other than to run on it. Before I say anything else, I want to start with the fact that exercise is one of the pillars of health. It's in every expert consensus document and oodles of observational studies correlate exercise with longevity. But 60-year-old people who cycle or run more than 10 hours per week for decades are not normal exercisers. For instance, if you do an hour ride 5 days per week, to get to that level of exercise, you have to do a 5-hour ride on the weekend or two 3-hour rides on Saturday and Sunday. That's a lot. I co-wrote a book called The Haywire Heart in which my chapters were basically narrative reviews of the cardiac effects of all this exercise. The one, single observation that got me interested in this topic is that endurance athletes have a heightened relative risk of getting atrial fibrillation (AF) and, electrically and structurally, their atria remodeling resembles that of a person of obesity and hypertension. Isn't that wild? But AF is not the topic of this paper. Endurance athletes can also develop both myocardial fibrosis (scar) and ventricular arrhythmias (VA). The VA story is less fleshed out than the AF story, perhaps because it is less common. But again, you have the paradox: endurance exercise correlates with scar seen in healed myocarditis and other cardiomyopathies. Something amazing for health — regular exercise — perhaps can be detrimental in huge doses. And we all know that myocardial scar can predispose to ventricular arrhythmias—which, can in rare cases, lead to sudden cardiac death. Now to the Ventoux study: The Leeds group recruited about 100 endurance athletes and 27 controls with the purpose of studying MRI, CMR, and monitoring for VA with implantable loop recorders (ILRs) over 2 years. All participants had to agree to an ILR implant, which is likely an important factor in the study's interpretation. This study is a great effort. I congratulate the authors. The average age was about 60. All were men. None could have any symptoms or heart disease. The athletes were pretty strong. The functional threshold power, what can be held for an hour, was around 244 watts. That's not close to elite but it's not Mickey Mouse either. Another factor was that the Leeds group is a well-known and experienced in imaging. So, the MRI imaging can be trusted, and an expert in imaging told me these are likely real findings. And here are the topline results: Nearly half (47%) had myocardial fibrosis on MRI, the vast majority had inferior lateral location of scar. The main way the authors display the results is with two groups—those with scar and those without scar. About 1in 5 athletes overall had some ventricular arrythmia on the ILR during 2 years of follow-up Those with fibrosis had a 4.7% higher relative risk of VA. (But crucially, the 95% CI were wide from 80% higher risk to 13x higher). Athletes with fibrosis were slightly older than athletes without fibrosis (62 years versus 57 years) Athletes with fibrosis exhibited a greater prevalence of premature ventricular contractions (PVC) during exercise testing than athletes without fibrosis (71% vs 42%; P =.003) All three of the athletes with VT longer than 30 seconds, which we call sustained VT, had fibrosis on cardiac MRI. And all 3 who experienced sustained VT were symptomatic and developed an episode of nonsustained VT before the onset of VT. One athlete received an ICD due to presyncope, one was scheduled for an EP study (the results of which were not known), and the third was advised to cease competing due to recurrent VT during exercise but declined further investigation Of those who experienced a ventricular arrhythmia, 78% of the athletes had myocardial fibrosis on CMR compared with 22% athletes who did not ( P < .001). Two other predictors of having VT were left ventricular (LV) dilation and exercise-induced PVCs. Late gadolinium enhancement (LGE) at the right ventricular (RV) insertion point was super common. This is also well known in athletes but it had no statistically significant correlation with VA. But confidence intervals (CI) were wide again. Both groups had normal left ventricular ejection fraction (LVEF). Both had similar and normal RV function. There were no significant differences in T2 times between athletes with and without ventricular arrhythmia. No athlete had T2 values indicative of acute myocardial edema, and no athlete fulfilled Lake Louise Criteria for acute myocarditis. The authors concluded that fibrosis incidence was high and associated with VA, and RV insertion point LGE was not associated with VA and…sit down for this conclusion: 'Further studies are needed to establish whether myocardial fibrosis itself is arrhythmogenic or in the case of athlete's indicative of a myopathic process.' I like this conclusion, and it's different from many of the posts I have seen on social media—which go too far in scaring people about exercise. There is much to say about these observations, but, sadly, most are questions rather than answers. First: Why was the fibrosis presence so high? Almost half of people. I think the issue here is a systemic bias in this study: it's a selection bias. Or more specifically , self-selection . A researcher who does a lot of normal volunteer studies told me that, after the fact, many asymptomatic volunteers admit to symptoms they wanted checked. Now…Recall this study was done in the UK, where getting checked out may not be so easy. So you enroll 100 people who have to agree to get ILRs. To me this is a special population, who are probably more than just curious; they are probably concerned. Endurance athletes tend to be obsessive and read a lot about the heart issues in athletes. I would almost consider getting an ILR a collider bias. You are not looking at a general population of endurance exercisers but rather those endurance exercisers interested in monitoring for some reason. Second issue: There is the issue of ILR monitoring . These things are always on. They, like pacemakers, pick up everything . They are far more sensitive than a 2-week ECG or even an Apple Watch. So you have a population enhanced by concern wearing a monitor that misses nothing. In my device clinic, I probably get three nonsustained VT alerts per day — 99% of which I say, just follow. I've become increasingly convinced that short episodes of atrial and ventricular tachycardia are probably normal in older adults. Indeed, in VENTOUX, only 3 of the 100 individuals had symptoms that required clinical action—an ICD, an EP study, results unknown, and an exercise restriction. All three were symptomatic, so clinically, I agree 100% with the authors, that there is nothing in this data that would suggest screening with MRIs and ILRs. Let me repeat: VENTOUX does not support screening athletes with MRI. You can and should wait for symptoms. The third matter is the degree of LGE. The scar burden was only 2%. These are tiny scars, and most serious VT we take care of in EP comes from much larger scars. But while it is true that any scar is abnormal, we don't know if it is from exercise or healed myocarditis. And we don't know the benefit/harm ratio of lifelong exercise. Lifelong exercise is protective against diabetes, hypertension, coronary artery disease (CAD), but it may cause small LGEs in a very small proportions of patients. Notice that all these are comments are questions. This study is interesting. Intensely interesting. It's a great effort, but to offer more actionable results, beyond, don't ignore symptoms , which is an easy thing to tell people, we would need larger samples that were crucially more random in sampling. Though they tried, VENTOUX is not all a random sampling of heavy exercisers. Imagine a study wherein you went to an endurance race, and you signed up one out of every five 30-year-olds and followed them serially, as they did in Framingham, with CMR every 5 years for two or three decades. Then we might know more. But such a study is vast and super expensive. For now, I recommend regular exercise as if it was a heart pill or AF pill. You take it daily. Todos los dias. The vast majority of people I see don't exercise enough. What bike racers like to do is not at all for health . It is for fun, or for sanity; whether it is a net harm remains to be seen. But even if it were, I am not sure most of us would stop doing it. Because rodents must run on the wheel. An Australian group of researchers have published a massive systematic review and meta-analysis of daily steps in and health. Lancet Public Health published the study, which was covered in 176 news outlets as of this morning. Most of the news stories say something like the 10,000-step myth has been busted and 7000 is sufficient. The news' stories are wrong. And I will briefly cover this study because it bookends the extreme exercise study quite well. The research team had 57 studies from 35 cohorts included in the systematic review and 31 studies from 24 cohorts included in meta-analyses. They looked at dose response (in steps) and many outcomes including all-cause mortality, cardiovascular disease incidence, dementia, falls, and type 2 diabetes. The comparator or control group was 2000 steps. Compared to that those who reported or documented with counter 7000 steps per day had a: 47% reduction in all-cause death (HR 0.53 [95% CI, 0.46–0.60]; I 2 =36.3; 14 studies) 25% reduction in CV disease incidence (HR 0·75 [0·67–0·85]; I 2 =38·3%; 6 studies) 47% reduction in CV mortality (HR 0·53 [0·37–0·77]; I 2 =78·2%; 3 studies) 37% reduction in cancer mortality (HR 0·63 [0·55–0·72]; I 2 =64·5%; 3 studies) 14% reduction type 2 diabetes (HR 0·86 [0·74–0·99]; I 2 =48·5%; 4 studies) 38% reduction in dementia (HR 0·62 [0·53–0·73]; I 2 =0%; 2 studies) 28% reduction in falls (HR 0·72 [0·65–0·81]; I 2=47·5%; four studies) My issue with this study is that when you go to the main figure, the plots with hazard ratio (HR) on the y-axis and step counts on the x-axis, you see a clear dose response of steps and specific outcomes. 2000 steps is where the HR is 0. If it's less than 2000 steps, the outcomes are actually higher, but as the step count increases the HR drops. The authors pick 7000, I assume because that is where the slope of benefit seems to plateau, but when you look at the curves, the HR keeps dropping with more steps. The authors quantify the added benefit of > 7000 steps per day in Table 8 of the supplement. For all-cause death, there is an added 10% lower HR with 10,000 vs 7000. Same for cancer mortality and depressive symptoms—an added 10% lower relative risk. So I don't think any myths were busted. 7000 is fine. 7000 steps per day is associated with lower bad health outcomes. But for many, including all-cause death, 10,000 is better. The added benefits reached statistical significance. So if a patient asks, the number is still 10,000. Though 7000 is also good. And 5000 is better than 4000, which is better than 3000. After reviewing this study, the thought about European vs American life popped into my head. We have probably 2000 people working at my hospital. Less than 5 of them walk to work; less than 10 of them ride their bike. When I visited the team at Basel, Switzerland last fall, it looked like more than 75% walked or cycled to work. Very few American cities are set up for walking. That's sad. So Americans have to make an effort to be active. I think it's worth it. And I recommend it in the clinic. For my height, 7000 steps is about 3 miles. 10,000 steps is nearly 5 miles. The optimal dose is the longer one. But some is better than none. Everyone Deserves a Shot at the American Dream: Sinus Rhythm Let me say a few words here about rate vs rhythm control, because this may actually be the number one issue in all of electrophysiology. The stimulus for writing such a review piece I think comes from the PRAGUE 25 trial of lifestyle modification vs AF ablation. I have opined on that in my July 11 podcast. In sum, AF ablation led to less AF than risk factor (RF) modification alone, though 35% (or 1 in 3 patients) in the risk factor modification group had sinus rhythm (SR) without ablation. And RF modification led to more weight loss, better glycemic control, and better fitness as measured by VO 2 max. PRAGUE 25 also found no statistical differences in AF burden nor quality of life measures. The sits in the literature as a 'positive' ablation trial, but I actually think, healthwise, it is a 'positive' trial for RF modification. The Medscape article cites a 'hybrid' approach wherein all patients who pursue rhythm control also get risk factor modification, which I totally agree with, and I have to say, is underused, at least in my zip code. The absolute wrong thing to do is ablate the AF and not help the patient lose weight and improve cardiometabolic health. Because if you do this, you have merely reduced a surrogate marker — AF episodes. Health is not improved if obesity, hypertension, diabetes, and poor exercise tolerance remain. You succeed as a proceduralist but fail as a doctor in this scenario. The Medscape article goes on to celebrate the benefits of SR over AF. The next logical step is to laud rhythm control over rate control. And here I have a problem. And I somewhat disagree with friend and colleague Eric Prystowsky, MD. Eric is well known for his criticism of AFFIRM and how that trial set EP back years. But I had the pleasure of speaking in Calgary and met the late Dr. George Wyse, the principal investigator of AFFIRM. AFFIRM is one of those landmark trials that deserves your attention. Published in 2002, a total of 4060 patients with AF were randomized to rate or rhythm control. Mortality was the primary endpoint. Patients in AFFIRM were like those we see every day. 70 years old. Most with hypertension, a third with CAD. At 5 years, 23.8% in the rhythm control arm died vs 21.3% in the rate control arm. The HR was 1.15, or 15% worse for rhythm control. CIs were 0.99-1.34 so the P value for arm was just outside 0.05 but the upper bound or worse case was a 34% higher rate of death in the rhythm control arm. More patients in the rhythm-control group than in the rate-control group were hospitalized, and there were more adverse drug effects in the rhythm-control group as well. AFFIRM was largely interpreted as showing no differences in the two strategies. But, really, there was a strong trend for worse outcomes in the rhythm-control arm. One of the major changes in knowledge that came from AFFIRM is the importance of maintaining oral anticoagulation (AC). In AFFIRM, patients in the rhythm control arm who were maintaining SR could stop their oral AC. This led to a difference in AC use 85% vs 70% rate vs rhythm control. In fact, AFFIRM largely changed the view that patients with AF with either strategy should remain on oral AC. The Sherman et al substudy in JAMA Internal Medicine found that patients who remained on warfarin were 68% less likely to have stroke. A large proportion of ischemic strokes (113 of 157) occurred in patients in whom anticoagulation had been stopped (on the basis of re-established normal sinus rhythm) and who had a subtherapeutic international normalized ratio. Eric's point about AFFIRM is it led to too many patients with AF being told that there was no reason to try to get into SR, and if you don't try, and you leave patients in AF, it becomes impossible to restore SR after a year or so. The other problems (or criticisms) of AFFIRM were that patients had to be able to tolerate rate-control. So highly symptomatic patients were excluded. AFFIRM should never have been applied to these patients. Many of these symptomatic patients were younger, and it is a serious error to just leave a symptomatic younger person with AF forever in AF. Another criticism of AFFIRM was that it only included antiarrhythmic drugs (AAD), and amiodarone was the most common one used. AAD were all that was available at the time. We now know that AF ablation is far more effective at rhythm control than drugs. So there is a like a bridge to SR early on, and many patients can be put into SR with rhythm control. Proponents of aggressive rhythm control also cite the EAST-AF trial, a rhythm vs rate control trial, which strongly favored rhythm control. But EAST-AF suffered from serious performance bias issues wherein the rhythm control arm got oodles more interactions with the health system. Here is my take of the decision: AFFIRM still applies. If you have an older person with minimal to no symptoms from AF, rate control is not only fine but maybe preferred. But if there are a) symptoms, and b) clues that rhythm control is possible (e.g., the LA size is not ginormous, or the patient can cooperate, and maybe the AF is not more than 2 years persistent), I try rhythm control. But I tell patients that while there is benefit from SR (in terms of quality of life) rhythm control is hard. It costs a lot, not only in money, but investment in their time and effort. Patients have to know that RF modification is crucial, they will also have to spend a few days in the hospital (for cardioversions, maybe drug initiation or ablation). Remember, when you are getting cardioversions and AAD and ablations you are not at work or on a bike. You are being a patient. It's fine. It's an investment but patients need to know that rhythm control is unlike a gallbladder operation or an appendectomy. Rhythm control is a process that requires a friendship with a cardiologist. It's not one and done. There are also risks to rhythm control. Drug side effects and ablation complications do occur. My friends, be careful flying close to the sun with rhythm control. One of the biggest mistakes I see in general cardiology is leaping to cardioversion without a plan. CV of AF is fine, but you have to have a plan for what will happen in a week or month when the patient is back in AF. CV doesn't modify the problem of AF. It just resets the heart. In the end: EP is here to help. Get us involved. Especially when there are symptoms. And doubly especially, when there is heart failure. But don't dismiss AFFIRM. It is important trial that shows that rate control is not a terrible strategy in selected patients. GLP-1 RAs Protective Against Stroke, Neurodegeneration? A GLP-1 study purports to show benefit in cerebrovascular health. It actually shows how observational studies can mislead you. The title of the study is, 'Neurodegeneration and Stroke After Semaglutide and Tirzepatide in Patients With Diabetes and Obesity.' It was published in JAMA Network Open . The goal of the Taiwan group was to evaluate the association of semaglutide and tirzepatide with the incidence of dementia, Parkinson disease, ischemic stroke, intracerebral hemorrhage, and all-cause mortality compared with other antidiabetic drugs in adults with type 2 diabetes and obesity. It's an important question. The best way would be to randomize, but that would be hard and costly. So this was a retrospective cohort study using electronic health records from the TriNetX US network, 2017-2024. The two groups in this study were those on GLP-1, either semaglutide or tirzepatide, vs those on any other diabetic medicines, such as metformin, sulfonylureas, DPP4, SGLT2 inhibitors, and others. This was a large study. About 30,000 in each of the two groups. The groups are not randomized. A doctor chose which of the two groups of drugs to use. So, since it's not randomized the authors did propensity matching. Age 57. Half female. BMI on average 40, and 70 % with hypertension. Here were the main results: During a 7-year follow-up, GLP-1 RA users had a lower risk of dementia (HR, 0.63; 95% CI, 0.50-0.81) lower risk of stroke (HR, 0.81; 95% CI, 0.70-0.93) lower all-cause mortality (HR, 0.70; 95% CI, 0.63-0.78) and had no significant differences in the risk of Parkinson disease or intracerebral hemorrhage. The authors concluded: 'These findings suggest potential neuroprotective and cerebrovascular benefits of GLP-1 receptor agonists beyond glycemic control, warranting further trials to confirm these outcomes.' Maybe these drugs are beneficial for cerebrovascular health, especially in young people (age 57 and diabetes and BMI of 40) But this is a hopelessly confounded study where healthier patients got the more pricey and newer drug. How do I know that? There are two clues. First, is that the Kaplan-Meier curve diverges immediately and continues in parallel. That's what you expect when healthier patients get one treatment. Immediately better outcomes. If the GLP-1 was better than other drugs, you'd see gradually increasing benefit. Second reason: the mortality benefit is huge. A 30% reduction in death. In the SELECT trial of semaglutide vs placebo in patients with heart disease and obesity, semaglutide only reduced CV mortality by 15%. In the SUSTAIN-6 trial of semaglutide vs placebo, semaglutide had no sig reduction in death or CV death and required a composite endpoint to drive positive results. My overall take, therefore, is that GLP-1 drugs induce weight loss. They do modify disease in patients with obesity and diabetes and patients with obesity and atherosclerotic disease. But whether they reduce important cerebrovascular outcomes like dementia cannot be answered by these confounded observational studies. I am not sure it's worth doing these studies because the only value is to show readers the signs of bias in non-random, retrospective comparison studies.
Medscape
7 hours ago
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EMA Launches Review of Tecovirimat Effectiveness for Mpox
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